摘要:

THE SEVENTH ANNUAL GENERAL MEETING OF THE FARADAY DIVISION of The Chemical Society was held at 9.00a.m., on 14 September 1978, in the Physics Lecture Theatre, The University, Canterbury with Professor F. C. Tompkins D.Sc., C.Chem., F.R.I.C., F.R.S. in the Chair. 1 Minutes The Prcsident commenced by reporting that the Minutes of the 6th Annual General Meeting were to be published in the index to the 1977 Farady Transactions which was no1 yet in print but which would appear in due course. The Minutes, which had therefore been tabled, were confirmed. 2 Annual Report In 1977, Professor F. C. Tompkins retired after 30 years as Editor of the Faraday Transactions. To inark this event and also his impending retirement as Honorary Secretaiy of the Faraday Society and Division after a similar period, member5 contributed to\vards a gift of Bristol crystal.The decanter and glasses, together with a cheque and a book containing the signatures of those who had contributed, were presented to Profesor Tomphins by the President before ;I gathering of Council members and past Presidents. Dr. D. A. Young took over the post of Scientific Editor of the Faraday Transactions in August 1977. Two General Discussion5 were held in 1977. In March, Discussion 63 ’Radiation Effects in Liquids and Solids’ was held in Leicester when 120 persons attended of \?.horn o\er 70 were from overseas. The second, Discussion No. 64 ‘Ion-Ion and Ion-Solvent Interactions’ was held at Sl. Catherine’s College, Oxford in September and attracted nearly 200 participants of whom half were from oier- seas representing 19 countries.This Discussion marked the 20th anniversary of the meeting on ‘Interactions in Ionic Solutions’ and the 50th anniversary of the meeting on ‘The Theory of Strong Electrolytes’ and the Division welcomed 5 of the participants in the 1927 Discussion as guests at the Dinner. For the first time at a Faraday Discussion, a poster session was held which allowed the presentation of contributions which were not for publication in the Discussion proceedings. The Symposium in 1977 was held in Southampton in December on ‘Electrocrystallization, Nucleation and Phase Formation’ which attracted 100 participants of whom one third were from overseas. The Faraday contribution to the Annual Congress held at University College London, comprised an informal discussion on ‘Pair Interactions between Molecules’ and at thc Autumn Meeting in Aberdeen an informal discussion on ‘The Chemistry of the Atmosphere‘.An informal meeting on ‘High Resolution Laser Spectroscopy’ was held in Cambridge in December which led to the formation of a new subject group for High Resolution Spectroscopy. Collaboration with the Institute of Physics was maintained in 1977 with the co-sponsorship of two meetings, ‘The 3rd National Quantum Electronics Conference’ and ‘The 50th Anniversary of Electron Diffraction’. Group meetings again played an important part in the activities of the Division and included the following topics : Molecular Beam Kinetics (Molecular Beams Group) Competing Channels in Elementary Processes (Gas Kinetics Group) The Nature of the Glass Transition in Polymers (Polymer Physics Group) Quantum Theory of Chemical Dynamics (Theoretical Chemistry Group) Polymer Surfaces (Polymer Physics Group) 3rd Interdisciplinary Surface Science Conference (Surface Reactivity and Catalysis Group) Hot Neutron Scattering (Neutron Scattering Group) Mechanism of Environmental Corrosion Cracking (Electrochemistry Group) Fundamental Electrochemistry (Electrochemistry Group) 6th Polymer Meeting Point (Polymer Physics Group) Lipid Bilayers : Structure, Dynamics and Interactions with Small Molecules (Colloid and Interface Science Group) 1 ii ANNUAL GENERAL MEETING Low Angle Neutron Scattering from Polymers (Neutron Scattering Group) Advanced Instrumental Methods in Electrode Kinetics (Electrochemistry Group) 5th International Symposium on Gas Kinetics (Gas Kinetics Group) Processing, Structure, Properties and Performance of Polymers (Polymer Physics Group) 4th European Crystallography Meeting (Polymer Physics Group) Experimental Neutron Scattering (Neutron Scattering Group) Surface Phenomena in Analysis (Electrochemistry Group) General Aspects of Chemisorption and Catalysis (Surface Reactivity and Catalysis Group) Statistical and Dynamic Behaviour of Chain Molecules (Statistical Mechanics and Thermo- dynamics Group) Concentrated Dispersions (Industrial Sub-committee with Colloid and Interface Science Group) High Temperature Battery Systems (Electrochemistry Group) Molecular Beams in Combustion Research (Molecular Beams Group) Polymer Materials at Low Temperatures (Polymer Physics Group) Molecules Adsorbed at Surfaces (Theoretical Chemistry Group) Magnetism Density Distribution (Neutron Scattering Group) The 1977 Bourke Lecturer was Professor R.G. Gordon of Harvard University who lectured on ‘Molecular Dynamics of Collisions’, ‘Solid State Chemistry’ and ‘Intermolecular Forces’ at the University of Strathclyde, University College of Wales, Aberystwyth and the University of Bristol respectively. A London Symposium on ‘Molecular Beams in Chemistry’ was sponsored by the Faraday Division which incorporated the Centenary Lecture of Professor D. R. Herschbach.Dr. J. N. L. Connor (University of Manchester), distinguished for his contributions to the under- standing of scattering processes in gases, was awarded the 1977 Marlow Medal. Regular communication with members was continued with the distribution in February of Newsletter No. 4. The total number of members of the Division in 1977 was 4414, there having been a small increase in the number of overseas members during the year. Professor Tompkins thanked members for the gift made to him on his retirement as Editor and Secretary. 3 Treasurer’s Report The Treasuer reported that it had been possible to increase further the support to scientific meetings and in 1979 the maximum sums available would be: General Discussions 2720 Symposium 5500 Annual Congress 2250 Autumn Meeting 2250 As from 1978 Divisions would carry forward to the following year the whole of any unspent balances accruing at the end of the year.The Society was in the process of changing its financial year from the academic to the calendar year and as a result the 1977-8 accounting period was 15 months ending in December 1978. 4 Elections to Council The President announced that the postal ballot for Ordinary Members of Council, conducted among members of the Division, had resulted in the following two persons being elected :Professor D. A. King and Professor J. H. Purnell. The members of Council of the Faraday Division of the Chemical Society to take office from July 1979 were as follows : President PROF.J. S. ROWLINSON Vice-presidents who have held ofice as President PORTER PROF. D. H. EVERETTPFI)F. SIR GEORGE DR.T. M. SUGDEN PROF.F. C. TOMPKINS PROF.R. P. BELL ... ANNUAL GENERAL MEETING Vice-presidents PROF. A. D. BUCKINCHAM DR. H. A. SKINNER PROF.P. GRAY PROF.J. G. WAGNER PROF.G. J. HILLS PROF.D. H. WHIFFEN PROF.N. SHEPPARD Ordinary Members PROF. W. J. ALBERY PROF.P. MEARES PROF.J. H. BAXENDALE PROF.J. H. PURNELL PROF.F. FRANKS DR. J. P. SIMONS PROF. D. A. KING PROF.F. S. STONE PROF. G. R. LUCKHURST DR.D. A. YOUNG Honorary Treasurer PROF. P. GRAY Honorary Secretary PROF.G. J. HILLS The President thanked the retiring members for their services: Professor G. Gee, Dr. W. J. Dunning and Professor B. A. Thrush.The President paid tribute to Professor R. G. W. Norrish, past President of the Faraday Society, and to Professor M. Magat, Vice-president of the Faraday Division, both ardent supporters of the Society who had died in 1978. 5 Review of Future Activities The President drew attention to the programme of future activities of the Division which had been tabled. Two further meetings in the series with the Societe de Chimie Physique, Deutsche Bunsen Gesellschaft and Associazione Italiana di Chimica Fisica were planned, the first to be heid in Fontevraud, France in September 1978 and the second in Italy in 1980. Arrangements for the 1978 General Discussions on ‘Kinetics of State Selected Species’ to be held in April in Birming- ham and on ‘Organization of Macromolecules in the Condensed Phase’ to be held in Cambridge in September were well advanced and the topics for the 1980 General Discussions had been chosen: ‘Phase Transitions in Molecular Solids’ in the spring and ‘Photoelectrochemistry’ in September.The 1979 Bourke Lecturer was Professor 0. Kedem of the Weizmann Institute, Israel. 6 Unification The President informed members that the Faraday Division was satisfied with the unification proposals and urged all members to take advantage of the facility to vote by proxy. NOTICES TO AUTHORS-NO. 7/1970 Deposition of Data-Supplementary Publications Scheme Preamble The growing volume of research that produces large quantities of data, the increasing facilities for analysing such data mechanically, and the rising cost of printing are each making it very difficult to publish in the Journal in the normal way the full details of the experimental data which become available.Moreover, whilst there is a large audience for the general method and conclusions of a research project, the number of scientists interested in the details, and in particular in the data, of any particular case may be quite small. The British Library Lending Division (B.L.L.D.) in consultation with the Editors of scientific journals, has now developed a scheme whereby such data and detail may be stored and then copies made available on request at the B.L.L.D., Boston Spa. The Chemical Society is a sponsor of this scheme and has indicated to the B.L.L.D. its wish to use the facilities being made available in this “Supplementary Publications Scheme”.Bulk information (such as crystallographic structure factor tables, computer programmes and output, evidence for amino-acid sequences, spectra, etc.), which accompany papers published in future issues of the Chemical Society’s Journal may in future be deposited, free ofcharge, with the Supplementary Publications Scheme, either at the request of the author and with the approval of the referees or on the recommendation of referees and the approval of the author. The Scheme Under this scheme, authors will submit articles and the supplementary material to the Journal simultaneously in the normal way, and both will be refereed. If the paper is accepted for publication the supplementary material will be sent by the Society to the B.L.L.D. where it will be stored.Copies will be obtainable by individuals both in the U.K. and abroad on quoting a supplementary publication number that will appear in the parent article. Preparation of Material Authors will be responsible for the preparation of camera-ready copy according to the following specifications (although the Society will be prepared to help in case of difficulty). (a) Optimum page size for text or tables in typescript: up to 30 cm x 21 cm. (b) Limiting page size for text or tables in typescript: 33 cm x 24 cm. (c) Limiting size for diagrams, graphs, spectra, etc. :39 cm x 28.5 cm. (d) Tabular matter should be headed descriptively on the first page, with column headings recurring on each page.(e) Pages should be clearly numbered. It is recommended that all material which is to be deposited should be accompanied by some prefatory text. Normally this will be the summary from the parent paper and authors will greatly aid the deposition of the material if a duplicate copy of the summary is provided. If authors have the facilities available the use of a type face designed to be read by computers is encouraged. Deposition The Society will be responsible for the deposition of the material with the B.L.L.D. The B.L.L.D. will not receive material direct from authors since the Library wishes to ensure that the material has been properly and adequately refereed.iv Action by the Society The Society will receive a manuscript for publication together with any supplementary material for deposition and will circulate all of this to referees in the normal way. When the edited manuscript is sent to the printers the supplementary material will be sent for deposition to the B.L.L.D. who will issue the necessary publication number. The Society will add to the paper, at the galley proof stage, a footnote indicating what material has been deposited in the Supplementary Publications Scheme, the number of pages it occupies, the supplementary publication number, and details as to how copies may be obtained. Availability Copies of Supplementary Publications may be obtained from the B.L.L.D. on demand by organisations which are registered borrowers.They should use the normal forms and coupons for such requests addressing them as follows: Mr. J. P. Chillag, British Library Lending Division, Boston Spa, Wetherby, West Yorkshire, LS23 7BQ, U.K. Non-registered users may also obtain copies of Supplementary Publications but should first apply for price quotations. These are available from the Loans Office at the above address. In all correspondence with the B.L.L.D. or the Society authors must cite the supple- mentary publication number. International Collaboration A similar scheme (known as the National Auxiliary Publications Service) is being operated in the U.S.A. by the American Society for Information Science. Similar schemes are also being contemplated in other countries. The provision of reciprocal arrangements for the exchange of supplementary data between the various national deposition centres is being investigated. V NOTICE TO AUTHORS-NO.9/1974 Nomenclature For many years the Society has actively encouraged the use of standard I.U.P.A.C. nomenclature and symbolism in its publications as an aid to the accurate and unambiguous communication of chemical information between authors and readers. Although the I.U.P.A.C. rules for naming organic compounds have now gained wide acceptance amongst chemists, mainly because they have been in existence for a number of years, those for naming inorganic compounds are of more recent origin and for this reason their acceptance is less general.In order to encourage authors to use I.U.P.A.C. nomenclature rules when drafting papers, attention is drawn to the following publications in which both the rules themselves and guidance on their use are given. ‘Nomenclature of Organic Chemistry, Sections A, B, and C‘, Butterworths, London, 2nd Edition, 1971. ‘Nomenclature of Inorganic Chemistry’, Butterworths, London, 1971. ‘Manual of Symbols and Terminology for Physicochemical Quantities and Units’, Butterworths, London, 1970. In addition to the above publications, provisional rules for the naming of organometallic compounds, amino-acids, carbohydrates, carotenoids and steroids, and rules of stereo-chemistry are available from the: I.U.P. A. C. Secretariat, Bank Court Chambers, 2-3 Pound Way, Cowley Centre, OXFORD OX4 3YF.It is recommended that where there are no I.U.P.A.C. rules for the naming of particular compounds or authors find difficulty in applying the existing rules, they should seek the advice of the Society’s editorial staff. vi NOTICE TO AUTHORS-NO. 10/1976 Authentication of New Compounds (1) It is the responsibility of authors to provide fully convincing evidence for the homo- geneity and identity of all compounds they claim as new. Evidence of both purity and identity is required to establish that the properties and constants reported are those of the compound with the new structure claimed. (2) In the context of this Notice a compound is considered as new (a)if it has not been prepared before, (b)if it has been prepared before but not adequately purified, (c) if it has been purified but not adequately characterised, (d)if, earlier, it has been assigned an erroneous constitution, or (e) if it is a natural product synthesised for the first time.In preliminary communications compounds are often recorded with limited characterising data ; in spite of (c) above later preparations of such compounds are not considered as new if the properties previously reported are confirmed ; the same applies to patents.* (3) Referees are asked to assess, as a whole, the evidence in support of the homogeneity and structure of all new compounds. No hard and fast rules can be laid down to cover all types of compounds, but the Society’s policy remains unchanged in that evidence for the unequivocal identification of new compounds should normally include good elemental analytical data ; an accurate mass measurement of a molecular ion does not provide evidence of purity of a compound and must be accompanied by independent evidence of homogeneity.Low-resolution mass spectroscopy must be treated with even more reserve in the absence of firm evidence to distinguish between alternative molecular formulae. Where elemental analytical data are not available, appropriate evidence which is convincing to an expert in the field will be acceptable, but authors should include, for the referees, a brief explanation of the special nature of their problem. (4)Spectroscopic information necessary to the assignment of structure should normally be given.Just how complete this information should be must depend upon the circum- stances; the structure of a compound obtained from an unusual reaction or isolated from a natural source needs much stronger supporting evidence than one derived by a standard reaction from a precursor of undisputed structure. Authors are reminded that full spectro- scopic assignments may always be treated as a Supplementary Publication where their importance does not justify their inclusion in the published paper. (5) Finally, referees are reminded of the need to be exacting in their standards but at the same time flexible in their admission of evidence. It remains the Society’s policy to accept work only of high quality and to permit no lowering of present standards.* New compounds should be indicated by underlining the name (for italics) at its first mention (excluding headings) in the Experimental section only, and by giving analytical results in the form : (Found: C, 63.1 ; H, 5.4. C,,H,,NO, requires C, 63.2; H, 5.3 %). If analytical results for compounds which have been adequately described in the literature are to be included, they should be given in the form: (Found : C, 62.95 ; H, 5.4. Calc. for CI3Hl3NO4: C, 63.2 ; H, 5.3 %).Analyses are normally quoted to the nearest 0.05 %. vii Publication of Theoretical and Computational Papers The Primary Journals Committee has been considering future policy towards the publication of papers with a heavily computational content, particularly where these involve standard methods, such as semi-empirical or ub initio calculations of molecular electronic properties using readily available computer programs.Many such papers report what would be con-sidered ‘routine work’ in other areas of chemistry and have often included extensive detail. A specialist sub-committee formulated a set of proposals which were circulated to a large representative sample of theoretical chemists and met with general acceptance. These, with the comments on them, form the basis of this notice. The Primary Journals Committee recognises that computational work can play a valuable role in chemistry, and will probably continue to do so on an increasing scale. It accepts the time-honoured principle that the first criterion for publication of a paper by the Society should be the worthiness of the chemical problem considered, rather than the particular techniques employed by the author.For example, the use of a new computing algorithm, or the modification of a program, would not usually, on its own, provide sufficient justification for publication. The Primary Journals Committee recommends to authors the following guidelines for the preparation of computational papers, so that the material can be presented concisely and effectively. (i) Papers should be submitted to the appropriate journal: a paper containing in- novations in theory to Faraday Transactions 11, one in which the computations are incidental to the chemistry to Perkin, Dalton or Faraday I Transactions.Papers concerned maidy with computational details are unlikely to be accepted. (ii) The purpose of the paper and the precise objectives of the calculations performed should be clearly stated: the results obtained should be reported only in so far as they relate to those objectives. (iii) Many papers use a routine procedure based on a well documented method, be it semi-empirical or ub initio. It is then sufficient to name the particular variant, referring to key papers in which the method was developed, to cite the computer program used, and to indicate briefly any modification made by the author. A review of theoretical background would be out of place, but an author should say why he considers the method adequate for his purposes.(iv) Extensive tabulation of numerical results, such as the magnitudes of atomic orbital coefficients, electron populations, contour maps of molecular orbitals and electron densities, and peripheral material of a similar nature, is normally unnecessary. Lengthy line-by-line discussion of such material is, as a general rule, quite unaccep- table. Where an author considers that there is a special need to make such material available to other workers, as with highly accurate computations, for example, then this may be deposited with the British Library as a Supplementary Publication. Such material should be submitted with the main paper, clearly distinguished from it, and referred to in the main text. Guidelines can never provide sufficient criteria for acceptance or rejection of a paper. Critical assessment of the theoretical methods used in a computation, and of their suitability for the purpose in hand, will continue to be entrusted to specialist referees who must also decide whether the results are new and advance science. viii PRINTED IN GREAT BRITAIN AT ABERDEEN UNIVERSITY PRESS

ISSN:0300-9238    

DOI:10.1039/F297874BP001

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Glass Transition in the Hard-Sphere Model Relaxation Effects BY LESLIEv. WOODCOCK Department of Physical Chemistry, Lensfield Road, Cambridge CB2 1EP Received 2nd December, 1976 The dependencies of a kinetic glass transition volume (V:) for hard spheres, and the corresponding amorphous close-packed volume (V:) at infinite pressure (NkT/pV, = 0), upon the number of spheres, the initial volume and the rate of compression, are examined in the context of two recently published conflicting viewpoints. Both V," and V: exhibit characteristic upper-bounds at the instantaneous-quench limit and decrease with reduced compression rates. The behaviour of V," and Vt for slower compression rates, beyond the present resources of MD computation time, is uncertain. The possibility of lower-bounds to YE and Vi associated with a quasi-thermodynamic transition as conjectured earlier, remains plausible.A molecular dynamics computation has been reported in which the density of 500 hard spheres, with periodic boundary, was gradually increased to amorphous close-packing. This, together with equation-of-state and existing transport data, failed to give any indication of the usual second-order discontinuity associated with glass formation, but seemed to suggest that the metastable fluid, for densities exceeding the equilibrium freezing density, might be approaching a quasi-thermodynamic transition of third-order in the vicinity of V/Vo= 1.41, where Vois the volume of crystalline (f.c.c. or h.c.p.) close-packing. Gordon et aZ.,2on the other hand, reported an analysis of an earlier 32-sphere MD densification computation by Alder and Wainwright with contradictory conclusions. They found that there was an operational, kinetic, glass transition at V/Vo= 1.24 with the usual second-order discontinuity.Gordon et al. further concluded that any glass-transition-like behaviour in the hard-sphere model is purely a kinetic phenomenon and that the hard-sphere fluid equation-of-state extrapolates in a well-behaved manner to a singularity with infinite pressure at a particular volume of amorphous close-packing (V/Yo = 1.1328). We should be cautious when drawing conclusions from single MD densification computations, particularly if N (the number of spheres within the periodic box) is small.The equation-of-state, or the volume at a prescribed pressure, of the amor- phous hard-sphere solid will, in general, depend upon N, the boundary conditions, the initial starting volume ( Vi), the densification mechanism, and the compression rate. Even then, it may only be defined as an ensemble average taken over the 6N-dimensional position and velocity space (rl . . . r,, v1 . . . v,), at V,, where Vimust exceed the equilibrium fluid freezing volume. If we adopt a particular well-defined densification mechanism, for example that described previously,l then the volume of the amorphous state at a given pressure is a precisely defined ensemble average. The close-packed volume at infinite pressure, with the superscript k intended to denote the dependence on kinetic relaxation time through the variable compression rate, is ','= (~NkT/pv()=o) ,Ua)i (1)(N, 6,dv/dt))(fi...fN, U1.. 11 GLASS TRANSITIONS OF HARD SPHERES where the angular brackets denote an ensemble average at V, and dV/dt is the com- pression rate, which could itself be an implicit function of volume depending on the densification mechanism. The ensemble averaging at Vi is emphasised because in the definition of an operational kinetic glass transition, the volume, will exhibit a discontinuity in its derivatives with respect to pressure. Any attempt to compute the pressure, however, by the conventional procedure of permitting the dynamics to proceed at a fixed volume and measuring the collision rate, will result in further relaxation and any operational transition will be obscured until the diffusional relaxation time exceeds the time-averaging scan in the pressure calculation.In the ultimate thermodynamic limit that dV/dt goes to zero, and Ngoes essentially to infinity, the system will reversibly crystallise in the most stable lattice arrangement with a perfectly sharp first-order discontinuity in Y at constant P,and Vf will become equal to Yo. This limit has not been approached with MD computations for hard- spheres, although there is some evidence that soft-spheres and Lennard-Jones molecules can, in certain circumstances, crystallise on practicable MD time-scales when N is small (< 500). Fig. 1 represents a purely schematic suggestion of how IIIIiIIi 123456789 log (compression ratelarb.scale) FIG.1.-A postulated representation of the behaviour of the excess volume at NkT/PVo = 0 as a function of the compressionlcooling rate. The plateau showing a lower bound to the residual volume of homogeneous amorphous packing is based upon the conjecture that a quasi-thermo- dynamic limit exists for amorphous phase space at high density. V! might be expected to behave as a function of the compression rate, for large N, over the entire range. At the opposite extreme to the thermodynamic limit, the instantaneous-quench limit for which d V/dt -+ -GO, V: must exhibit an upper-bound due to the mechanical instability of loose-packed configurations at very high pressures. It is likely that there will be a reasonably narrow range of the compression rate below which crystallisation always occurs and above which a homogeneous amorphous solid is obtained.Questions of interest in the theory of glass formation concern the behaviour of Vifor the intermediate range. Will it exhibit a plateau, as depicted in fig. 1, associated with a quasi-thermodynamic limit as postulated before,' or will it approach Yoas a smoothly decreasing monotonic function of the compression rate ? In order to obtain information Viand Vk,as defined above, in the high compression rate region, several further MD densification computations have been carried out for iV = 32, 100 and 500, for Vi ranging from 2.0 Voto low density gas values, and for compression rates scanning four orders-of-magnitude.The densification mechanism has been modified slightly from that used previously.' At a densification step, the dynamics are stopped and the spheres are uniformly expanded until the nearest pair come into contact. In the present modification this is effected every AT, collisions, instead of a fixed time increments At as in the previous procedure. Owing to the L. v. WOODCOCK A 1500 I1000 I 1500 collisions collisions AIt! A a An I.!.!? A A 100 p AA -8 AL 14000 16000 18000 160,000 /80,000 ] l00,000 collisions collisions I -25 4 ann na A 2 A AA collisions FIG.2.-Data from some typical MD hard-sphere densification computations showing the dependence of volume, relative to the crystal close-packed volume, on the collision rate in the vicinity of a kinetic glass transition.The glass transition is observed as a break in the change in volume with time measured by the number of collisions as shown by the solid circles. The triangles give the actual time in program units, whereby the unit of time is defined from the mass of a sphere, the energy in units of kT, and length by the side of the MD box (S),i.e., (mSL/kT)*,during the course of the com- putations beginning with T = 0 at the initial volume Vi. N is the number of spheres and AT^ is the number of collisions between densification steps. (a) N = 32, AT^ = 1, Vi = 14.14V0; (b)N = 32,AT^ = 10, Vi = 14.14Vo; (c) N = 32, AT^ = 100, Vi = 3.535Vo ; (d) N = 32, AT== 1000, Vi = 2.828Vo ; (e) N = 100, AT^ = 50, Vi = 2.828Vo ; (f) N = 500, AT^ = 50, Vi = 2Vo.GLASS TRANSITIONS OF HARD SPHERES strong dependence of the collision rate on density, the former technique is inefficient at very low and very high densities. Some typical results for single runs are shown in fig. 2. A kinetic glass-transition volume V: is identified by a break in the rate of change of volume with respect to the number of collisions. It would be desirable to obtain Vi from petit ensemble averages of pressure over large numbers of starting configurations at Vi but this is not possible, at present, owing to the heavy demands upon computer time. Fig. 3 shows the dependence of Vi on the compression rate, each point representing a single run.As an artefact of the chosen densification mechanism, neither d V/dt, nor dpldt, are independent of volume but dV/dt is reasonably constant over an appreciable volume range at the onset of Vi. The interesting observations from these data are that Y$seems to be approaching an upper-bound, which is close to the random “parking” volume (VR -2.12YJ for hard-spheres, and occurs at lower volumes as the compression rate decreases. A 001 0.I I -dV/dt at V: FIG.3.-Operational glass transition volumes plotted against the compression rate at the onset of V:; the units of volume and time are as in fig. 2; the horizontal dashed line represents the random “parking ” volume [ref. (6)]. 0 N = 32, A N = 100, N = 500. The corresponding relaxation time-dependent amorphous close-packed volumes Vt have also been obtained by extrapolating to infinite time at constant densification rate using the previously described meth0d.l The results are shown in fig.4. Y:/V,-,also seems to exhibit an upper-bound, close to the volume obtained for the random close-packing of macroscopic spheres.’. * The statistical errors arising from the 1.19 -a -a 1.18 -a1’17 1.16 = -------.-----ImA Ad .Ye 1.15- A m 114- a om 1.13 - a a 1.12 -@ I 1 I L. V. WOODCOCK small numbers of spheres are quite large here, but there is, however, a definite reduc- tion in V:/V, with decreasing compression rates. The form of this dependence and its behaviour for longer relaxation times cannot be established from the sparse statistics available to date.Thus, the behaviour of the supercooled hard-sphere fluid in the long relaxation time limit, barring crystallisation, is still a matter of uncertainty. There is, neverthe- less, an inconsistency arising from the argument that there should not be a quasi-thermodynamic discontinuity associated with the transition from fluid to glass for hard-spheres. If this were true, the pressure of the amorphous phase must be a well-behaved function of volume from zero density to some amorphous close-packed density exhibiting no discontinuities. Therefore the virial expansion for the hard- sphere fluid pressure would have to reflect this by having a first-order pole at some particular value of V,, assuming it to be continuous through the equilibrium freezing volume Vf. Now, if the occurrence of Vgwere to be a " purely kinetic phenomenon ", then it would follow from the definition of Vi [eqn (I)] that the complete virial equation of state has a singularity with infinite pressure at V,.This implies that the two asser- tions, (i) that the fluid equation of state is continuous up to some amorphous close- packed volume V, and (ii) that the glass transition is purely a kinetic phenomenon, are incompatible. It is quite plausible that the virial equation of state has a pole at spheres. In (NkTlpYO) FIG.5.-Excess entropies, relative to the ideal gas, of amorphous and crystalline phases of hard The extrapolated super-cooled fluid entropies are calculated from the Pad6 approximant [ref.(12)], the Carnahan-Starling equation [ref. (13)] and the D2-closure [ref. (14)]. brium freeing temperature [ref. (lo)] and Tgis the approximate glass transition temperature [ref. (l)].the equili- is 7'' GLASS TRANSITIONS OF HARD SPHERES Yo, but this still does not eliminate the possibility of an inevitable quasi-thermo- dynamic transition of high-order, at compression rates exceeding the critical com- pression rate for crystallisation, as envisaged in fig. 1. Finally, Gordon et aL2 raise the question of whether the hard-sphere system exhibits a temperature “T,”, which they define as the temperature for which the excess entropy of the supercooled liquid over the crystal at the same temperature and pressure, sometimes called the “ configurational entropy ” and denoted by Sc,9 would become zero.If the metastable amorphous hard-sphere fluid is a perfectly continuous extrapolation of the equilibrium fluid through the freezing transition point then, as discussed previously,l it will be represented by the virial equation of state. If we calculate the entropy of the supercooled fluid from an accurate empirical equation of state based upon extrapolated virial series, knowing the temperature of fusion together with ASf,* and the crystal equation of state,ll T2 can be obtained. Fig. 5 shows the excess entropies predicted by the Pad6 approximant of Ree and Hooveryl the Carnahan-Starling nearest-integer recursive representation,l and the D2-closure.14 The temperatures of zero S, predicted by these three approximate virial equations are NkT,/pV, = 0.033, 0.029 and 0.035 respectively.Whatever may be the temperature T2for which S, passes through zero, according to the virial expansion, however, it is evident that it is not nearly approached in MD quenches ; the amorphous hard-sphere solid always maintains a high residual entropy of approximately Nk. It is inconceivable, as envisaged in some interpretations of glass-transition phenomena,,. that two quite different phases of the same system could have identical entropies over a finite temperature range (0-T,). Even the two alternative crystal phases of hard spheres, f.c.c. and h.c.p., for example, will have a definite, though as yet unknown, small entropy difference for all temperatures.The low temperature entropy of any vitreous state relative to the most stable crystal phase of a system, whether real or purely classical, as in the case of MD computer models, should always be a positive quantity in accord with the Nernst Heat Theorem. L. V. Woodcock, J.C.S. Faraday II, 1976,72, 1667. * J. M. Gordon, J. H. Gibbs and P. D. Fleming, J. Chem. Phys., 1976, 65,2771. B. J. Alder and T. E. Wainwright, J. Chem. Phys., 1960,33, 1439. W. G. Hoover, S. G. Gray and K. W. Johnson, J. Chem. Phys., 1971,55,1128. A. Rahman, M. J. Mandell and J. P. McTague, J. Chem. Phys., 1976, 64, 1564. The random “parking ” volume is dehed by analogy with the random car parking problem in 1 dimension [H.Solomon, Fifth Berkeley Symposium on Mathematical Statistics and Prob- abiZity (Universityof California Press, 1954), vol. 3, p. 1191. It corresponds to the density for which a maximum number of spheres can be randomly placed in a fixed volume without overlap. The vdue quoted has been determined numerically. G. D. Scott and D. M. Kilgour, J. Phys. D,1969,2,963. J. L Finney, Proc. Roy. Soc. A, 1970, 319,479. G. Adam and J. H. Gibbs, J. Chem. Phys., 1965,43, 139. lo W. G. Hoover and F. H. Ree, J. Chem. Phys., 1968,49,3609. l1 B. J. Alder, W. G. Hoover and D. A. Young, J. Chem. Phys., 1968,49,3688. l2 F. H. Ree and W. G. Hoover, J. Chem. Phys., 1964,40,939. l3 N. F. Carnahan and K. E. Starling, J. Chem. Phys., 1969,51, 635. l4 L. V. Woodcock, J.C.S. Faraday II, 1976,72,731. (PAPER 61221 1)

ISSN:0300-9238    

DOI:10.1039/F29787400011

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Branching Ratios for the H+DCl and D+HC1 Reaction Systems BY FRITZs. KLEIN*AND IAN VELTMAN Isotope Department, Weizmann Institute of Science, Rehovot, Israel Received 14th February, 1977 The branching ratios for the two reaction systems : HD+C1 HD+ C1 f kn f (a)DfHC1 ( and (b)H+DC1 ( kx \ kxL ' DCI+H WCI+D have been measured as a function of temperature using a fast-flow discharge reactor coupled with a quadrupole mass spectrometer for analysis. The branching ratios obtained were : (a)kJkx = (9.7f1.0)exp (-4440k 590 J mol-'/RT), 215 < T < 438 K and (b)kJkx = (4.3k0.7) exp (-435Ok290 J mol-lIRT), 138 < T < 451 K. The results were interpreted in terms of transition state theory and a simple statistical model. Studies of the chain reactions involved in the hydrogen, halogen and hydrogen halide system have played an historic and central role in the development of chemical kinetics.l Recently these reactions have achieved new significance with the advent of hydrogen-halogen chemical lasers.2 In spite of the substantial effort devoted to understanding these systems, there still exists uncertainty concerning isotope exchange reactions of the type kx H+XH' -+ WX+H' exchange where H and H' are H and D, while X is C1, Br, or 1.In this paper we present the results of an experimental study of the relative rates of abstraction and exchange for the H +DCl and D +HCl systems using a fast-flow- discharge reactor coupled with a quadrupole mass-spectrometer for analysis. Wood has published the results of experiments involving the photolysis of H2+DCl mixtures at temperatures between 300 and 498 K.By measuring the quantum yields of HD and D, he obtained values for k,/kxfor the H +DCl reaction and concluded that not only was k, > k,, but also the activation energy for abstraction was greater than that for exchange by -9.2 kJ mol-l. Thompson et aL4 have queried these results. Why, they argued, should the reaction cross-section for exchange predicted by this work be very much smaller than that for abstraction? A priori one would expect the chlorine atom to present a much larger target for attack by the incoming H atom than the small D atom. These workers conducted a comprehensive quasi-classical trajectory study of 13atom-diatomic molecule reactions in the H2+C12 system.Their calculations yielded results in reasonable agreement with experiment in all cases except for H +ClH isotope exchange reactions where their derived pre-exponential factors were -103-104larger than indicated by Wood's 17 BRANCHING RATIOS work. Their results also predict that k, > ka at all temperatures. A mechanistic analysis involving “hot atom” D and H chain carriers failed to resolve this discrepancy. Molecular beam experiments on the D+HX system (X-CI, Br, I) revealed exchange cross-sections of the order of 0.01-0.1nm2. If the steric factor for exchange is as small as suggested by Wood (-3 x then it seems surprising that any exchange was observed in this work. Quasi-classical trajectory calculations simulating the conditions of the beam experiments are in reasonable agreement with observation.Some earlier thermal and photolysis work support the conclusion that k, > kx, whereas De Vries and Klein’s photolysis experiments on D, +HCI mixtures imply that k,/k, < 0.9 at room temperature and k,/k, < 10 at 425 K. There is much contradiction between the findings of these investigations. This lack of consistency is not confined to the experimental field, for it is also apparent that different theoretical approaches yield contradictory results. Statistical treat- ments 99 lo find abstraction predominant whereas quasiclassical trajectory studies 4* find exchange predominant. It is clear from this discussion that further experimental measurements of these rate constants are desirable in an effort to clarify the situation.The fast-flow discharge technique has proved a versatile and reliable tool in the investigation of the kinetics of atom reactions.’l The ability to work with high atom concentrations and short reaction times enable measurements to be performed before “ steady-state ” conditions are attained, and the perturbations arising from complex chain processes are thus minimized. This contrasts with pyrolytic and photolytic experiments where atom concentrations are low and reaction times long compared with the time taken to attain steady-state conditions. Consequently, interpretation of the experimental results is heavily dependent upon assumptions about the mechanism and the uncertainty of non-thermal reactions involving hot-atoms.The experiment described here is conceptually simple. The reacting gas mixture was sampled after a short time (10-40 ms, depending on temperature) and the ratio of diatomic products was estimated by means of mass spectrometry. This ratio, rcxp,was then corrected to account for subsequent reactions to obtain the ratio of abstraction to exchange rate constants, ka/kx. EXPERIMENTAL The apparatus comprised a cylindrical quartz reactor, 16 mm i.d., similar to that described previously.” Mixtures of H2+He and D2+He (>lo-fold excess of He) were pumped through the reactor by an Edwards ED200 rotary vacuum pump. This gave flow speeds of -10 m s-l at room temperature.Dissociation of the H2 or D2 was accomplished by a microwave cavity discharge (Raytheon Corp., 100 W) positioned on a side arm near the reactor. HCl or DCI was metered into the gas stream through an axially mounted Pyrex probe, the end of which was fashioned with four small holes (-2 rnm i.d.) to ensure rapid mixing of the reactants. The gases were sampled 170mm down-stream through a pin-hole leak into an EAI 250 A quadrupole mass spectrometer. The thee gas flows were measured by calibrated manometer flow-meters. The total reactor pressure (in range 105-135 N m-2) was monitored using a variable reluctance transducer (Celesco). The reactor was heated by means of a resistance wire mesh wrapped around the tube. It was cooled by passing cold nitrogen gas from evaporating liquid nitrogen through four metal tubes interspersed with copper sheets attached lengthwise to the outside of the reactor.A calibrated copper-constantan thermocouple was mounted inside an axial sheath so F. S. KLEIN AND I. VELTMAN that it measured the temperature close to the end of the probe. Temperature differences along the reactor were <3"C. MATERIALS HCl was Matheson Lecture bottle grade, degassed by freeze-thaw-pump cycles and stored over P205 before reaction. Hz was Technical grade, dried by passage through a trap cooled in liquid nitrogen. He was Technical grade, dried by passage over P2OS. DCl was from Merck, Sharp and Dohme (nominal purity 99 %). Mass spectrometric analysis indicated the presence of -6 % HCl, probably originating from exchange within the apparatus.It was degassed by freeze-thaw-pump cycles. HD was from Merck, Sharp and Dohme and used for calibration without further purification. Drifilm (General Electric) was used as wall poison. MEASUREMENT PROCEDURE Both exchange and abstraction products were observed for both reaction systems. In addition reaction was observed between H and D atoms and reactor surfaces which had been freshly exposed to DCl or HCl. The mass spectrometer signal (peak maximum) was measured, after amplification, by a digital voltmeter (Hewlett-Packard 3440A) coupled with a 100 shot digital averager (home built). HD was measured via m/e 3 (HD+),HCl via m/e 36 (H35Cl+) and DCl via m/e39(D37Cl+).The mass spectrometer was calibrated against known flow rates of products and sensitivity ratios obtained were used to convert peak height ratios to the concentration ratios, re,,. The main causes of systematic experimental error are : (i) heterogeneous exchange and reaction of atoms with hydrogen-containing compounds present on reactor wall surfaces ; (ii) heterogeneous reaction between atoms and the highly polar hydrogen chloride adsorbed on the reactor walls ; (iii) co-pyrolysis of Hz+ DCl or Dz+ HCl on the mass spectrometer filament ; (iv) effects of background (mainly hydrogen chloride) from gas adsorbed on the mass spectrometer walls, and (v) the presence, especially in the case of DCl, of isotopic impurity. To gain a true picture of the homogeneous reaction it is therefore necessary to devise an experimental procedure than can take account of these likely sources of error.The procedure used here comprised the following measurements : (A) The m/e 3, 36 or 39 peak was measured with HZ,DC1 or D2, HCl and He flowing while the discharge was not operating. [A(3), A(36), A(39)]. (B) The discharge was switched on and the nz/e 3 and 36 or 39 peaks were measured. [B(3), B(36), B(39)]. (C) With the discharge still operating, the DCl or HCl flow was turned off and the m/e 3 and 36 or 39 peaks measured. [C(3), C(36), C(39)l. The 36 or 39 peak was monitored over a period of >100 s. Alimiting value estimated from these readings was used to ensure that all reactant hydrogen chloride had diffused out of the probe.(D) The discharge was switched off and with only H2 or Dz and He flowing the m/e 36 or 39 peak was measured. [D(36), D(39)]. In the case of the D2+HC1 system, A(39) estimated the extent of isotope scrambling occurring in the mass spectrometer. In the case of the Hz+DCl system A(36) measured additionally (and more importantly) the presence of HC1 impurity in the DC1 sample. The total production of HD and HCl or DCl arising from homogeneous reaction and from heterogeneous factors (i) to (v) above could be estimated from B(3), B(36) and B(39) respectively, i.e., and B(36) = B(39) = A(36)+ W(36)+ H(36) A(39)+ W(39)+ H(39) (1) for the H2/DCl system (2) for the D2/HCl system B(3) = A(3)+ W(3)+ H(3) (3) for both systems, where W represents the product formed through wall reactions and H that formed through homogeneous reaction.A(3) was found to be negligibly small compared with W(3) and was not used in the final analysis. The extent of wall reaction (discussed in more detail later on) was estimated via measure- ments C(3) and C(36) or C(39). BRANCHING RATIOS Thus we have: C(3) = W(3) (4)C(36) = W(36)+D(36) (5) for the Hz/DCl system and C(39) = W(39)+D(39) (6) for the Dp/HCl system. Here D(36) and D(39) are measures of the residual 36 and 39 peaks arising from the back- ground effect (iv). D(36) and D(39) are used rather than A(36) and A(39). Since no hydrogen chloride is flowing in measurement C, C(3) estimates W(3) directly, as effect (iv) was non-existant for the volatile, non-polar HD.From these measurements the values of the homogeneous components H(36), H(39) and H(3) could be estimated using the following equations, derived from eqn (1) to (6), and since A(3) Q W(3) : H(3) = B(3)- C(3) H(36) = [B(36)+D(36)]-[A(36)+ C(36)] and H(39) = [B(39)+ D(39)I- [A(39)+ C(39)l. (9) The ratio H(3)/H(36) and H(3)/H(39), after correction for mass spectrometer sensitivity, thus provided a measure of the product ratios, r,,, = [HD]/[HCI] and [HD]/[DCl],respectively, resulting from the homogeneous gas phase reaction. RESULTS Table 1 gives some examples of the raw data used to evaluate H(3) and H(36) and H(39) for the two systems. The method by which ka/kxwas obtained from these product ratios is described in the discussion below.The associated Arrhenius plots are shown in fig. 1 and 2. The experimental observations may be summarized as follows : the value of rexptended to be about a factor of two higher in the D+HCl system than in the H+DCl system. In both systems rexptended to peak as a function of temperature (-320 K for H+DCl and -400 K for D +HCl). However, the general trend indicated by these results was one of increasing abstraction with increasing temperature as also found by other '9 TABLEEX EXAMPLES OF DATA USED FOR ANALYSIS (i) H+ DCI T/K aA(36) B(3) B(36) C(3) C(36) D(36) H(3) H(36) H(3)/H(36) w % 451 23 6.8 31 1.6 5.5 5.3 5.2k1.5 7.5f2 0.70k0.3 3 301 30 29 65 1.7 22 12 27.2f1.5 26 22 1.06k0.1 29 195 61 9.5 102 2.7 51 25 6.8k1.5 15.0f2.6 0.45fi0.1 64 (ii) D+ HCI T/K A(39) B(3) B(39) C(3) C(39) D(39) H(3) H(39) H(3)/H(39) w % 215 3.9 27 13 6.4 4.9 3.3 21k1.5 7.7+2 2.7fi0.7 16 300 3.9 55 14 6.9 4.7 3.2 48k1.5 8.9+2 5.4k1.2 14 457 4.7 30 8.7 4.8 3.6 3.5 25f1.5 4.0k1.5 6.3fi2.5 16 a All peak heights are given in mV.The estimated error in the direct measurements [A(36) to D(36) and A(39) to D(39)] is -= 1 mV. 1 mV 2: 5 x inol dm-3 of gas at room temperature. w % is the estimated percentage of total hydrogen chloride produced originating from wall reaction. F. S. KLEIN AND I. VELTMAN 21 0.4 1 I L I 4 - 3 -0.6-3 -0.8---I .or-I I I I 1 I 3 a 5 & 7 lo3K/T FIG.1.-Arrhenius plot for the HfDC1 branching ratio.I I I 1 I lo3KIT FIG.2.-Arrhenius plot for the D+HCI branching ratio. DISCUSSION INTERPRETATION OF DATA In this section we discuss the extent to which reactions subsequent to elementary abstraction and exchange reactions cause the experimentally observed product ratio, rexp, to deviate from the branching rate-constant ratio and the procedure adopted to correct Yexp for these reactions. In table 2 a 27 step mechanism of 24 homogeneous steps and three wall terminations is listed. This mechanism describes both experimental systems. The four reactions of interest are 20,22 (abstraction) and 21,23 (exchange). The rate equations derived for this mechanism were integrated on an IBM 370/165 computer using Gear's l3 variable order, variable step-length method coupled with subroutines for automatic derivation of rate equations and jacobian matrix elements (Pij = d2 [XJd[X,]dt) froin mechanism input in encoded form.The rate constants used are as shown in table 2. The appropriate exchange rate constant (k21or kZ3)was varied until the value of r,, the ratio predicted by the nth iterative integration for reaction time t, was equal to within 1 % of the experimental value, rexp,i.e., initially where k, (i.e. k20or kz2)was assumed to be given by the reduced-mass adjusted simple collision theory expression given by Clyne and Stedman 2o for the reaction H+HC1 = N2+C1. BRANCHING RATIOS TABLE2.-MECHANISM AND RATE CONSTANT PARAMETERS * USED FOR KINETIC ANALYSIS A EjkJ mol-1 n wall 1 D-++Dz 3.1 x 10-3 S-1 0 1.5 b wall 2 3 wall H--+&H2 Cl --+ * c12 4.o~10-3 s-1 1.6~10-3 s-1 0 0 1.5 1.5 b b 4 D+D+M+Dz+M 2.2~109 dm6 moP2 s-l 0 o c 5 H+H+M+ H2+M 2.54~lo9 dm6 mok2 s-l 0 0 C 6 C1+ C1+ M -+ Cl2+ M 1.40x lo9 dm6 mok2 s-l 0 0 d 7 D+H+M+HD+M 4.74~lo9 dm6 mol-2 s-l 0 O e 8 D+ C1+ M 3DCl+ M 1x lolo dm6 moF2 s-l 0 O f 9 10 11 12 H+ C1+ M -+ HCl+ M H+D2 -+ HD+D D+H2 -+ HD+H Cl+D2 -+ DCI+D 1 x 1O1O dm6 mol-' s-' 2.291 drn3mol-1 s-l 4.786 x lo2dm3 mol-1 s-l 8.3 x lo9 dm3 mol-1 s-l 0 23.7 20.7 22.7 O 3.21 2.51 O f g g h 13 Cl+H2 + HCl+H 1.2~1O1O dm3 mol-l s-l 18.0 0 i 14 15 D+ Clz -+ DCl+ C1 H+Clz -+ HClfC1 4.77~1011dm3 mol-1 s-l 3.7~loll dm3 mol-1 s-' 8.4 7.5 O O jk 16 Cl+HD 3 HCl+D 5.71 x lo9 dm3 mol-l s-l 22.0 0 1 17 Cl+HD -+ DCI+H 7.04~lo9 dm3 mol-1 s-l 20.0 0 I 18 D+DCI + D2+C1 4.44~lo8 dm3 mol-1 s-' 13.0 0.5 rn 19 H+HCl+ HZ+C1 6.20~lo8 dm3 mo1-1 s-l 13.0 0.5 n 20 D+HCl-+ HD+C1 4.44~lo9 dm3 mol-l s-l 13.0 0.5 m 21 D+HCl3 DCl+H - dm3mol-l s-l - - 22 H+DCl+ HD+C1 6.20~lo8 dm3 mol-'s-l 13.0 0.5 rn 23 H+DCI 3 HCl+D - dm3 mol-l s-' - - 24 Cl+DCl-+ ClZ+D 1O'O dm3 mol-l s-' > 170 0 0 25 CI+HCI + Clz+H 1O1O dm3 mo1-l s-l >170 0 0 26 27 H+HD+HZ+D D+HD 3Dz+H 2.76~lo2 dm3 mol-1 s-l 0.935 dm3 mol-'s-l 24.2 19.5 2.51 3.21 p p a Quoted in the general form : ki = Ai Tni x exp (-Ei/RT); b assuming a wall recombination coefficient of 3x ref.(14); Cref. (15); dref. (16); ek, = 2(k4k5)+;fan estimated upper limit ; ref. (17) ; derived from ref. (18) and (29) ; i ref. (18) ; J derived from ref. (4) and (19),taking k14 = &kI5,where E is the k14/k15 ratio predicted by theoretical work of ref. (4); kref. (19) ; from ref. (11) and (18) ; M derived from S.C.T. expression quoted in ref. (20) ; n ref. (20) ; O -200 kJ endothermic ; P derived from isotopic equilibrium constants and ref. (17). For subsequent iterations (kx)n+~= (kx)n * (rnlrexp) (1 1) that is, use was made of the almost linear dependence of r on ka/kx.As no direct determination of initial atom concentrations was made, these were estimated in the following manner :for the first iteration [Dl01 = [HDIObs +[Dcl]~bs+ 1, for D/HCl and 1, for H/DC1 (12) where [HD]ob,, [DCl],bs and [HCl]ob, were concentrations estimated using absolute mass spectrometer sensitivity measurements and A[D], A[H] corresponded to con- centrations of the atoms unreacted after time &.For the first iterations these were estimated using the approximate relations (I 3) AID1l = ([HDlobs+[DC1lobs) exp (-(k20+k2l)[HClIi+k,)f,) = ([HDlobs+[HC1]oba) exp (-{(k22 $.k23)[DCl]I +k2) tr). (13)For subsequent iterations A[D]n or A[H]n was set to A[D]n-I and A[H],-,, respectively, the residual atom concentrations calculated from the integrations. "1 [HDIo~s [HCl]~bs =[HI01 F.S. KLEIN AND I. VELTMAN 23 These relations are only approximate and the choice of the " basis set " of rate constants used for the integrations is to some extent arbitrary because of the wide spread of literature values. However, in most cases the corrections and the probable global error in the derived rate-constant ratios are small, <25 %. Arrhenius plots of the derived rate constant ratios are shown in fig. 1 and 2. DISCUSSION OF MECHANISM At higher temperatures rexpunderestimates the rate constant ratio primarily due to reactions (12) and (13) in which Ci atoms react with D2 or HZ.The corrections caused by these chain propagating steps are larger for the H+DCl+H2 system because kI3> kI2at ail temperatures. For the D+HC1+D2 system the correction applied for rexpwas < 10 % for T < 370 K.These corrections also eliminate the slight decrease observed in the product ratios above 320 and 400 K for the H+DCl and D +HCl systems, respectively. The corrections at lower temperatures tend to decrease rexp,primarily through loss of product HC1 or DCl through reactions with H and D respectively [steps (19) and (18)]. For the H+DCl+H2 system these are as much as 30 %, but for D+HC1+D2 these corrections are quite small and could not account for the constancy of rexpwith temperature observed at T < 200 K. This trend, which deviates sharply from that indicated by the higher values, was not explicable in terms of loss of a source of DCl from reaction (14) caused by C12 removal at low temperatures by wall adsorption.It seems likely that the DCl produced in these runs exchanged to some extent with hydrogen-containing material adsorbed on the walls. It should be added that at the lower temperafures the estimations of the corrections are subject to higher uncertainty, because the rate constants used are based mainly on extrapolations from data at higher temperatures, and also because of uncertainties in the effects of adsorption of certain species, e.g. C12. WALL EFFECTS Here we further discuss the heterogeneous reactions described earlier. These reactions were observed to occur even though the reactor was frequently coated with a known efficient wall poison, Drifilm.12 In the D+HCl system the hetexogeneous reaction resulted mainly in the production of HD and occurred whether or not the reactor wall had been freshly exposed to HCl.We conclude that this was caused by reaction of D atoms with hydrogen containing material in the reactor, i.e. systematic error (i). In the H+DCI system large amounts of HCl were produced when the reactor walls had been freshly exposed to flows of DCl, whereas heterogeneous production of HD was much less marked. The ratio W(3)/W(36) was -0.1 at room temperature, whereas H(3)/H(36) N 1. Additionally, the fraction of the total HCl produced through wall reaction increased with decreasing temperature. This indicates some specific heterogeneous effect favouring the exchange reaction. It is interesting to speculate as to the reason for this.DC1 is a highly polar compound (dipole moment -1.084e.s.u.) and could thus readily form a hydrogen bond with oxygen- containing materials such as silica or glass. This would simultaneously weaken the D-CI bond and orientate the molecule in a way favourable to exchange : nnf s IHa + C1-D. . . O-Si--+ HCl+D=. . .O-Si-s 1 l+3D2. wall diffusion BRANCHING RATIOS This would explain the HC1-catalysed loss of H in H+HCl experiments reported by some workers 20* 22 and the ability of certain wall coatings, especially waxes, to suppress this effect.22 COMPARISON WITH OTHER EXPERIMENTAL STUDIES Least mean squares analysis of the Arrhenius plots gave (ka/kx)~+~~l= k20/kZ1= (9.7+ 1.0)exp (-4440+ 590 J mol-l/RT)215 < T < 483 K and (14) (ka/kx)~+~~l= k22/k23 = (4.3k0.7)exp (-4350+290 J mol-l/RT) 138 < T<451 K.(15) The errors quoted above are standard deviations and reflect the substantial experi- mental scatter which is a manifestation of the varying heterogeneous effects. The values go some way to vindicate Thompson et aZ.’s4 suspicion that the reported anomalously low exchange cross section from Wood’s photolysis work is in error. Wood’s simplified mechanism consisting of steps (ll)?(13), (18), (22) and (23) gives a good account of the formal kinetic processes occurring, although his derived steady-state expression appears to be in However, this has little effect on the interpreiation of the data. The ratio of quantum yields is given by and as kll 4k18, 6HD/4D2 -k22/k23.The possibility that hot atom reactions could account for the observed photolysis ratios was considered by Thompson et aZ.,4but they were unable to reconcile their theoretical results with these experiments. It would seem that the discrepancy between Wood’s results and those obtained here could be explained by; (a)production of HD through reaction of D atoms with the walls of the photolysis cell.3 This would have lead to production oi HD during the actinometry measurements when DCl was photolysed on its own ; (b) enhancement of HD production caused by trace impurities of O2 in the reactants. This effect is thought responsible for the anomalously high rate constants obtained in single pulse shock tube measurements of H2/D2exchange 179 23 and could conceivably lead to similar consequences in the H2+DC1 photolysis, although Wood’s purification procedure should have removed such impurity.Better agreement occurs between this work and the photolysis data of De Vries and Klein.’ They obtained kzo/kzl< 0.9 at 295 K and k20/k21< 10 at 425 K. This is to be compared with 1.6k0.4 and 2.8k0.4 derived from eqn (15). The earlier work of Leighton and Cross 24 on D2+HCl photolysis and Steiner and Rideal’s D2+HCl pyrolysis support the general conclusion that k2,/k2,> 1 at higher temperatures, but these experiments were rather qualitative in nature so that direct comparison is difficult. The most recent measurements by Endo and Glass 25 on D +HCl exchange rates are in reasonable agreement with our results, considering their experimental error of a factor of two.COMPARISON WITH THEORETICAL TREATMENTS Theoretical studies of the chemical dynamics of D+HX and H+DX systems include statistical phase space treatments of the abstraction to exchange ratio 9*lo and quasi-classical trajectory calculation^.^* The trajectory calculations predict F. S. KLEIN AND I. VELTMAN k,, = 4.06 x lo9exp (-4470/RT) dm3 mol-' s-1 kzl = 6.36 x 1O1O exp (-910/RT) dm3 mol-l s-l k22= 1.31 x 1O1O exp (-14 760/RT)dm3 mol-I s-I k23 = 8.69 x loioexp (-4000/RT) dm3 mol-I s-I giving, ka/kx(D+HCl) = kzo/kzl = 0.064 exp (-3560/RT) ka/k,(H+DCl) = k22/k23 = 0.151 exp (-10760/RT). The pre-exponential factors are in clear. disagreement with those found here [eqn (14) and (15)], and disagree qualitatively to the extent that A20/A21<A22/A23.The activation energy difference predicted by their surface for the D +HC1 reaction is in fair agreement with our value but that for the H+DCl reaction is -6.3 kJ greaterthan our experimental value. Heidner and Bott 26 have criticized these activation energies, they estimated the total rate of reaction (k,+ k,) between D and HCl,=o and concluded that k, (7+3)x lo7 dm3 mol-I s-l at room temperature. These results imply that the activation energy for exchange is >12.5 kJ mol-I, assuming A21 N lolodm3 mol-1 s-l. Truhlar and Kupperman have applied quantum mechanical phase space theory to H+DX reactions. For X = C1 they obtain k22/k23 = 5.98 at 300 K.This theory requires little knowledge of the interaction potentials of the reaction system except at large separations of stable reactants and products, and thus the absolute magnitude of the rate constants found by this method are too high due to the failure to account for potential barriers along the reaction coordinate. Because of this their value of k22/k23 is perhaps best compared with the experimental pre- exponential factor ratio with which there is good agreement. MODELS TO ACCOUNT FOR EXPERIMENTAL FINDINGS In view of the disagreement between these two theoretical approaches it was decided to do some further calculations of a semi-quantitative nature. These are described in this section and are (I) transition state theory using a L.E.P.S.potential energy ~urface.~' (11) Levine-Bernstein estimation of prior branching ratios." (I) L.E.P.S. TREATMENT The H +HCl abstraction reaction rates were derived from a colinear transition state configuration, which had shown reasonable agreement with experimental kinetic isotope effects in the reaction Cl+H2,28 i.e. the reverse of the abstraction reaction. The HClH potential energy surface was chosen by adjusting the Sat0 parameters, ki, so that the resulting temperature effect of the branching ratio agreed with that of the experimental ratio.28 All of several colinear transition state configurations which were tried showed vibrational bending frequencies of <200 cm-l. The resulting calculated branching ratios were found to be one to two orders of magnitude lower than the experimental values, see fig.3. Similar branching ratios have been obtained by Thompson et aL4 The addition of an Eckart tunnel correction 29 increased this ratio in the D +HCl system and decreased it for H +DCl, but even for D +HCl the calculated values differed considerably from experiment. We may conclude then that unless very high tunnel effects contribute to the rate of abstraction, a colinear HCIH transition state exchange model does not agree with experiment. The effect of bending the H-Cl-H transition state configuration was tested next. It was found that small deviations from linearity had little effect on the rate results, whereas accute H-Cl-H angles (<goo) gave potential energy surfaces with un- realistically high energy barriers along the reaction coordinate. Transition states with an approximately rectangular configuration gave bending frequencies in the BRANCHING RATIOS 0.0.0 1 k,.-0.005----*\ *\* i\: '4 '% 'a, +\ 0.001 I I 1 *\a range 400-700 cm-l. The calculated branching ratios tor such a model are also shownin fig.3, in fair agreement with the experimental lines. The addition of Eckart tunnel corrections increased the difference between the two isotopic systems. Such corrections generally overestimate the tunnel effect.29 If then only 10 % of the calculated effect is added, the resulting branching ratios will agree with the experiment within the standard error.F. S. KLEIN AND I. VELTMAN It appears from this study that the abstraction reaction requires a colinear HHC1 transition state, which has already been shown to fit the opposing reaction, Cl+Hz. The exchange reaction rates can be rationalized according to the LEPS model, by a potential energy surface based on an approximately rectangular transition state. TABLE3.-L.E.P.S. MODELS DHCl HDCl DClH HClD model B kl 0.18 0.16 k2 = k3 0.18 0.10 angle a 0 0 A Y/kJ mol-l 18.7 (AYZ == 31.3) 13.1 v1/cm-l 991 1338 2195 2952 v2/cm-l 521 410 149 153 iv3/cm-l 1898 1405 1190 1672 model C kl 0.077 0.18 kz = k3 0.28 0.16 angle a 0 A Y'kJ mo1-1 20.9 (AYz = 33.5) 13.0 v /cm-981 1311 2129 2406 v2/cm-543 427 681 688 iv3/cm-1986 1486 1186 1359 (11) LE VINE-BERNSTEIN ESTIMATION OF PRIOR BRANCHING RATIO, r,q, Levine and Ber nstein O have proposed a simple way of estimating prior branching ratios, based on the statistical assumption 30 and rigid rotor-harmonic oscillator (RRHO) approximation.They obtain, (17) where ET is the relative translational energy of the atom and diatomic molecule, E, and Ex are the total energies available to abstraction and exchange products respectively, and the y's are "structural factors " given by Y = @/haeBe (18) where ,up is the reduced mass for relative translation of products, me and Be are the stretching frequency and equilibrium rotational constants for the products. For the H+DCl and D+HC1 systems ya/yx = 0.32 and 0.31 respectively.Since we are dealing with a thermal system we now need to develop an equation for rtx(T). This means integrating the numerator and denominator of eqn (17) for a Boltzmann distribution of incident relative translational energies, ET, i.e. PT(E*)E? exP (-E,IW dET Kx(T) = (19)t)fPT(ET)Ef exp (-ET/RT) dET E: where pT(ET)is the translational density of states given by 31 P=(E~)= (,.4/2~i31~4 (20) BRANCHING RATIOS P is the reduced mass of reactants, and E: and Ei are " barrier height " constraints to the abstraction and exchange paths respectively (note that, since the integration is over incident reactant relative translational Energies, the denominators of the distribu- tion functions have been assumed to cancel). We now make the approximations Ea -&-Ha and Ex-&-Hx (21) where Ha and H, are the heats of abstraction and exchange respectively.Putting 8, = Ha/RT; 0, = H,/RT; a, = E:/RT; a, = E",/RT; p = (ET-E")/RT; p' = (ET-Eo)/RT and changing the limits of integration we have r;,(T) =(:)e-(a*-ax) (B+aa)* (p+aa-0a)3 e-B dp Irn03 (p' fa,)' (p' +ax-e-8' dp' 0 (22) If we assume that the halogen atom formed in the abstraction reaction is in its ground 'P3state and that the D and H atoms formed in the exchange process are in the ground 'S, state, we must make allowance for electronic degeneracy ra~x(~)= (;)( :)V(E: 9 E:, ea, 0x1 exp -(aa-ax> (23) where q(E&Ei, Oa,83 is the ratio of integrals in eqn (22). These integrals were evaluated using the method of Laguerre quadrature (32 points) with the IBM 2 3 4 5 6 7 8 103~/~ FIG.4.-H+ DCl reaction. Levine-Bernstein prior branching ratio estimation E:* e* "A "t (( Esct99 * -.-.- 16.7 14.6 11.8 9.6 3.12 3.7 4.36 4.36 .... .. . 11.3 5.5 5.2 4.24 12.6 7.1 4.5 4.33 experimentline 4.3 4.37 F. S. KLEIN AND I. VELTMAN SPEAKEASY facility. For H +DCl : Haand H, were taken as -6.7 and +1.67 kJ mol-', respectively, and for D +HCl as -8.4 and -1.67 kJ mol-l. Fig. 4 and 5 show the results obtained in comparison with the experimental Arrhenius line for sets of reasonable values of E,"and E:. The procedure adopted was to choose an Ei value (11.3, 12.6, 14.6, 16.7), adjust E: until the slope of the line, calculated by least mean squares analysis between 100 and 500 K, agreed with the experimentally determined value.In one calculation for D+HCl, Ei was set to zero and E," was varied. Using this procedure it is seen that r:: does not have a strong dependence on the value of Ei (or Ei). I I I I 1 I 11 4.0 t-103 KIT FIG.5.-D +HCl reaction. Levine-Bernstein prior branching ratio estimation. E:* E:* "A " t " Eact "* ----16.7 12.0 2.2 4.44 -.-.-.-14.6 9 8 2.5 4.42 ------12.6 7.5 2.8 4.41 ......... 11 3 6.1 3.2 4.44 -e ..-...-7.5 0 4.9 4.44 ---experiment line 9.7 4 45 * kJ mol-I; t pre-exponential factor A In interpreting these results one must be wary of the inaccuracies of this method, especially the break-down of semi classical density of states approximations at low energies.32 However some useful comparison can be made with the experimental results. Although the close agreement between prior and experimental branching ratios for the H+DCl is perhaps fortuitous we can conjecture that these results are ''not-surprising ".For the D +HCl reaction, though, r&is predicted lower than BRANCHING RATIOS for H+DCl, whereas experimentally it is higher. We can thus say that these results are "surprising " and take it as indicative of some dynamic constraint operating in the system. The most likely effect is one of rotational screening of the chlorine atom by the rotating H-atom. For D+HCl the H rotates relatively rapidly around the central Cl atom and the approaching D moves relatively slowly.For the other system the reverse is true and the approaching H atom can thus penetrate the D "shell " with more ease, leading to relatively more exchange. This concept is compatible with the molecular beam observations of measurable exchange cross sections in the D+HCl reaction as they used D beams of relatively high collision energy (-38 kJ mol-l) colliding with HCl molecules thermalised at 250 K, i.e. with a rotational energy of -2.1 kJ mol-1, CONCLUSIONS We conclude from this study that the cross-section for the isotope exchange process H +ClH is not as small as at once thought but is still smaller than that for abstraction. The results obtained (ka/kx)D+HCI = (9.7_+1.0)exp (-4440+590 J mol-l/RT) and (ka/k&+DC1 = (4.3*0.7) exp (-4350f290 J rnol-l/HT) can be rationalised in terms of (a)a rectangular HClH transition complex with inclusion of quantum mechanical tunnelling or, (b) the relative availability of product quantum states for the two reactions paths in the Levine-Bernstein statistical model.The results of analysis in this case indicate that relative reactant atom diatomic translation and rotation should influence the choice of reaction paths in reactive molecular collisions. These results do suggest the desirability of better potential energy surface calcula- tions and further studies with three dimensional trajectory calculations, and also of further experiments, especially with molecular beams to determine the effect ot relative translational energy and diatomic rotation on the branching ratio.One of us (I. V.) wishes to thank the Weizmann Institute for its hospitality and for the award of a Junior Weizmann Fellowship for 1976. We also thank Prof. Aron Juppermann for helpful discussions. N. Semenoff, Chemical Kinetics and Chain Reactions (Clarendon, Oxford, 1935).(a) J. V. V. Kasper and G. C. Pimentel,Phys. Rev. Letters, 1965, 14, 352 ; (b)Proceedings of the Army Symposium on High Energy Lasers (Redstone Arsenal, Alabama, 3-4 Nov., 1975). G. 0.Wood, J. Chem. Phys., 1972,56, 1723. D. L. Thompson, H. H. Suzukawa Jr. and L. M. Raff, J. Chem. Phys., 1975,62,4727. J. D. McDonald and D. R. Herschbach, J. Chem. Phys., 1975, 62,4740. L. M. RaE, H. H. Suzukawa Jr. and D. L. Thompson,J. Chem.Phys., 1975,62,3743.'H. Steiner and E. K. Rideal, Proc. Roy. SOC.A, 1939,173,503 and 531.* A. E. De Vries and F. S. Klein, J. Chem. Phys., 1964,41, 3428. D. G. Truhlar and A. Kuppermann, J. Phys. Chem., 1969,73,1722. lo R. D. Levine and R. B. Bernstein Chem. Phys. Letters, 1974, 29, 1. 'l Y. Bar Yaakov, A. Persky and F. S. Klein, J. Chem. Phys., 1973,59, 2415. l2 J. P. Wittke and R. H. Dicke, Phys. Rev., 1956,103, 620. l3 C. W. Gear, IFIP Proc. A, 1968, 81. l4 B. J. Wood and H. Wise, J. Phys. Chem., 1962,66, 1049. l5 D. W. Trainor, D. 0.Ham and F. Kaufman, J. Chem. Phys., 1973,58,4599. l6 H. Hippler and J. Troe, Chem. Phys. Letters, 1973, 19, 607. l7 G. L. Pratt and D. Rogers, J.C.S.Faraday I, 1976, 72, 1589. l8 A.A. Westenberg and N. de Haas, J. Chem. Phys., 1968,48,4405. l9 A. F. Dodonov, G. K. Lavrovskaya, I. I. Morosov, R. G. Albright, V. L. Tal'rose and A. K. Lyubinova, Kinetiku i Katuliz, 1970, 11, 821. 2o M. A. A. Clyne and D. H. Stedman, Trans. Faruday SOC.,1966, 62, 2164. 21 E. A. Guggenheim and J. Weiss, Trans. Faraahy SOC.,1938,34,57. F. S. KLEIN AND I. VELTMAN 22 J. E. Spencer and G. P. Glass, J.Phys. Chern., 1975,79, 2329. 23 S. H. Bauer and E. Ossa, J. Chem. Phys., 1966,45,434. 24 P. A. Leighton and P. C. Cross, J. Chem. Phys., 1938, 6, 345. 25 H. Endo and G. P. Glass, Chem. Phys. Letters, 1976, 44, 180. 26 R. F. Heidner, 111 and J. F. Bott, J. Chem. Phys., 1976, 64,2267. 27 J. C. Polanyi, J. Quant. Spectr. Radiative Transfer, 1963, 3, 476. 28 A. Persky and F. S. Klein, J. Chem. Phys., 1966,44, 3617. 29 H. S. Johnston and J. Heicklen, J. Phys. Chem., 1962, 66, 532. 30 R. D. Levine and R. B. Bernstein, Molecular Reaction Dynamics (Oxford University Press, New York, 1974), p. 102. 31 R. D. Levine and J. Manz, J. Chem. Phys., 1975, 63,4280. 32 P. J. Robinson and K. A. Holbrook, Unimolecular Reactions (Wiley, London, 1972). (PAPER 7/252)

ISSN:0300-9238    

DOI:10.1039/F29787400017

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

On the Direct Inclusion of Polarization into an Electron Gas Model for Closed Shell Atom-Ion Interactions BY JOHNLLOYD? AND DAVIDPUGH" Department of Pure and Applied Chemistry, University of Strathclyde, Glasgow G1 1XL Received 1st April, 1977 A general extension of the Gordon and Kim method has been made which allows polarized atomic charge densities to be used. Comparisons with ab initio studies for He+Li+ and He+Na+ show that good agreement can be obtained without any empirical modification of the polarization energy but that an exchange correction is necessary in the He+Li+ system. Results for He+ Li+, He+Na+ and He+ F-are compared with the modified Drude model and show smaller differences than might be expected. The investigation of closed shell intermolecular forces is currently receiving much impetus from density functional formulations of the short range interaction.A bibliography to many recent applications may be found in the study by Parker et al? in addition, further reports on atom-molecule 6* and atom-ion ** pairs have appeared, together with an attempt to include open shell configurations in the general scheme.lO The methods used are of importance since they combine the desirable feature of relative computational simplicity with a low degree of empiricism and, therefore, form an ideal basis for the interpretation of experimental data. In only a few studies, however, have the characteristics of the method been investigated in any detail. It is clear from the analyses given by Gordon and Kim l1 and Nikulin 2* that the method produces a nun-bounded approximation to the self- consistent density functional formulation of Hohenberg and Kohn l2 and tends to succeed by a cancellation-of-errors effect.Cohen and Pack l3 have shown for noble gas pairs that the exchange correction introduced by Rae 49 l4 in the Gordon and Kim method,2 results in much better agreement between electron gas and Hartree- Fock-SCF-LCAO potentials. Although there have appeared several variants on the magnitude of this c~rrection,~. 13-15 it is apparent in all cases that the resulting potentials tend to overestimate the repulsive part. This has the result of making the simple addition of a dispersion energy series a reasonable procedure since some cancellation of errors will occur.These characteristics are obviously of some importance if optimum use is to be make of the calculated potentials. The case of atom-ion systems has not been investigated so deeply. As is well knownY1" the effect of charge overlap will be to make a simple multipole series for induction forces l7 unreliable at short-to-intermediate separations. Both Gordon and Kim 18*l9 and Gianturco have considered this aspect and suggested modifica- tions to the long range potential which will compensate for this. They assumed implicitly, however, that a superposition of free component densities was adequate to generate the repulsive portion, and did not consider a self exchange correction in any way. This may be ieasonable for heavy components at short internuclear t Present address : C.S.I.R., Pretoria, South Africa. 32 J.LLOYD AND D. PUGH distances, but over the intermediate range, where the overlap is small, it is of question- able validity since relatively large changes may arise from polarization. To consider these problems in more depth we report here some extended Gordon and Kim calculations on the interaction of helium with Lif, Na+ and F-. The effect of polarization on the atomic charge density is included by determining wavefunctions for the atom in the presence of a point charge of appropriate sign. This accounts for the bulk of polarization effects, while still allowing a superposition of component densities to be used in construcfing the potentials.The small size of helium makes the potentials sensitive to a self exchange correction, and the availability of ab initiu studies 2oy 21 allows direct comparisons to be made. Since the atomic charge density is now cylindrical rather than spherical a generalisation of the method is required and we present this in the following section. This is followed by a discussion of the results obtained; in the final section we provide a brief summary of the major conclusions arising from the study. COMPUTATIONAL METHODS The effect of an ion on the atomic charge densities has been simulated by determining sets of wavefunctions for helium in the presence of & 1 a.u. charges over a range of separations. The calculations were made using a modified version of the ATOM-SCF program of Roos et aZ.22with the full basis set of Clementi 23 plus additional polarization functions of p, d and ftype Slater orbitals.It is necessary to include the sign since the atom is not isotropic with respect to this quantity.l7 Tests showed that multiple exponent optimization of only one function of each type was required to produce around 0.3 % convergence in the moments when the separation was large ; this set has been used for the +1 a.u. charge. The -1 a.u. set has been determined in an unpublished study on higher polarizabilities and hence is slightly larger. A full discussion of the method is given by Roothaan et aZ:24 it has previously been used in determining higher polarizabilities of closed shell atoms 25-27 and rnolec~des.~~ To determine the interaction potentials the energy is first written as a sum of well defined components +EEXCH +ECORR +E~SPEI = EKIN + GOUL-(1) The kinetic, exchange and correlation contributions are calculated from the total charge densities exactly as in ref.(2), the total density still having C,, symmetry. EDIspis a dispersion energy series, and we follow the practice of earlier studies 6*7* l5 in keeping this term sepaiate from the correlation energy since the latter is zero when the overlap goes to zero. VcouLis used to designate the coulomb interaction because it will be necessary to separate this into additional terms and it removes any possibility of confusion with other components. We begin the evaluation of the Coulomb term, as in ref.(2), by extracting the zero monopole-monopole interaction to give : where, pa(rl) and Pb(r2) are the total charge densities of the atom and ion respectively: they can be expanded as sums of spherical harmonics 29 11-2 CLOSED-SHELL INTERACTIONS The Eegendre polynomials P;(cos 0) can be defined such that R,,lies on the Z-axes and provides the simplification that pa = pb. Since the ionic charge distribution retains spherical symmetry this leads to : The first term of the summation, 1, = 0, involves only spherical distributions and can be evaluated using standard techniques as a contribution vsph. For higher terms we continue the reasoning behind removing the monopole term and extract ion-induced dipole interactions to give : where, and r> is the greater of rib and r2b for a fixed value of rib.The advantages of this method are that the first integral in eqn (6) depends only on the atom and can be obtained directly from the SCF results as a contribution, V&, providing terms which will be included in vsph are subtracted. -ESCF( [is:VpoL= EscF(q,R) co)+ -p:”(r)r2 dr+s,paOJ0(r)rdr .1 Inspection of the second integral in eqn (6) shows that it contains the desirable feature of automatic cancellation when the overlap is small. It is, therefore, suitable for numerical quadrature and in practice a 24 x 24 Laguerre-Legendre crossed quadrature gives about 3 figure accuracy. The integrals of eqn (6) depend only on the ion and can be stored numerically with the atomic densities and polarization energies.Thus one obtains a scheme involving numerical quadratures of tabulated functions which depend only on the atom or ion involved and can be applied in a wide variety of situations. The calculations reported here use the polarized atomic wavefunction plus the ionic functions of Clementi.23 1, was set to an arbitrary figure of 5, and for inter- mediate points not covered by accurate functions we have used exponential interpola- tion for each component potential. This procedure is quite accurate, since the variation is very regular as a function of distance, and it is only the sum which shows rapid changes. An asymptotic dispersion-induction series is used to represent the potential at distances >7 a.u.DISCUSSION OF RESULTS As might be expected the interactions are dominated at large distances by induction forces, while at shorter separations the kinetic energy rapidly becomes dominant. A comparison of the results with those of Gordon and Kim is given in table 1 where we have tried to maintain equivalence by using only the kinetic, exchange, correlation and spherical parts of the coulomb energies. It can be seen that the positive ion examples differ significantly, with the present results being larger J. LLOYD AND D. PUGH 35 by several orders of magnitude; in all cases the rate of change with separation is smaller. The latter effect is partly due to the non-spherical Coulomb terms which will tend to compensate changes in the spherical part and also explains the apparent similarity between He +F-results.Note that only total interaction potentials have been used here and that the component potentials show even greater differences. TABLE1.-~MPARISON OF THE ELECTRON GAS POTENTIAL OF GORDONAND KIM,18 &(I), AND THE PRESENT RESULTS, E~(11). ALLVALUES ARE IN ATOMIC UNITS. 1 a.u. of energy = 4.3559~ J; 1 a.u. of length = 5.2917~ m He +Li+ He+Na+ He+F-R EG(II) R EGU) EG(II) R EGU) Ea(I1) 10-1 0.306~ 10-1 2.5 0.149~10-1 0.394~ 10-1 2.5 0.66 x 10-1 0.164 3.0 0.416~ 3.0 0.331 x 10-2 0.104X 10-1 3.0 0.472~10-1 0.472~10-1 4.0 0.459~10-2 0.391 X 10-2 4.0 0.668~10-4 0.224~ 10-3 4.0 0.750~10-3 0.306~10-2 5.0 0.819~10-4 0.211 X 10-3 5.0 -0.163~ 10-4 -0.250~ 10-6 5.0 -0.545~ 10-4 0.753~10-4 6.0 -0.208~10-3 -0.219X 10-5 In comparing the present study with ab initio calculations we are faced with exactly the same type of problems encountered in the noble gas pairs.Cohen and Pack l3 used the total number of electrons to generate an exchange correction, while in earlier studies 6*7*l5 we have argued that only valence electrons need be considered. Both approaches tend to produce similar final results, however, since in the former case the correlation energy is linked to the dispersion energy l3 while in the latter it is considered as part of the " Hartree-Fock " energy and mutual cancellation of errors occurs. For the comparison given here we have followed the practice of our earlier studies but for completeness will mention any significant differences which arise if Cohen and Pack's correction is used.It is of value to compare the He +Li+ potentials at two levels. If dispersion terms are neglected then we can consider the Hartree-Fock study of Krauss et aL2' Applying known values of the dispersion coefficients up to Clo30* 31 allows the configuration interaction calculations of Hariharon and Staemmler 21 to be used as the basis for comparison. A summary of these is given in table 2 ; several interesting observations can be made. TABLE2.-cOMPARISON OF SCF AND ELECTRON GAS RESULTS FOR He+ Li+ IN ATOMIC UNITS ECIR EHF EG Eb E; 2.5 1.6708 x 4.781 x 2.085 x 1.676~ 1.604~ 10-4 1.60 x 10-43.O 1.23 x -4.658~ 1.226 x 2.58~ 3.5 2.259~ -4.509~ -2.426 x -2.732 x -2.678 x 4.0 -2.171 x -3.166~ -2.348 x -2.466 x -2.450 x 4.5 -1.591 x -1.943 x -1.701 x -1.754 x -1.775 x 5.0 -1.101x 10-3 -1.240~10-3 -1.159~ -1.185~ -1.219~ 6.0 -5.44 x 10-4 -5.609~ 10-4 -5.513 x -5.60 x 10-4 -5.82 x 10-4 7.0 -2.98 x -2.913 x -2.904~ 10-3 -2.94 x 10-4 -3.06 x 10-4 Em -2.41 x -5.138~ -2.633~ -2.842~ -2.744X rm 3.65 3.20 3.70 3.60 3.63 om 3.01 2.65 3.05 3.05 3.O EHFHartree-Fock study of Krauss et aLZ0 &-Present results without correction. Eb-Present results plus exchange correction for 4 electrons (C,= 0.1324). Ei-EG of present study plus EDISP.EcI-Configuration interaction study of ref. (21). om, rm and Em are defined in the caption to fig.1. CLOSED-SHELL INTERACTIONS The introduction of an exchange correction based on 4 participating electrons can be seen to provide a dramatic improvement between electron gas and HF potentials. This is, in fact, independent of the correlation energy and, if it is neglected, one simply obtains a slightly steeper repulsive portion with closer agreement on the well depth (-2.402 x a.u.). A difference of 9.3 % in the potential minimum is hardly significant, however, compared with a 92 % overestimate when no exchange correction is used. The exchange-corrected results for He +Li+, therefore, appear to follow closely the observations on He+He.49 l3 Comparison of the total interaction energies given in table 2 would appear to confirm this since the agreement is very good at all points, and supports a simple addition of dispersion energies.TABLE3.-cOMPARISON OF SCF AND ELECTRON GAS RESULTS IN ATOMIC UNITS FOR He+Na+ R EHF EO E', 2.5 7.852~ 7.641x lo-* 1.173 x 10-1 3.O 2.071x 2.753x 4.327 x 3.5 3.941 x 4.360~ 1.023x 4.0 -3.90 x 10-4 -8.699x 1.274~ 4.5 1.139~ -1.493x -7.200x 5.0 -9.60 x 10-4 -1.189~ -9.110~ 10-4 6.0 -5.30 x 10-4 -5.783 x -5.425x 7.0 -2.90 x 10-4 -2.960 x -2.914~ Ern -1.153 x lob3 -1.514~ -9.32 x 10-4 rm 4.70 4.40 4.85 urn 3.87 3.80 4.20 EHp-Hartree-Fock study of Krauss et aL20 &-Present results without correction. EG-Present results corrected for a 10 electron exchange interaction (C,= 0.3175). * Values based on analyticalfit given in ref. (20).The He +Na+ results are not so unambiguous, as is shown in table 3. Correcting the exchange potential for 10 electrons produces a potential well which is too narrow and slightly too shallow. Neglecting this correction on the other hand gives again too narrow a well but overestimates the well depth by a larger amount. Exclusion of the correlation energy, assuming a 12 electron exchange, moves the uncorrected results into closer agreement with the HF study but the general observation is unchanged since the two forms still bracket the SCF results over a _+20% range in the well depth, E,, with little variation in positional parameters. Thus, the observed behaviour does not quite follow that for He+Ne 4* l3 and we find here a smaller sensitivity to exchange corrections than might be expected.Use of a point charge to simulate the ion will tend to overestimate polarization at short distances since it includes only the penetration of a screened core and neglects inter-electron effects. This, combined with the smaller exchange correction and larger density on Na+ is more than adequate to explain the observed differences in the He+Na+ potential, even though the distances involved are larger. If one con- siders the relative simplicity of the present method compared with the ab initio calculation, for example the calculation takes no longer than for He+Li+, then the overall agreement between electron gas and SCF results is very reasonable. It should be remembered at this point that no empirical parameters have been used to calculate the potentials.The correction for self exchange may be considered to be arbitrary 32 but it is certainly not empirical; inspection of the results indicates that the correlation term is of little importance. We, therefore, believe that the J. LLOYD AND D. PUGH comparison with the SCF studies is of significance and shows that if polarized charge densities are used one obtains atom-ion potential which approximate closely more rigorous studies, without the introduction of further empiricism into the model. With these observations in mind it is of further value to compare the present results and the modified Drude model of Kim and Gordon.lg These are shown graphically in fig.1-3 with some experimental The Drude model calculations differ significantly from the present study since they do not include exchange corrections, use only free component charge densities and base the attractive long range interaction on a coupling of harmonic oscillators perturbed by the short range potential. Values for the dispersion energies in He +Na+ and He + F-used here are only rough estimates based on their dipole polarizabilities ’* 25* 26 but it can be seen that the general agreement in all cases is quite good. R1a.u. Gordon and Kim’s modified Drude model. FIG.1.-The He+Li+ potential. -present work; Experimental estimates for the parameters of the potential are as follows : a, (internuclear distance at zero potential) = 3.7k0.1 a.u.; Ym (internuclear distance at the potential minimum) = 4.0k 0.1 a.u. ; Em (interaction energy at the potential minimum) = (2.0k0.1)x a.u. This apparent agreement can only be explained by different errors cancelling to produce similar results. We have already shown that such an effect exists between our formulation and the similar concept of Cohen and Pack. The correspondence with Kim and Gordon’s study is, therefore, surprising and leads to an unexpected conclusion. Because the method of ref. (19) uses readily available wavefunctions it is simpler and of more general applicability than the present scheme. Provided a greater degree of empiricism is acceptable then it would appear that not only is it, therefore, preferable but further may be taken with more confidence than might have previously been supposed.CLOSED-SHELL INTERACTIONS Of $ ‘r;i\ -0.5 m53 -1 -0 -I .5* FIG.2.-The He+Na+ potential. -present work ; 0 Gordon and Kim’s modified Drude model. Experimental estimates give om = 3.8 a.u., r, = 4.5 a.u. and E, = -1.5 x a.u. \ R1a.u. 1 4 P b 8 -OS{ FIG.3.-The He+F-potential. -present work ; 0 Gordon and Kim’s modified Drude model. J. LLOYD AND D. PUGH CONCLUSIONS By extending the Gordon and Kim method to include polarized atomic charge densities we have been able to make direct comparisons between SCF and electron gas calculations. For He + Lif the results clearly show an exchange correction as the only modification required, and that any uncertainty on the role of the correlation energy will be of little significance.The He+Na+ comparison was less clear, suggesting that for heavier components the potentials ale not very sensitive to this correction. In both cases studied the general correspondence was sufficiently good to indicate that a high degree of consistency is present in the electron gas model when the charge densities are a reasonable approximation to the physical situation present. V. A. Gaydaenko and V. K. Nikulin, Chem. Phys. Letters, 1970, 7, 360. R. G. Gordon and Y.S. Kim, J. Chem. Phys., 1972,56,3122. V. K. Nikulin, Zhur. Techn. Fiz.,1971, XLl, 41 (Sou. Phys-Tech. Phys., 1971, 16, 28). A. I. M. Rae, Chem. Phys. Letters, 1973, 18, 574.G. A. Parker, R. L. Snow and R. T. Pack, J. Chem. Phys., 1976, 64, 1668. J. Lloyd and D. Pugh, Chem. Phys. Letters, 1976, 39,468. J. Lloyd and D. Pugh, J.C.S. Faruday 11, 1977, 73, 234. V. K. Nikulin and Yu. N. Tsarev, Chem. Phys., 1975, 10,433. F. A. Gianturco, J. Chem. Phys., 1976, 64,1973. lo W. A. Harrison and R. Sokel, J. Chem. Phys., 1976, 65, 379. l1 Y. S. Kim and R. G. Gordon, J. Chem. Phys., 1974,68, 1842. l2 P. H. Hohenberg and W. Hohn, Phyx. Rev. B, 1964, 136, 864. l3J. S. Cohen and R. T. Pack, J. Chem. Phys., 1974, 61, 2372. 14A. I. M. Rae, Mol. Phys., 1974,29,467. l5 J. Lloyd and D. Pugh, Chem. Phys. Letters, 1974,26, 281. l6 J. 0.Hirschfelder, Chem. Phys. Letters, 1967, 1, 363. l7 A. D. Buckingham, Adv. Chem. Phys., 1967,12,107.Y.S. Kim and R. G. Gordon, J. Chem. Phys., 1974,60,4323. l9 Y. S. Kim and R. G. Gordon, J. Chem. Phys., 1974,61,1. ‘O M. Krauss, P. Malconado and A. C. Wahl, J. Chem. Phys., 1971,54,4944. 21 P. C. Hariharan and V. Staemmler, Chem. Phys., 1976, 15,409. 22 B. Boos, C. Solez, A. Viellard and E. Clementi, IBM Res. J., 1968, 518. 23 E. Clementi, Supplementary tables to IBM, J. Res. Dev., 1965, 9, 2. 24 H. D. Cohen and C. C. J. Roothaan, J. Chem. Phys., 1963,43, 534. 25 H. D. Cohen, J. Chem. Phys., 1965,43, 3558. 26 H. D. Cohen, J. Chem. Phys., 1966,45, 10. ’’R. E. Sitter and R. P. Hurst, Phys. Rev. A, 1972, 5, 5. 28 A. D. McLean and M. Yoshimine, J. Chem. Phys., 1967,46, 3682. 29 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, Molecular Theory of Gases and Liquids (J. Wiley, N.Y., 1964), chap. 12. 30 J. T. Broussard and N. R. Kestner, J. Chem. Phys., 1973, 58, 3593. 31 R. F. Stewart, J. Phys. B, 1973, 6,2213. 32 E. A. Mason and H. W. Schamps, Ann. Phys. (N.Y.),1958,4,233. 33 S. Green, B. J. Garrison and W. A. Lester Jnr., J. Chem. Phys., 1975, 63, 1154. (PAPER 7/579)

ISSN:0300-9238    

DOI:10.1039/F29787400032

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Sorption Kinetics and Time-Lag Theory Part 1 .-Constant Diffusion Coefficient ASH,RICHARD AND ROBERTBY RICHARD M. BARRER" J. B. CRAVEN Physical Chemistry Laboratories, Chemistry Department, Imperial College, London SW7 2AY Received 7thApril, 1977 The time-lag concept has been applied to sorption kinetics so as to provide an alternative procedure for obtaining diffusion coefficients D, in systems where D is a constant. From the graph of fractional approach to equilibrium, ~(f),[0 < ~(t)< 13 against tv(v > 0), the quantity Z = Jr~(t)d(tV) (obtained as the area beneath the curve) may be found as a function of tv. For large t the curve of Zagainst tvapproaches a linear asymptote which intersects the axis of tv to give a time-lag, Lv, where L~ = ?(t)l d(t9.J~u-Explicit equations for Lv have been obtained and investigated for various geometries of the diffusion media.Those for v = 3 and v = 1 in particular often lead to simple relations from which D can be evaluated. Both constant pressure and constant volume sorption kinetics have been considered, in the latter instance when Henry's law governs the sorption isotherm. 1. INTRODUCTION The rate of sorption by porous solids or organic polymers is an established means of determining diffusion coefficients in these media. Measurements are usually made under conditions of constant pressure or of constant volume. In either case one may construct a plot of (Qt- Qr)/(Qm -Q,) as a function of .Jt (or of t), where Q, and Qm are, respectively, amounts of diffusant (uniformly distributed) taken up by the sorbent at time t = 0 and when equilibrium has been attained and Qt is the amount taken up (but not uniformly distributed) at time t.In the earlier stages of a kinetic run it is often found that d -[(Q,-Qi)/(Qm-Q,)] = k, a constant. dJt Then, for constant pressure sorption by slabs, solid cylinders, spheres and rectangular parallelepipeds and a constant diffusion coefficient, D, we have k = 2(A/V)(D/n)* (2) where A and V are, respectively, the total external surface of the particles across which transport takes place and the volume of the particles. Thus knowledge of A and V (or AIV) permits calculation of D from k. Analogous equations have been derived for constant volume sorption Even if the diffusion coefficient is concentration dependent, eqn (1) is still frequently ~bserved.~ Less frequent use is made of the complete plot of (Qt-Q,)/(Q, -Q,) against .Jt (or t) in evaluating diffusion coefficients. It is with an aspect of this matter that this paper is concerned.40 R. ASH, R. M. BARRER AND R. J. B. CRAVEN 2. TIME-LAG EXPERIMENT Fig. l(a)and (b)shows the form of a plot of q(t) against t, where q(t) = (Qt -Q,)/ (Qo0-Q,). This form would be obtained for a kinetic run under constant pressure or constant volume conditions. The area beneath the curve up to time t’ is then given by t’ T(t’)= q(t)dt (3) 0 and the shaded area of fig. l(u) by [l -q(t)]dt. Provided t’ 03 L = lim [l-~(t)] dt = 1 [l-q(t)] dt (4)t’+m 0 0 exists, the plot of T(t) against t will have the form shown in fig.l(c). This curve approaches an asymptote of unit slope which gives an intercept (or time-lag), L, on the axis of t. Thus, for large t, T(t) = t-L, (5) where L is the shaded portion of fig. I@). This result is of complete generality, being independent of all assumed equations of flow and of the conditions under which the equilibrium was attained, and valid for any distribution of particle sizes and shapes. Therefore, if transport is formulated in terms of Fick’s equations, eqn (4) and (5) are true for all functional dependances of the diffusion coefficient. t FIG.l.-q(t) and T(t)as functions of t. In practice we find it is often difficult from the kinetic runs to be sure of (Qa,-Q,).In these circumstances the time-lag procedure may be modified to allow estimation of (Qol-Qf). Errors in placing the dashed line in fig. l(b)introduce errors in the cross- hatched area, which is L. However if T,(t’) = (Qr-Q,)dt is plotted against tsb’ there is no error in evaluating Tl(t’) associated with uncertainty in (Qco-el),and T,(t’)approaches asymptotically a straight line of slope (Qa -Q,). This straight line makes an intercept 9on the axis of t where 9= (Qo0-Qi)L. In what follows we derive L, and hence know 9,for a number of situations, so that by using plots SORPTION KINETICS AND TIME-LAG THEORY of Tl(t’)against t’ not only may (Qm -Ql) be estimated but also from 9 or L the diffusion coefficients are very simply derived.3. CALCULATION OF L FOR PARTICULAR SYSTEMS Solutions for the concentration, C, as a function of time and positional co-ordinates are available for many cases of interest, or may be readily derived. From these solutions by integration through the volume of each medium, one obtains q(t). Finally by integrating { 1-q(t)) between t = 0 and t 3 ca[eqn (4)] expressionsfor L may be obtained. All results are for a constant diffusion coefficient. 3.1 CONSTANT PRESSURE SORPTION KINETICS The boundary conditions are : C = C1at the outer surfaces for t > 0 ; C = Ci, a constant, within the medium at time t = 0. V and A are, respectively, the volume of the medium and the total surface area through which transport takes place.(i) The slab (-I < x < I) with both plane faces permeable ;no transport across the remaining surface* : L = Z2/3D= (V/A)2/3D. (6) This equation also holds for a slab of thickness I with one plane face permeable and no transport across the remaining surfaces. (ii) The solid cylinder (0 < r < b) with no transport across the plane faces : L = b2/8D = (V/A)2/2D. (7) (iii) The solid sphere (0 < r < b): L = b2 J15D = 3(VJA)2/5D. (8) (iv) The solid cylinder (0 < r < by -I < x < I) with transport across plane and curved surfaces : and The Pn are the positive roots of .Io@)= 0, where .To is the Bessel function of the first kind and of zero order. An alternative procedure, outlined in the Appendix, gives the following expression for L involving only a single summation : Eqn (10) and (1 1) allow the evaluation of L as a function of Z/b.The limiting cases when Z/b 4 1 and I/b S 1 are given by eqn (6) and (7) respectively. (v) The rectangular parallelepiped (-a < x < a,-b < y < b, -c < z -c c) with transport across all surfaces : * In eqn (6),A = 2 x area of one plane face. R. ASH, R. M. BARRER AND R. J. B. CRAVEN in which al,m,n = (h2/4)[(22+ 1)2/a2 +(2m+ 1)2/b2+(2n+ 1)2/~2] and L = (4a2/n2D)(8/n2)’x An alternative procedure, outlined in the Appendix, gives the following expression for L involving only a double summation : in which Po E [(22+ 1)2 +(2m+ l)2/(b/a)2]lfn. From eqn (13) and (14)it follows that for the cube of edge 2a the time-lag is given by d = (4a21n2D)(8/n2)3x 1 21+ Q2(2rn+1)2(2n+1)2[(2~+ 112+(2m+112+(2n+ where y = [(21+1)2+(2m+ I)’]%.(vi) The infinite cylinder of rectangular cross-section ( -a < x < a, -b < y < b): The alternative procedure of the Appendix gives : L = (32a2/D)2 111-2{ cosh (a, :) -1} ’ (17)l=O a;: (ao :) sinh (ao :)\ J in which a. = (22+ 1)n. In (i) to (vi) nothing has been said concerning the relative magnitudes of C, and C,. Hence, all the results are valid for sorption or desorption. Further, the time-lags are independent of C1 and C,. Eqn (6)-(8)in particular provide simple means of evaluating diffusion coefficients, D, when these are independent of concentration, position and time.The remaining results [eqn (lo), (11) and (13)-(17)]involve infinite series whose sums are readily computed. In fig. 2, DL/b2 for the solid cylinder [eqn (11) and (12)]is shown as a function of Z/b for three different ranges of Z/b. Fig. 3 shows DL/a2 for the rectangular parallelepiped [eqn (13)and (14)]as a function of c/a for various values of bla, the two sections of the figure covering different ranges of cia. The upper curves of fig. 3(a) and (b) approximate to the graph of DL/a2 as a function of c/a for the infinite cylinder of rectangular cross-section: -a < x < a, -c < z < c, and are analogous to the curves of fig. 2. SORPTION KINETICS AND TIME-LAG THEORY 60 l0:O 20!0 3& LdO 5dO 6do 7d0 8Ob 9& Id0 (a) 0.0 2.0 LO 6.0 ELO 10.0 12.u ILU 1.w rao 2ao (b) 0.0 02 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 20 (C) Ilb FIG.2.--Solid cylinder with transport across plane and curved surfaces : DL/b2 as a function of ilb; (---), I/b + CO. If D is a function of concentration, then each equation for the time-lag will lead to some kind of integral diffusion coefficient for the interval (Q, -el). By making this interval small, D will change little with concentration over each such interval and so the time-lag equations for a series of interval kinetic runs will also give D as a function of concentration.For beds of particles, each one acting as an individual sorption medium, the results will apply provided all the particles are of the same shape and size. The new time-lag method applies to the study of trace-ion diffusion in exchanger beds (since here also D is constant) and to tracer diffusion of molecules in the sorbent containing untraced molecules of the same kind.3.2 CONSTANT VOLUME, VARIABLE PRESSURE SORPTION KINETICS Here C1= C,(t)in the transient state; we write lim Cl(t) = C,. Further, we t-t, assume the distribution coefficient between sorbed and gaseous diffusant to be a constant, kl, so that Cl(t)= k,4(t), C, = k14, where 6 is the gas-phase concentra- tion. The bed of sorbent is assumed to consist of n identical particles each of volume u. Conservation of diffusant gives : Vi$o+Qi = K4(t)+Qt = vi4,+Q, (18) in which Vl = Vg/n,where Vgis the constant external volume containing gaseous diffusant from which sorption takes place and Qt, Ql and Q, (as previously defined) each refer to a single particle of sorbent.Thus Qi = uCi with Ci the constant concentration of diffusant within the sorbent at time t = 0 and Q, = vCm,where, for each case below, c, = (aC,+ Ci)/(l +a) (19) with lim Cl(t) = Co. a = Vl/klv= V'/klnu and has been termed the "effective- t-ro volume ratio ".3 R. ASH, R. M. BARRER AND R. J. B. CRAVEN 45 10.0 4.0 2.5 .IS 1.0 0.8 cla FIG.3.-Rectangular parallepiped : DL/a2as a function of c/a; figures on curves are values of b/a. SORPTION KINETICS AND TIME-LAG THEORY (i) The slab (-I < x < I) with both plane faces permeable ; no transport across the remaining surface : with the Pn the non-zero, positive roots of ap+tan p = 0, and Eqn (21) may be simplified using a result given by Jaeger and Clarke ’or by using the alternative procedure given in the Appendix.In either case the result is : L = (Z2/3D)[a/(l+a)]. (22) (ii) The solid cylinder (0 < r < b) with no transport across the plane faces : The Pnare the non-zero positive roots of aPJo(P)+2J1(P)= 0 where Jo and J1 are Bessel functions of the first kind, of zero and first order respectively, so that The procedure of the Appendix gives : L = (b2/8D)[a/(l+a)]. (iii) The solid sphere (0 < Y < b) : with the Pnthe non-zero, positive roots of tan P = 3P/(3+ap2)and The procedure of the Appendix gives L = (b2/15D)[a/(l+a)]. As Vl (or YJ3 00, we obtain constant pressure kinetics and eqn (22), (25) and (28) reduce to eqn (6)-(8)respectively.4. VARIATION OF TIME-SCALE The time-lags of section 3 relate to plots of q(t) against t. However, as explained in section 1, frequent use is made of curves of q(t) against ,/t in interpreting experi- mental data. The immediate effect of replacing t by Jt is to condense the time-scale ; conversely,a plot of y(t) against ts (or some higher power oft) results in an expansion of the time-scale. It is of interest to evaluate the time-lags* corresponding with these various plots for the systems of section 3. (i) As an example of the method employed in calculating the time-lag, we consider the system of section 3.1(i). For this case : * The dimensions of these intercepts on the abscissa are tv.It is nevertheless convenient to refer to them as “time-lags ”. R. ASH, R. M. BARRERANDR. J. B. CRAVEN We require the result : Ve-"' d(tv) = ;T(v) (30)a where T(v) is the gamma-function defined by : r(v) = so e-x xV-l dx. The integral is finite for v > 0. Then, using eqn (29) and (30) we obtain : The following particular results are readily deduced from eqn (31) : L, = 7Zt(3)/n%D* (32) where <(p)is the Riemann Zeta function. 5(3) = 1.2020569032 [cf. ref. (S)] L1= Z2/3D [cJsection 3.l(i)] (6) L, = 9913 <(5)/2xS DS [5(5) = 1.03692775511 L2 = 414/15D2. Below we give the general results obtained in this manner for the remaining systems of section 3.1 plus, in each case, the particular result for v = .4; : a3 (ii) L, = 4vT(v)(b2/D)' n= 1 2bJn a L, JD n=l N 0-08088) (33)= -c {h}; (n= 1 {i}6(iii) L, = -pr(v)(b2/n2D)'71 m=l 0316b O3L, = -c cn3D3m=O n=l Fig.4 shows (D*L,)/b as a function of Z/b over two ranges of 216. As Z/b 40, the curve approaches asymptotically the straight line of equation (D+L+)/b= (7t(3)/71%) (ZIb) = (D+/b)L+(slab), where is given by eqn (32) (75(3)/n* = 0.48100). As co I/b + 00, (D+L+)/b+ 2n+ c (1/Pn3) = L,(cy*.) D+lb n=l = 0.28671 SORPTION KINETICS AND TIME-LAG THEORY where L3ccul.,is given by eqn (33). 0303 l=O m=O n-0 1 (21-t 1)2(2m+ 1)~(2n+ (2m+1)2 (2n+1l2 (b/a)2 +m]1 Fig. 5 shows (D+Li)/aas a function of c/a for various fixed values of b/a over two ranges of c/a.When c/a 3 0, (D*L3)/a+ (Di/a)Lt<srab)= 0.48100(cla) where L+(slab)is given by eqn (32) with I = c. When c/a + 00 : which is the result for the infinite cylinder of rectangular cross-section 4ab. If in addition to c/a -+ co : (i) b/a 3 00, then (DiLi)/a + (7/7$)9(3) = 0.48100 (ii) bla is small, (D*L+)/a3 (7/7c*)t(3) (b/a)and so 0 as b/a 3 0. Finally, if b/a + 00, the upper curves of fig. 5(a) and (b) approximate to the curves for the infinite cylinder of rectangular cross-section : -a < x < a, -c < z < c, and are analogous to those for the solid cylinder (c$ fig. 4). FIG.4.-Solid cylinder with transport across plane and curved surfaces : D*L+/b as a function of l/b; (---), Z/b + 00; ( . . .), D*L+/b = (7[(3)/&)(Z/b) = 0.48100 (Z/b).0.3 (b) 0.1 0.21 cla FI~.5.-Rectangular parallepiped : D+L+/uas a function of c/a; figures on curves are values of b/u; (. . .), D+L*/u = (75$(3)/7r3)(clu). SORPTION KINETICS AND TIME-LAG THEORY General and particular values of L for the systems of section 3.2 are readily evaluated using the relevant equation given for 1 -y(t) in conjunction with eqn (30) and hence will not be given here. 5. DISCUSSION Fig. 6 illustrates the relation between y(t) and t [and also between T(t)and t]for sorption of krypton and methane by a sample of trisethylenediamine cobalt(rI1) fluorhectorite" [designated Co111(en),-FH150] at 195 K. A volumetric adsorption apparatus of standard design was employed.Although the pressure of diffusant was not held constant, the adsorption isotherms for Kr and CH4 at 195 K (both of type I in the B.E.T. classification) were sufficiently rectangular in character for the concen- tration of diffusant just within the boundary of the fluorhectorite to remain reasonably constant for the duration of the experimental run, thus approximating to the boundary condition of section 3.1. 0.0 1'0 20 30 4.0 5.0 6.0 7.0 8.0 9.0 10-3 tls 0.0 1,O 2.0 3.0 4.0 50 6.0 7.0 8.0 9.0 10-3 tls FIG.6.-q(t) and T(t)as a function of t for sorption by CoIII(en)j-FH150 at 195 K : (a) Kr, (6)CH4. (Upper scale refers to upper curve). * cation exchange capacity, 150 mequiv. per 100 g. R. ASH, R. M. BARRER AND R. J. B. CRAVEN The crystallites of fluorhectorite are in the form of hexagonal platelets, diffusion taking place perpendicular to the edges of the platelets with zero diffusion parallel to the edges of the platelets.The closest approach to this situation among the various systems investigated is that of the solid cylinder (of radius b) with transport across the curved surface and no transport across the plane faces [section 3.1, (ii)]. From the values of L, a mean value of (D/b2)was determined using eqn (8) : diffusant Lls lo4 (Dlb21ls-1 Kr 365 3.42 CH4 319 3.92 The successful application of eqn (1) and (2) (the Jt relationships) for evaluating D depends upon the accuracy with which the initial part of the curve of uptake as a function of time (or Jt) can be found.In beds of particles with extreme size distributions the linear region may be limited, and for systems in which the approach to equilibrium is very rapid it is difficult to make measurements. On the other hand, for measurable rates where the linear Jt position of the curve can be determined the method has the advantage of being independent of size and shape distributions of individual porous particles in the bed. Methods of measuring D based on curve-fitting for the plots of q(t) against t [fig. l(a)] are tedious and do not take account of the effects of size and shape distribu- it t.0 t FIG. 7.-Pressure against time curves for constant volume/variable pressure sorption kinetics : (a) adsorption, (b)desorption. SORPTION KINETICS AND TIME-LAG THEORY tions.They also suffer, in the case of rapid uptakes, from the experimental difficulty of establishing the first part of the curve of y(t) against t,just as in the case of the Jt procedure. The present method, as noted in section 3, also cannot take account of size and shape distributions of porous particles in a bed of powder. However, because the determination of the cross-hatched area [fig. I@)] is less sensitive than curve fitting to the exact form of the curve of y(t) against t, the method for uniform particles has advantages over curve-fitting procedures, besides, at least for the first three cases of section 3.1, being much simpler. It could be particularly useful for sorption in beds of uniform particles in the Henry’s law range at constant volume and variable pressure.Here it is only necessary to monitor pressure change in the system, using, for example, a transducer coupled to an automatic recorder. Then 1 -y(t) = [p(t)-p,]/(pO -pa)wherepo,p(t)and pa,are respectively, pressures at t = 0, t and at equilibrium. The value of L is given by the shaded area in fig. 7 divided by (po --pa). As a = (pa,-pi)/ ((0 -pa) [so that a/(l +a) = Cp, -pi)/(po -pi)] wherepi is the pressure in equilibrium with Ciy there is no need even to evaluate the amounts sorbed or to calibrate the various volumes of the apparatus in order to find D, using the result of section 3.2 for the appropriate particle shape. Two further points concern the results for constant volume, variable pressure sorption kinetics [eqn (22),(25) and (28)].First, for each of the three systems we have L, = L,Ca/(l +a)] (34) where L, is the time-lag associated with the constant-volume kinetics and Lp that associated with the constant-pressure kinetics. Since a > 0, it follows that Lp > L,. Eqn (34) suggests a possible general relationship between L, and Lp,although the generality has yet to be proved. An analogous relationship-also not yet proved in general-exists between the initial slopes of the y(t) against Jt plots for the two types of kinetics when D is a constant. Secondly, with a = Vg/klnu, denote the time-lag for this value of a by LU1. If the amount of sorbent is increased to mnu so that, as a result, Vgis reduced to Vo-(m-l)nu, the new value of a, denoted a,, is given by [a-((m-l)/kl]/m.The time-lag, L,,, for this value of a, is given by : Lm = Lp[a-(m-1)lk1l/[(a+1) -(m-1)/k11 so that L,, decreases with increasing m, ie., with increasing amount of sorbent for fixed total volume of sorbent plus gas phase. We may consider constant-pressure sorption kinetics for a parallelepiped in terms of the time-lags, Lpl, Lp2,Lp3, associated with sorption taking place through 1, 2 or 3 pairs of parallel faces respectively. Eqn (17) may be written where Lpl = a2/3D [cf: eqn (6)]. Since the summation in eqn (35) involves only positive terms, we have Lpl > Lp2. Similarly, eqn (14) [in conjunction with eqn (16)] may be written R. ASH, R. M,BARRER AND R. J. B. CRAVEN Thus Lpl > Lp2 > Lp,.Combination of eqn (35) and (36) indicates clearly the progressive modification of the time-lag, Lpl,for " 2-face " adsorption as the number of faces available to sorption is increased. If, in a given kinetic run, for the complete curve one determines both the slope, k, of the linear Jt portion and the value of L, then combination of eqn (2) and (6),for example shows that k2L= 4/3n so that if one of these quantities is known the other can be found. Provided the sorbent consisted of uniform particles, differences between observed and calculated values of k2Lcould possibly be used to indicate a distribution of particle sizes or the existence of functional dependences of D (i.e., dependence on concentration or on positional coordinates or time) although concentration dependence in D is best investigated by the interval method already referred to in section 3.APPENDIX ALTERNATIVE PROCEDURE FOR THE CALCULATION OF TIME-LAGS The procedure is originally due to Jaeger.6s If the Laplace transform, y', of y is of the form withf(p)/g(p) regular at p = 0, then y is given by y = f(o)(t -L)+transient terms do) in which Thus, for large times, y is asymptotically a straight line with slope f(O)/g(O) and intercept (time-lag), L, on the time axis. Use of this procedure can lead to an alternative, simplified, expression for L.l0*l1 Here we employ the procedure to calculate the time-lags for some of the systems of sections 3.1 and 3.2. In all the systems considered, the diffusion coefficient, D, is a constant.Section 3.1 : For all the systems, the time-lag, L,characteristic of the sorption kinetics has been shown to be independent of C1and C,. It is, therefore, sufficient to make the calculation of L for one particular set of allowed boundary conditions. For ease of calculation, we choose the case when C1 = 0, corresponding with constant-pressure desorption kinetics. (a)THE SOLID CYLINDER: 0 < r < b, 0 < x < Zl ; all surfaces (plane and curved) at zero concentration (for t > 0) ; C(x,r, 0) = C,,0 < r < b, 0 .c x < Zl. The equation of transport is Applying the Laplace transformation to eqn (A.1) gives the subsidiary equation : SORPTION KINBTICS AND TIME-LAG THEORY where q = (p/B)).The solution of eqn (A.2), subject to the given boundary condi- tions, is in which J, is the Bessel function of the first kind of order v and the Pnare the positive roots of Jo(p)= 0. The amount of diffusant in the cylinder at time t is given by : Application of the Laplace transformation to eqn (A.4) gives which, in conjunction with eqn (A.3) leads to the result : in which /I= (f3n"/b2+4')+. 1 1 f(P)T = s' q(t) dt [cf. eqn (3)]; hence T = -ij = --0 P P2dP) with 4 O3 1 ~[COS~ -11 ~[COS~@ZJ (BE,) -l] = 9 zip-p3Z1sinh @Zl) +(/?:/b2)(Br,)sinh @Z1) and g(p) = 1. Therefore, g(0) = 1 and g'(0) = 0. From eqn (AS) we find that f(0) = 1 [so thatf(O)/g(O) = 1, as required by eqn (5)] and Substituting 21 for I, in eqn (A.6) gives eqn (1 1).For this first example, the method of calculating the time-lag has been presented in some detail. Essential results will be stated, without proof, for the remaining systems. R. ASH, R. M. BARRER AND R. J. B. CRAVEN (b)THERECTANGULARPARALLELEPIPED:0 < x < al,O < y < bl,O < z < c, ; all surfaces at zero concentration (for t > 0) ; C(x, y, z, 0) = Ci,0 < x < al, 0 < y < bl,O < 2 < c,. {sin [czz~pnx]sin [(2m +1)ny][1--(sinh [P(q -z)] +sinh (pz)}bl (21+ Q(2m+ 1)p2 sinh (Pc,) in which with Po = [(21+ 1)2+(2m+ l)2j(bl/al)2]~n. Writing 2a for al, 2b for 6, and 2c for c, in eqn (A.7) then leads to eqn (14). (c) THE INFINITE CYLINDER OF RECTANGULAR CROSS-SECTION: 0 < x < al, 0 < y < bl; all surfaces at zero concentration (for t > 0) ; C(x,y, 0) = Ci,0 < x < a,, 0 < y < b,.co 1-(sinh [a(b,-y)] +sinh (ay)}isinb2' "'3[+ C = (4/n)(Ci/D) lgo (21+ 1)&2 sinh (ab,) in which with a. = (21+1)71. Writing 2a for a, and 2b for b, in eqn (A.8) leads to eqn (17). Section 3.2: Solutions of the relevant subsidiary equations for the three systems considered are, in order : a(C0-CJ cash (4~)(i) E = -a cosh (41)+sinh (qZ)/(qI)+ci}; SORPTION KINETICS AND TIME-LAG THEORY where Z,is a modified Bessel function of the first kind of order v ; (iii) C = -(blr)a(C~-Ci) sinh (qr)l(qb) 3 a siiih (qb)/(qb)+ -2[cosh (qb) -sinh (qb)/(qb)]I .3(qb) Then, proceeding as before, the results of eqn (22), (25) and (28) are obtained.l J. Crank, Mathematics of Diflusion (Oxford Univ. Press, London, 2nd edn, 1975).R. M. Barrer, Trans. Faraday SOC.,1949, 45, 358. P. C. Carman and R. A. W. Haul, Proc. Roy. Soc. A, 1954,222,109.4R.M. Barrer, in Molecular Sieve Zeolites-11, Amer. Chem. Soc., Advances in Chemistry Series No. 102, 1971, 1. R. M. Barrer and D. J. Clarke, J.C.S. Faraday I, 1974,70,535. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids (Oxford Univ. Press, London, 2nd edn., 1959).'J. C. Jaeger and M. Clarke, Phil. Mag., 1947, 38, 504. H. B. Dwight, Tables oflntegrals and Other Mathematical Data (Macmillan,4th edn., 1961).H. S. Carslaw and J. C. Jaeger, Operational Methods in Applied Mathematics (Oxford Univ. Press, London, 2nd edn, 1953). loJ. C. Jaeger, Trans. Faraday Soc., 1946, 42, 615. l1 J. C. Jaeger, Quart. Appl. Math., 1950, 8, 187. (PAPER 7/612)

ISSN:0300-9238    

DOI:10.1039/F29787400040

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Far Infrared Multipole Induced Absorption in Compressed Gaseous C2F6 BY GRAHAMJ. DAVIES P.O. Telecommunications H.Q., Martlesham Heath, Ipswich IP5 7RE AND MYRONEVANS*7 Physical Chemistry Laboratory, South Parks Road, Oxford OX1 342 Received 9th May, 1977 The very weak far infrared collision induced absorption band of hexafluoroethane (C2F6)has been detected using a polarising interferometer and a Rollin [He(l) cooled] In/Sb crystal detector. The absorption, centred at +16cm-1 (296 K) and markedly asymmetric with a high frequency shoulder, is treated in terms of a multipole induced dipole that gives rise to sets of rotational AJ = 2 (quadrupolar) and AJ = 4 (hexadecapolar) lines. These were dynamically broadened using the j-diffusion model of Gordon into bands the relative intensities of which were estimated by a self consistent field molecular orbital calculation of the non-zero elements of the quadrupole and hexadecapole moments.This analysis reproduces the experimental data fairly well, but the calculated cross section is too small. The local charges on each atom of the C2F6molecule estimated with m.0. theory were used to represent its electrostatic field as an alternative to the multipole expansion which tends to diverge at the very short intermolecular distance (R)needed for the R-l0 hexadecapole field to become effective. Recent advances in theoretical and experimental techniques now make it possible to study the very far infrared pressure induced rotational bands of relatively heavy and unsymmetrical molecules;1-4 (Frost has extended the theory to symmetric top molecules).The frequency range over which measurements can be made has been extended to near 2 cm-l by the use of helium cooled Rollin detectors and a polarising interferometer. Non-dipolar molecules absorb in this region (2-200 cm-l) because intermolecular electrostatic fields distort the overall symmetry of a given molecule's electron cloud, producing upon "collision " a small dipole moment that changes in magnitude and direction rapidly with time. Thus compressed gaseous mixtures of rare gas atoms absorb, whereas the components when separated and moderately pressurised do not. A pair of colliding helium atoms, for example, will not possess a resultant electronic cloud of dipolar asymmetry, whereas a helium-neon pair will modulate the electromagnetic field over a broad band of far infrared frequencies commensurate with the most probable frequencies at which interatomic collisions occur. Atomic induced absorption is of a purely translational origin, a mechanism that persists in molecular fluids such as hydrogen and nitrogen as the absorption AJ = 0, where J is the rotational quantum number.A dipole moment set up between a pairof colliding molecules will in addition absorb by rotational means since even without relative translation of molecular centres, the effect on each other of their rotatory t Present address : Edward Davies Chemical Laboratories, Aberystwyth, Dyfed SY23 1NE. 57 A CzF6 HEXADECAPOLE electrostatic fields will not cancel.A practical means of dealing with these inter- molecular absorption mechanisms is to treat them separately. The rotational absorption is dealt with by expanding the field in terms of multipole tensors, which all vanish only in the case of spherical symmetry such as that of atoms. Pseudo-spherical molecules such as SF retain the higher multipoles (those above and including the hexadecapole for Ohsymmetry), and thus display a weak, rotational induced absorption band at moderately high number densities., The first non- vanishing multipole in a homogeneous diatomic such as N, is the quadrupole, which produces a dipole on molecule A which is modulated by the rotational motion of the inducing molecule B.The symmetry of the quadrupole moment is such that it rotates twice as fast as the molecule itself and thus produces quantum absorptions with the selection rule AJ = 2, in contrast with the AJ = 1 rule for the rotation of a permanent dipole. Similarly, the first non-zero multipole moment for Tdsymmetry-the octopole, produces AJ = 3 absorptions, and the hexadecapole AJ = 4. It is important to emphasize that point multipoles are mathematical limits of point charge distributions, and for highly asymmetric molecules may have tenuous links only with physical reality. Furthermore, the range of a hexadecapole field, falling off as R-l0 (where R is the intermolecular vector) is so short that for it to be effective a pair of molecules would need to have their van der Waals shells nearly overlapping.This necessity tends to undermine the whole basis of the multipole series expansion, which diverges in this region. For this reason, the very concept of a point octopole or hexadecapole induced dipole has been questioned since the experimental evidence is sparse, especially for the latter, and up to now necessarily confined to simple molecules of high symmetry. In this paper we explore the far infrared quadrupole and hexadecapole induced band of CzF6, a non-dipolar symmetric top molecule for which a small point quadrupole component Qzz and a large hexadecapole component CDzzzz can be defined mathematically. In the staggered conformation this molecule has Df symmetry and thus no non-zero elements of the octopole.It has one independent element of the quadrupole and two of the hexadecapole, of which iDzzzz is the largest. In the sparsely populated eclipsed conformation, it has some finite elements of all three of the above multipoles. Thus the AJ = 4 set of transitions should be a major contributor to the high frequency part of the observed band. The formal quantum mechanical equation for hexadecapole-induced dipole absorption in symmetric tops is developed here from the general Frost equation,' and the set of lines broadened into a band using a version of thej-diffusion model of collisional broadening. Values of Qzz and CDzzzz are calculated here by an ab initio self consistent field molecular orbital method ; in this way the relative contributions of quadrupole and hexadecapole can be estimated.Due to the continued absence of satisfactory theoretical methods, phenomena such as translational induced absorption (of the rare gas type), and induced absorption due to overlap distortion of electron clouds, have not been treated here explicitly, our main aim being to detect some evidence of a hexadecapole-induced absorption. We furthermore confine our treatment to bimolecular impacts, using measurements at a maximum pressure of t20 bar of gas. EXPERIMENTAL The power absorption spectra were recorded in a few hours using a Michelson/N.P.L. interferometer modified to operate in the polarizing mode as described by Martin and Puplett.6 In principle the interferometer is that shown in fig.3 of the above reference and is reproduced below (fig. 1). Polarizer PI is a circular grid wound with 10pm tungsten wire with 50pm spacing. G. J. DAVIES AND M. EVANS This grid can be spun about its centre point, thus acting as a polarizing chopper blade. P2 is of similar construction to P1but is held in a fixed configuration, i.e., either with the windings perpendicular or horizontal. The beam divider D, is again of similar construction but in this case the windings are at 45" to those of P2. The "roof-top "reflecting mirrors were manufactured to be 90"+1' arc. A collimated beam is plane polarized at PI in the plane normal to the page. It is then divided by the wire grid polarizer D into a beam A, polarized with its E vector at 45" to the paper and beam B polarized at 90" to A. The "roof-top "reflectors act as polarization rotators and beams A and B are recombined at D.The recombined beam finally passes through polarizer P2 (the analyser) which has its axis parallel to or at 90" to that direction. The beam reaching the detector is planepolarized with an amplitude which varies periodically with path-difference in the same way as the normal Michelson interferometer. DETECTOR ,ft. SOURCE FIG.1.-Schematic diagram of polarizing interferometer. For a monochromatic source, that is Ip= I,/2 [1+cos 61 and It = Io/2 [l--0s 61 where 6 = (2n/A)x, (where x is the path difference) and I. is the intensity of the plane polarized beam incident on D. The case Ipis for parallel P1 and P2and It is for crossed P1and Pz.Polarizing grids have reflection and transmission coefficients close to 100 % for their respective planes of polarization,* from frequencies close to zero up to 3d cm-l where 1 /d is the spatial frequency of the wires. Grids have the advantage of eliminating "hooping " as found with Mylar or Melinex beam splitter^.^ The response curve for the interferometer should be flat with frequency falling off at 1/(2d) cm-'. This, however, does not take into allowance the frequency response of the mercury arc lamp which rises with increasing frequency; so although we do not see the full benefits of a flat frequency response at low frequency we do gain over other far infrared modulation techniques (amplitude and phase modulation) with an increased response at wave numbers <10 cm-I plus an increased spectral range with a single beam divider due to the elimination of "hooping ".The interferometer was not evacuable as we were interested in the range 2-35cm-I where water vapour absorption can be considered to be negligible, particularly so below 30 cm-l which was our main region of interest. The detector was a liquid helium cooled InSb Rollin detector.1° The resolution was 2cm-' in all cases. The relative intensities of the bands are not what is expected from a bimolecular collision mechanism, possibly because of three-molecule effects, another hidden proper mode, or a slight amount of dipolar impurity, causing an extra absorption proportional to the molecular number density N, A C2F6 HEXADECAPOLE instead of N2. However, in this paper we have used an analysis aimed at explaining the bandshape, which will not be affected critically by a constant background absorption. The relative intensities may also be affected by the use of a slightly convergent beam through the metre tube instead of an accurate collimator. Data above 35 cm-l are not available because of beam divider characteristics, The sample was contained within a specially constructed metre cell with TPX (poly-4-methyl pent-1-ene) windows, placed between P2and the detector.Matheson research grade CzFs was used without further purification. The cell was purged beforehand with a few atmospheres of rare gas until no change in absorption could be registered upon evacuation.RESULTS AND INTERPRETATION The absorption of compressed gaseous C2F6 is shown in fig. 2 at three number densities. It is an asymmetric broad band peaking at -16 cm-l with a marked high frequency shoulder, so that there is a continuing significant contribution at near 40 cm-l, the limit of our interferometer. The data are for 4.93, 9.59 and 17.8 bar N = 1.29, 2.69 and 5.63 molecules ~m-~ respectively), and the reproducibility between successive runs at the same pressure is satisfactory. The integrated intensities of all three bands are much greater than the 0.002+_0.0008cm-2 of a much narrower forbidden a,, torsional vibration mode recently observed at one atmosphere pressure of CzF6 at 67.5 cm-, : a band from which a barrier to internal rotation of 1333-1367 cm-I (-16 kJ mol-l) was estimated.It is unlikely, therefore, that more than a small fraction of the high frequency wing can be assigned to this intramolecular al,, mode, and furthermore, with such an internal barrier, the C2F6 molecules would populate overwhelmingly the staggered conformation at any one instant at ambient temperature of 296 K. ./cm-FIG.2.-Absorption of compressed gaseous CzF6 at 296 K. 0-17.8 bar; 0-9.59 bar ; 0-4.93 bar. (1) -- - AJ = 2 curve broadened by j-diffusion. (2) ---AJ = 4 curve broadened by j-diffusion. (3) -Curve (l)+curve (2), normalized to the data at 17.8 bar. The stick spectrum is of sume of the AJ = 2 absorptions(K = 0). Thus it is reasonable to treat the experimental data in the 2-4Ocm-' range in terms of a mechanism of quadrupole and hexadecapole induced dipole absorption, generated by bimolecular collisions of C2F6 molecules in DJdsymmetry.The equation linking the observed absorption cross section with the theoretical is thus : a(?)di = (AQ+Ao)N2 band G. J. DAVIES AND M. EVANS 61 where AQand Amare sums over all J of coefficients defined by Fr0st.l AQis written out explicitly el~ewhere,~ and A@is a summation over J of eqn (1A) in the Appendix, derived from the general Frost equations. These are not used in their full, exceedingly complex form in this paper since we are interested specifically in AJ = 4 and AJ = 2 transitions only. A full treatment would include the great number of allowed AJ = 1, 2, 3 and 4 for hexadecapole-induced absorption in symmetric tops, and the extra AJ = 1 transition for the equivalent quadrupole-induced absorption. The approximation used here seems to be satisfactory insofar as the observed broad band is fitted well.Throughout it is assumed that AK = 0. Since AQ and A@are sums of quantum line absorptions, and the data, in common with nearly all induced absorptions l2 at these pressures and frequencies, are broad bands, a mechanism of collisional line broadening is needed if a meaningful comparison is to be made between theory and experiment. It is important to distinguish this broadening from trans- lationally induced absorption (rare gas type) : the former produces no extra absorption of its own but merely gives each AJ = 2 or AJ = 4 line a finite half width.Birnbaum et al. found the latter to be confined in extent to below 10 cm-l in a molecule as light as COz. With CzF6,its contribution would be found at even lower frequencies, out of our range. The broadening is treated rather simply here in terms of Gordon's naive but very useful j-diffusion model l3 so that the CzF, molecules are assumed to undergo periods of free rototranslation, interrupted at a mean interval of time z by elastic impacts that randomise into a Boltzmann distribution each molecular angular momentum vector, and randomise completely the molecular orientation. The quantum equations such as (1A) of Appendix 1 may be made to follow the Gordon hypothesis as shown elsewhere,14 the relevant transform being given for the hexa- decapole component in eqn (3A). The final expression for a(co), the theoretical absorption coefficient (o= 2nVc) as a function of angular velocity may be expressed as a continuum : a(o) = (E~-&,)a2 C(w)/n(o)c (2) where C(o) = C,(O) + Ca(co), is a sum of two broad bands.In eqn (2), (go -8,) is the total dispersion, n(o)the frequency dependent refractive index (effectively unity), and c the velocity of light. C@(o)is related to C(t)of eqn (3A) by : r(i-Z-lr)-7-lA2 (3)C,(co) = (1-2-1q2 +C2h2 with : with C(t)= lmf(C2) cos Rt dR. 0 CQ(o) is similarly defined e1~ewhere.l~ Thus we have C,(W) as a continuum representation of the set of AJ = 2 lines, and C,(w) as that of the AJ = 4 set.The mean time between collisions (2) may be estimated roughly from kinetic theory as 5.5, 11.5, and 24.0 ps for our three number densities. A CzF6 HEXADECAPOLE In order to evaluate eqn (2), an estimate of Qzz and Qzzzz was made with a self consistent field molecular orbital calculation of the charge on each atom of the C2F6 molecule using a standard Harwell algorithm,15 the results of which are tabulated below. TABLE1.-S.C.F.M.O. CALCULATION OF C2Fs ATOMIC CHARGES (&a symmetry) coordinates*/A 1010 charge (e)atom X Y z le.s.u. C 0 0 f0.78 2.642 F 0 f1.25 k1.22 -0.879 F 1.09 k0.63 k1.22 -0.879 F -1.09 f0.63 21.22 -0.879 * (0, 0,O) = mid point of C-C bond.From table 1 we have the dipole and octopole components all zero, and : Qzz = -0.53 x e.s.u. @zzzz = -2@xxzz = 8@,,,, = + ei[35z4-5($ +yz +22)(6z:)+3($ +y’ +z;)~]i = -18.53 x e.s.u. The radial averages of the intermolecular potential UAA(R) were calculated by the method of Buckingham and Pople.16 DISCUSSION In eqn (1) Q is the tensor magnitude related approximately to the scalar azzZz by l7 @ = (7/12)* @zzzz, so that the calculation outlined above yields : ama,(Q) : amax(@) = 1 : 0.3. Thus the type of absorption band predicted is similar to that observed, i.e., a low frequency peak with a substantial shoulder. This is true despite the fact that the S.C.F.M.O. calculation shows the molecule to have a large hexadecapole moment and a very small quadrupole (commensurate with that of nitrogen or hydr~gen).~ The set of AJ = 2 lines 1-3* 18-20 peaks at 19.4 cm-l (see fig. 2), but it is interesting to observe that broadening with j-diffusion shifts amax(Q)[or amax(@)to lower frequencies], a result first observed by Frenkel and Wegdam 21 for the AJ = 1 transitions in linear molecules.Furthermore, the peak frequency approaches that of the observed (16 cm-l) as z, the mean time between elastic collisions, approaches zero, the rotational diffusion limit. However, in this limit the return to transparency is far too slow, a characteristic 22 of mechanisms which imply singularities in the intermolecular torque at every impact. The experimental band is fitted best (fig. 2) with a z of 0.3 ps, an order of magnitude lower than the approximation from kinetic theory.The overall calculated bandshape is close enough to the observed to conclude that the origin of the band is predominantly quadrupole induction, with a high frequency contribution corresponding to AJ = 4 transitions, attributed here to a point hexadecapole source. However, there is a basic interpretative difficulty in that the data consists of an asymmetric but simple broad band which may arise from more than the two sources treated here. Our conclusions above may be modified, but probably not greatly so, by a more detailed study involving the following factors. G. J. DAVIES AND M. EVANS 63 (i) Overlap absorption arising from the interpenetration of van der Waals shells (repulsive regime).lg* 2o Ho et aZ.2 found this to contribute a 1 % part of the total integrated intensity in compressed gaseous C02,which has zero dipole and octopole, as in C2F6 (D3d).The theory of this effect is at a primitive stage,23 and parameterised for H2 only. (ii) Translational absorption (AJ = 0, non resonant), which is a low frequency effect of very low absorption cross section, especially for the massive, slow-moving C2Fs. (iii) Cross relaxation between sets of overlapping J + J+n lines (n = 2,4, . ...). Very little is known about this,3 especially when rotational constants are very small (as in C2F6) where the effect is more significant and difficult to measure. (iv) Angular dependence of the intermolecular potential (Le., non-Lennard-Jones), when the radial averaging used here would be an approximation. (v) Contributions from the octopole components of the C2F6 molecules not in D3d(staggered) conformation.These are probably small since the internal rotation barrier is substantial at ambient temperature. Although our (Q, a)point multipole expansion results are in fair agreement with the observed broad-band absorption, the theoretical absolute integrated intensity is much too small as in many other studies of this l2 When the mean distance travelled between collisions becomes small, of the order of the molecular van der Waals field, it may be more realistic to use a local charge basis 24 for the field in the neighbourhood of a molecule, and this may go a long way towards explaining this difference in absorption cross section.M. W. E. thanks the S.R.C. and the Ramsay Memorial Trust for Fellowships. We thank the director of research at the P.O. for permission to publish. APPENDIX 1 The equation for hexadecapole induced AJ = 4 absorption in symmetric top molecules may be written as : I.7+5+4 --4nR-l' exp [-U,,(R)/kT] dR x4niE:2 [{I -exp [-hcij4(J)/kT])exp (-EJkhc/kT)S4[S(I, K)] x K=-J 1 {C(J,4,J' ; K, 0, K')2[40(W+1)a; + (280/9)(2J+1)2 a2C(J, 2, J' ; K, 0, K')2] + C(J, 4, J'; K, 3, K')2 [20(2J+ l)ac+(40/3)(2J+ 1)2 d2C(J, 2, J; K, 0, K)2]) (1A) where C(J, 4, J'; K, 0, K')2 35(J + K + 4)(J + K + 3)(J+ K +2)(J-K + 4)(J-K + 3) (J- K +2)(J- K + 1)(J+ K + 1) ; 8(2J + l)(J + 1)(2J + 3)(J + 2)(2J + 5)(J + 3)(2J + 7)(J + 4) I C(J, 2, J'; K, 0, K')2 -3(J- K + 2)(J-K + 1)(J + K +2)(J + K + 1)-7(2J + 1)(2J+ 2)(2J+ 3)(J+ 2) A CzF6 HEXADECAPOLE C(J, 4, J'; K, 3, K')2 8(J +K +7)(J+K +6)(J+ K +5)(J+K +4)(J +K +3) (J +K +2)(J +K +1)(J-K +1) * (2J +8)(2J +7)(2J +6)( 2 J +5)(2J +4)(2 J +3)(2J +2)(2J +1) 1 In eqn (lA), N is the molecular number density (molecules ~rn-~),@ the relevant scalar 25 of the hexadecapole moment, 2 the rotational partition functionY3 UAA(R) the Lennard-Jones type intermolecular potential and J and K the usual rotational quantum numbers.The energy EjK is defined by : EjK = BJ(J+ l)+(A-B)K2 ; the wavenumber V,(corresponding to AJ = 4), by : v4 = 4B(2J+ 5) ; and the nuclear spin weighting factor S(I,K) is defined as previ~usly.~ A and B are the usual rotational constants, given for C2F6by : A = 0.0935 cm-l ; B = 0.0610 cm-l.The Clebsch-Gordan coefficients C(J, 4, J' ; K, 0, K') and C(J,4, J' ; K,3, K')were calculated using the general Wigner expansion 3* 2o available in standard texts. Finally, a, and S are the molecular polarisability and its anisotropy. APPENDIX 2 An orientational type correlation-function [C(t)]related to the Fourier transform of eqn (1A)may be calculated using techniques developed previously 3*l4 for other symmetric tops, based on the transform [eqn (2A)]: (7 = V4) J = (5-2OB)/8B, so that C(t)OcImK=-J0 f: (S(I,K)exp[-(A-B)K2hc/kT]xexp[-:?(8yB ---:)(8"B~ ---:)Ix 2[40( -&-4); +?(-&-4) S2C(J, 2, J' ;K,0,K')2 +1 C(J,4, J'; K,3, 20 --4 a;+---4 )S2C(J,2,J'; K,0,K')2[ (4".) Y(lB cos (2nVct)d5; (3A) where C(J,2, J'; K, 0,K')2 =(3[j-4B(1-2K)][ij -4B( 3 -2K)][ij -4B(1+2K)][--8(?-4B)(ij-16B)(ij- 12B)(ij- 8B) 4B(3+2K)1) C(J, 4, J'; K, 0,K')2 35[V +4B(2K +3)][ij +4B(2K +l)][ij+4B(2K-l>][J+4WK -311 X [V-4B(2K-3)][V-4B(2K-1)][?-4B(2K+ l)][ij-4B(2K+3)] 64(j- 16B)(V- 12B)(V-8B)(V)(V+4B)(V+8B)(V+ 12B)(v'-4B) G. J. DAVIES AND M. EVANS C(J, 4, J’; I<, 3, K‘)’ [~+4B(2K+9)1CV+4B(2K+7)][ij+4B(2K+5)][j+4B(2K+3)]x [ij +4B(2K +l)][ij +4B(2K -l)][ij +4B(2K-3)] [?-4B(2K +3)] ).=( 64(V +12 B)( V +8B)(ij+4B)(S)(ij-4B)(ij -8B)(3-12B)(V -16B) For each time instant t, the integration in eqn (3A)may be carried out using Simpson’s rule on a fast machine such as the CDC 7600.APPENDIX 3 COMPARISON WITH POINT CHARGE FIELD For a molecule of DSdsymmetry we show in this section that the multipole expansion of the potential (and thus the flux density) due to an array of point charges (the molecule) at a point P relative to a reference 0 in the molecule is a poor one, and further quantum theories should take account of this fact. Buckingham 26 has reviewed the relevant theory. Consider a distribution of point charges el at points (xi,yi,2,) represented by the vectors ri from an origin 0 in the molecule. We wish to find the potential 4 at P(X, Y,Z), or R,from 0,where R > ri for all i. We have, charge by charge: q5 = c$i= zeiRil (41 a = C ei[(X-xJ2+(Y-y,)’ +(z-z~)~]-+ with Rias the distance between el and P.In the Maclaurin expansion of 4 Bucking-ham has evaluated the terms up to the octopole, but here we need to include the hexadecapole term : where the a, 8, y, 6 subscripts are shorthand tensor notations explained by Bucking- ham. For example : Rf = (Ra-ria)(Ra-ria). (B) Here we use eqn (B) to complete the multipole expansion inclusive of the hexa- decapolar case for the general charge distribution of any symmetry : +R;,R,aa,) +~ay6~~-aab6~y)] . (C) Here q, pay @a@, !2a/jy, and Qapya are the symmetrised multipoles and the 6’s are Kronecker symbols. For symmetry Kielich has listed the non-vanishing elements of cap and Qapy,. If we now compare -aq5/aR from eqn (A)and (C) for a point 20 A away from the molecule it is found that the first value obtained for the radial component of the flux density is two orders of magnitude less than that from eqn (C) using point charge and elements of @as and @apya from our S.C.F.molecular orbital calculation. This is a discrepancy far more serious than hitherto found by 11-3 A C~FGHEXADECAPOLE Davies et aZ.24 For what it is worth, eqn (C) reduces, for exactly symmetric distri- butions to : -+5 COS~8-30~0~~a) e+3)+ . . . 8R where the multipoles are now defined by the fact that no components thereof are mutually independent. B. S. Frost, J.C.S. Furuduy 11, 1973, 69, 1142. G. Birnbaum, W. Ho and A. Rosenberg, J. Chem. Phys., 1971, 55, 1028 ; J. E. Harries, J.Phys. B, 1970,3,704. G. J. Davies and M. W. Evans, J.C.S. Furuduy II, 1975,71, 1275 ; 1976,72,40 ; A. I. Baise, J. Chem. Phys., 1974,60,2936. P. E. Clegg and J. S.Huizinga, I.E.R.E. Conf. Infrared Techniques (Reading, 1971). S. Kielich, in Dielectric and Related Molecular Processes (Chemical Society London, 1972),VOI. 1, pp. 192-387. D. H. Martin and E. Puplett, Infrured Phys., 1969, 10, 105, ’D. G. Vickers, E. I. Robson and J. E. Beckman, Appl. Optics, 1971,10,682. A. E. Costley and J. M. Ward, unpublished work at N.P.L. P. L. Richards, J. Opt. SOC.Amer., 1964,54, 1478. lo P. E. Clegg and J. S. Huizinga, I.E.R.E. Conf. Infrared Techniques (Reading, 1971). l1 D. F. Eggers, R. C. Lord and C. W. Wickstrom, J. MoZ. Spectr., 1976,59, 63. l2 See for example, J.P. Colpa and J. A. A. Ketelaar, Mol. Phys., 1958, 1, 14 ; H. M. Foley, Comments At. Mol. Phys., 1970, 1, 189. l3 R.G. Gordon, J. Chem.-Phys., 1965, 43, 1307 ; R. E. D. McClung, J. Chem. Phys., 1972, 57. 5478. l4 M: Evans, Spectrochim. Acta A, 1976,32, 1253. l5 C. S. Pangali, personal communication. l6 A. D. Buckingham and J. A. Pople, Trans. Furaduy SOC., 1955, 51, 1173. l7 C. G. Gray, J. Phys. B, 1971,4, 1661. l8 D. R. Bosomworth and H. P. Gush, Canad.J.Phys., 1965,43,751. l9 J. H. Van Kranendonk, Cunad.J.Phys., 1961,39, 189. 2o M. Evans, Mol. Phys., 1975,29,1345. 21 D. Frenkel and G. H. Wegdam, J. Chem. Phys., 1974,61,4671. 22 M. Evans, J.C.S. Furuday IZ, 1975, 71, 2051; 1976, 72,727. 23 I. Ozier and K. Fox, J. Chem. Phys., 1970, 52, 1416. 24 G. J. Davies, J. Chamberlain and M. Davies, J.C.S. Furuduy II, 1973, 69, 1223. 25 G. Birnbaum and E. R. Cohen, Cunud.J. Phys., 1976,54, 593. 26 A. D. Buckingham, Quart. Rev., 1959, 13, 183. (PAPER 7 /792)

ISSN:0300-9238    

DOI:10.1039/F29787400057

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Electron Spin Resonance Studies of Ion Association Part 7.-Effects of Slow Intramolecular Cation Exchange on the ENDOR of Potassium-2,5-di-t-butyl p-benzoquinone BYNEILM. ATHERTON" A. KENNEDYAND PAMELA Department of Chemistry, The University, Sheffield S3 7HF Received 9th May, 1977 Slow intramolecular cation exchange can manifest itself in ENDOR by reducing the relative intensity of the response from hyperfine lines whose breadths are sensitive to the exchange process, even when the rate is too slow to contribute sensibly to the e.s.r. line breadths. The effect has been studied in potassium-2,5-di-t-butylp-benzoquinone ion-pairs, and is satisfactorily understood using accepted theories of spin relaxation and solution ENDOR. Ion-pairs containing a radical-anion can participate in a variety of exchange-processes whose rates and mechanisms can be conveniently studied by e.s.r.line breadth analysis. Examples of such processes include electron (or atom) transfer, cation exchange, and intramolecular cation exchange.' The breadth of an e.s.r. transition is sensitive to such an exchange if, as a result of it, the Larmor frequency of' the unpaired electron is changed. The range of sensitivity begins when the rate of the exchange approaches the linewidth (slow exchange limit) and continues until the rate of exchange exceeds the change in Larmor frequency which it causes (fast exchange limit). The work reported here comes from enquiring whether the range of exchange rates which can be studied can be extended by using electron nuclear double resonance (ENDOR) spectroscopy, i.e., is there a range of rates where an exchange process manifests itself in ENDOR but not e.s.r.? In this initial study we consider intramolecular cation exchange at the slow limit. We anticipate that the ENDOR spectrum might respond to an exchange process in two ways : through linewidth effects, arising in the same sort of way that they do in the e.s.r. spectrum, and through variations in the strengths of the ENDOR signals for monitoring different hyperfine lines, arising from variations in the saturation behaviour between different lines. Exchange contributions to ENDOR linewidths have been discerned and analysed, but only in a few cases. Thus rates of conforma- tional interconversions in the ubiquinone,2 sila~yclopentadiene,~ and 0-and p-terphenyl radical-anions have been measured.These studies validate the theory of exchange contributions to ENDOR linewidths and illustrate the potential of the method. More recently, von Borczyskowski and Mobius have elucidated effects due to hindered internal rotation and ion-pair formation for the phenyl-naphthalene radical anions. There has been little previous discussion of exchange effects on the strengths of the ENDOR signals from different hyperfine lines and it is this particular point which is our concern here. We emphasise from the outset that we are concerned with the rate range where the e.s.r. spectrum is not obviously affected by the exchange process.For this initial study we chose ion-pairs of the 2,5-di-t-butyl- 1,4-benzoquinone (DTBQ) anion. This has been thoroughly studied by e.s.r.,6* and the Arrhenius 67 ION ASSOCIATION parameters for intramolecular cation exchange, which depend markedly 011 both solvent and counter ion, have been measured for several solvents and counter ions. The system is attractive in that the e.s.r. spectrum is simple and, in the slow exchange region, the lines are well-separated so that each hyperfine state can be studied independently by ENDQR. Against this the lines are inhomogeneously broadened by the t-butyl proton coupling so that analysis of the relaxation and ENDOR behaviour in terms of an eight-level system, which is what we present here, can only be qualitative.EXPERIMENTAL The ENDOR spectra of small radical-anions in ether solvents are normally only readily detected at low temperatures. We wished to examine cases where the exchange contributions to the e.s.r. line widths would be very small, or imperceptible; bearing in mind that we would be constrained to low temperatures we chose to look for ENDQR effects at the slow exchange limit. Warhurst and Wilde's data indicated potassium-dimethoxyethane(DME) and pot assium-methyl tetrahydro furan (MTHF) as suit able metal-solven t combinations and these were studied initially. For reasons to emerge 1 :1 (v/v) mixtures of these two solvents were also used. The kinetic data also indicate sodium-tetrahydrofuran as a possible combination but we have not been able to achieve uniformly clean reductions in this system.All samples were prepared using conventional techniques. ENDQR spectra were taken using a Varian E-700 instrument operating in conjunction with a V-4500 series e.p.r. spectrometer. Sample temperatures were controlled with a V-4540 unit. The free proton frequency was at about 14.6 MHz and the high frequency ENDOR transitions for the ring protons at =17.1 and 18.6 MHz. RESULTS AND DISCUSSION The e.s.r. spectra consisted of four well-resolved lines; at the temperatures at which ENDOR spectra were studied no linewidth effects ascribable to slow intra- molecular cation exchange were evident. In order to facilitate reference to the various transitions the energy levels are summarised in fig.1. The proton hyperfine coupling constants are assumed to be negative so the 1-5 transition, corresponding to the (MM) nuclear spin state, occurs at high field and high frequency ENDOR transitions are in the M, = ++manifold. uio4 Wn @ a vie Q YJ, @ we We We we 0 Wn 0 wc 0 wn 0 N. M. ATHERTON AND P. A. KENNEDY In MTHF, ENDOR signals from the ring protons were examined at 193 K. The ENDOR intensities were independent of which hyperfine line was monitored. As the temperature was increased the intensity of the ENDOR signals decreased, the decrease being the same for all e.s.r. transitions. This behaviour is quite normal for radical-anions :8 the strength of the a-proton ENDOR response falls off as the correlation time for rotational diffusion decreases.No dramatic effects ascribable to chemical exchange were thus discerned in this system. In DME at 183 K ENDOR signals from the ring protons were observed, but only for saturation of the 1-5 and 4-8 transitions, the two outer e.s.r. lines. No ENDOR was detected when monitoring either of the central lines, those which broaden at higher temperatures as the rate of intramolecular cation exchange increases. The intensity of the ENDOR signals which could be detected decreased as the temperature was raised, again presumably reflecting the shortening of the rotational correlation time. According to the results of Warhurst and Wilde7 the rate of intramolecular potassium exchange at 193 K in MTHF should be 0.73 x lo3s-l, while at 183 K in DME it should be 2.95 x lo3 s-l.In MTHF at 203 K, where ENDQR signals could no longer be detected, the exchange rate should still only be 1.86 x lo3 s-l, slower than in DME at 183 K where ENDQR could be detected only for saturation of the 1-5 and 4-8 transitions. It is clear that in DME at 183 K the exchange is catastrophically affecting the ENDOR response for the 2-6 and 3-7 transitions, while being much too slow to show as a linewidth effect in the e.s.r. However, it is an " on-off " type of effect and we sought a system showing a less drastic differentiation between the e.s.r. lines. We did the obvious experiment and examined a mixture of DME and MTHF, 1 : B v/v. At 183 K the ENDOR intensities for saturation of the 2-6 and 3-7 transitions were clearly less than those for the 1-5 and 3-8 transitions.Representative spectra are shown in fig. 2. It is difficult to make precise comparisons of the signal strengths because, of course, one has to move along the e.s.r. spectrum between the two recordings, but there is no doubt the difference in intensity is real. We have examined it carefully many times taking care to hold all instrumental sensitivity factors constant and it is quite reproducible. At 193 K no ENDOR could be detected for saturation of the 2-6 or 3-7 transitions while for the 1-5 and 4-8 transitions the signals had about half the peak-height/noise ratio of those in fig. 2 and were broader. Fig. 2 also shows a second derivative e.s.r. spectrum taken at 183 K and serves to emphasise that the exchange is too slow to be readily measured from the e.s.r.line- widths. To interpret these observations we consider that the spin-lattice relaxation in the system is determined by the three types of lattice-induced transition probabilities shown in fig. 1. The probability for pure electron spin transition, We,is taken to be independent of the nuclear spin state, consistent with the observation that there is no perceptible MI-dependent e.s.r. linewidth contribution. All probabilities W, for pure nuclear transitions are taken to be the same. The prime mechanism for these transitions is expected to be the electron-nuclear dipolar (END) interaction though the equality for the two nuclei can hardly be exact as they have different isotropic couplings and so, presumably, different dipolar couplings.However, since our aim is to understand the observations physically, this is a useful clarifying assumption and it is not critical. We neglect electron-nuclear cross relaxation. Again this affords considerable simplification and may not in fact be unrealistic, since it implies the limit of a long correlation time for rotational diffusion. The effect of the intra- molecular cation exchange is to introduce the non-zero transition probabilities W, ION ASSOCIATION E -17 18 I5 17 I8 I9MHt ZOOpT bi1 FIG.2.-(a) and (b) the high frequency ring proton ENDOR transitions of K-DTBQ in DME-MTHF at 183 K. The spectra were obtained from monitoring the 1-5 and 2-6 e.s.r.transitions for (a) and (b) respectively and equivalent spectra were obtained from the 4-8 and 3-7 transitions. (c)Second derivative e.s.r. spectrum at 183 K showing points monitored in ENDOR. For the high field lines the corresponding points on the high field sides of the lines were monitored. connecting levels 2 and 3 and levels 6 and 7. The contribution of the exchange to the W-matrix can be seen intuitively, but the result can easily be derived formally using Alexander’s method. lo We view the ENDOR effect as being due to the change in the saturation parameter for the e.s.r. transition which occurs when a nuclear resonance is excited by the r.f.’p 11-l2 Thus we can ascribe different strengths of response under fixed experi- mental conditions for different e.s.r.transitions as reflecting different saturation parameters for those transitions. The electrical resistance network analogy makes the effect of the exchange on the e.s.r. saturation parameters very clear. Fig. 3 depicts the equivalent network in a three-dimensional representation in order to emphasise the symmetry. It is clear that in the absence of exchange, R, = co,the resistances between the pairs of points 1-5, 2-6, 3-7 and 4-8 are all the same, i.e., all the e.s.r. transitions have identical saturation parameters. However when R, becomes finite the resistances across 2-6 and 3-7 become different from those across 1-5 and 4-8. In fact, the latter two resistances are independent of R,. To see that this is so consider measuring them by applying a potential across, say, 1-5 : since all R, are assumed, the same points 6 and 7 (and 2 and 3) will be at the same potential and so no current will flow in the R,.N. M. ATHERTON AND P. A. KENNEDY FIG.3.-The equivalent resistance network for the eight-level system of fig. 1 represented in three dimensions. Ri W;'. The foregoing argument is instructive and provides a basic physical understanding of the observed effects. However, to become more quantitative it must be recognised that the ENDOR response depends on more saturation parameters than just those for the e.s.r. transitions. Freed's analysis 9* l2 of ENDOR shows that the e.s.r. signal for the transition between states i and j when there is also present a r.f.field close to resonance with the nuclear resonance transition between states j and k is given by This is just a saturated Bloch lineshape : qo,gives the thermal equilibrium magneti- sation, deis the interaction of the microwave field with the electron magnetic moment, TLlis the linewidth, A, is the departure from exact resonance of the microwave frequency, Qj, fi is the saturation parameter for the e.s.r. transition and 5, is Here dn, Tn and An are the analogues of de,Teand Ae, for the r.f. field and the nuclear resonance transition, Qjk, jk is the saturation parameter for the nuclear resonance transition, and afj,jk is the cross-saturation parameter, which enters because states i and k become coupled when there are both microwave and r.f.fields present. Eqn(1) and (2) show very clearly that the ENDOR effect arises from the modification of the saturation parameter of the e.s.r. transition by the r.f. field. The equations are valid provided that coherence effects may be neglected, which is reasonable if the correlation time is relatively long.g* l2 In the absence of an r.f. field (or when it is far from resonance), 5, = 0 and the ordinary, saturated, e.s.r. signal is just Z:j(O). In the ENDOR experiment one observes the difference {Z;j([e)-Z:u(O)). If one monitors the peak of the e.s.r. absorption then As = 0 and the ENDOR signal is where ION ASSOCIATION Freed does not develop eqn (3) explicitly from eqn (1) and (2) but considers the case of strong saturation of the e.s.r. signal, Qij,ij Ted:9 1, to obtain 99 l2 where Eqn (3) and (5) show that the ENDOR signal should have a saturated Bloch lineshape with a relaxation-dependent intensity factor.The relative peak intensities (AD = 0) of the ENDOR signal for saturation of the 1-5 and 2-6 transitions predicted by eqn (5) and (3) have been evaluated. Initially eqn (5) was used and, as the observed ENDOR signals do not have dramatically different widths, we assumed d:Tflajk,jk&j,jk < 1 so that the peak intensity ratio is zi;6(o) = { a26,jk:5,15}2 (7)z’;5(o) a26,26 26,jk * The required saturation parameters were obtained from measurements on a resistance network analogue (fig. 3). This was constructed using 5 % oxide resistors with R, = 9.1 kV A-l and Re was varied to achieve various values of the conventional 99 l2 parameter b = Wc/Wn(=Rn/Re).For each value of b a series of values of R, was put in to correspond to various rates of chemical exchange. We parameterise the results with c, defined as c = W,..Wn = R,/R,. (8) It is clear from fig. 3 that the differential effect of the chemical exchange on the ENDOR signals will not be significant if c 4 1. The required values of a,,,,and ajk,jk were measured directly, and the i&j,jk using the relation &jJk = afj,ij+Qjk,jk-Qik.ik (9) which we have presented elsewhere. l4 With the simplified symmetrical model we have chosen all Qjk,jk are equal. Resistances were measured using a Sinclair Multimeter, Model DM2.Fig. 4 shows some results : the right hand side of eqn (7), which can be called the relative intensity of the ENDOR from the 2-6 transition, is plotted against c. Curves a and p show the results for b = 0.615 and b = 1 respectively. The reduction in relative intensity of the ENDOR for the 2-6 transition as the chemical exchange rate increases is predicted, but the effect is not very strong. The effect is slightly enhanced if strong saturation of the ENDOR is assumed, (d;TflQjk,jk Afjejk)9 1. The intensity ratio (7) is now multiplied by a factor (A)15,jk/l\$6,jk). Results for b = 1 are shown in fig. 4, curve y. In practice one usually adjusts the microwave power to maximise the e.s.r. signal Ted: = 1, rather than the when doing an ENDOR experiment.This implies t221j,1j strong inequality assumed in the derivation of eqn (5). Adopting this equality for the 1-5 e.s.r. transition, assuming Teis the same for both e.s.r. lines, and taking the limit of no saturation of the ENDOR, the ratio of the ENDOR peak intensities is This quantity is plotted against c, for b = 1, in curve 6 of fig. 4. The intensity ratio has also been calculated assuming strong saturation of the ENDOR signal for these N. M. ATHERTON AND P. A. KENNEDY conditions. The curve for b = 1 lies just slightly below curve 6 of fig. 4 and, as it provides no new enlightenment, it is not shown. Our analysis seems to account qualitatively for the observations, but we must enquire if the required relaxation rates are physically reasonable.Leniart, Coniior and Freed have made detailed studies of relaxation and ENDOR in semiquinones, and it certainly seems that the value of b could approach unity in our mixed solvent at low temperature, while if we take an optimistic view of their results the value of We might become as low as 4 x lo4 s-l. This then would have to be the value of W,to I 1.0 0.8 0.6 P v)3 -ct :m 0.4 L 0.2 2 4 6 8 10 C FIG.4.-Relative peak intensity of the ENDOR from the 2-6 transition as a function of relative exchange rate, calculated using saturation parameters measured from resistance networks. (a), eqn (5) (strong saturation of esr.), b = 0.615, no saturation of ENDOR; (p), eqn (5), b = 1, no saturation of ENDOR; (y), eqn (9,b = 1, strong saturation of ENDOR; (a), eqn (3) (maximum e.s.r.signal), b = 1, no saturation of ENDOR. get an ENDOR intensity ratio of 0.73, which is a little higher than we observe at 183 K. Even if we use the limits of error quoted by Warhurst and Wilde 'to our full advantage this rate seems too high by a factor of -8. Superficially this seems disappointing. However, the unresolved t-butyl proton splittings were ignored by Warhurst and Wilde7 in their analysis of the exchange rates from the e.s.r. spectra (as indeed they are by us in our discussion of the ENDOR spectra) and this seems a likely source of unassessed error in the rate data. The kinetic analysis also takes no account of any temperature dependence of the exchange independent part of the true linewidth.If we also remember that we are making intuitive interpolations to discuss the behaviour in the mixed solvents, we are left with the feeling that the magnitude of the effect we observe in the ENDOR can be accounted for without any additional hypotheses. As a final exercise we have calculated ENDOR lineshapes for our model 8-level system. The spectra were calculated by solving Freed's matrix equationsYg with all off-diagonal elements of the coherence matrix set equal to zero, i.e., coherence effects were neglected. The two nuclei were assumed to have couplings of 4 and 6 MHz respectively, the microwave and r.f. field strengths were 2 pT and 1 mT respectively, and we took We= 4 x lo4 s-I b = 1 , T; = 50 We.The high frequency ENDOR ION ASSOCIATION signals calculated for W' = lo4 s-1 and 5 x lo4 s-1 are shown in fig. 5. For the faster exchange rate the intensity ratio for the two e.s.r. transitions is -0.67 and there is a clear difference in width of the ENDOR signals, showing that there is some saturation of them. FIG. 5.4omputed high frequency ENDOR lineshapes for a 2-proton radical with (11 = 4 MHz, a2 = 6 MHz, at two values of the exchange rate. See text for values of other parameters. The markers on the axis indicate a frequency interval of 1 MHz. In conclusion, there can be an effect of intramolecular cation exchange on ENDOR intensities under conditions where it is not manifest in e.s.r. It seems that our present understanding of spin relaxation and ENDOR in solution can account for the magnitude of the observed effect.Analysis of the effect to obtain kinetic data is sufficiently non-trivial that it does not seem likely to become commonly used. We thank the S.R.C. for equipment grants and the award of a studentship to P. A. K. For a survey see, e.g., J. H. Sharp and M. C. R. Symons, in Ions and Ion Pairs in Organic Reactions, ed. M. Szwarc (Wiley-Interscience, New York, 1972), vol. 1, chap. 5, p. 177. h.1. R. Das, J. D. Connor, D. S. Leniart and J. H. Freed, J. Amer. Chem. Soc., 1970,92,2258. C. von Borczyskowski, K. Mobius and M. Plato, J. Magnetic Resonance, 1975, 17, 202. M. Piato, R. Biehl, K. Mobius and K. P. Dinse, 2.Naturforsch., 1976, 31a, 169. C.von Borczyskowski and K. Mobius, Chem. Phys., 1976,12,281. J. C. Chippendale and E. Warhurst, Trans. Faraday SOC.,1968,64,2332. E. Warhurst and A. M. Wilde, Trans. Faraday SOC.,1971, 67, 605. see, e.g., R. D. Allendoerfer, in Znt. Rev. Sci. : Phys. Chem. Ser. Two, ed. C. A. McDowell (Butterworth, London, 1975), 4,29. J. H. Freed, J. Chem. Phys., 1965,43, 2312. lo S. Alexander, J. Chem. Phys., 1962, 37, 967. R. D. Allendoerfer and A. H. Maki, J. Magnetic Resonance, 1970, 3, 396. l2 J. H. Freed, in Electron Spin Relaxation in Liquids, ed. L. T. Muus and P.W.Atkins (Plenum Press, New York, 1972), chap. 18, p. 503. l3 F. Blah, Phys. Rev., 1956, 102, 104. l4 N. M. Atherton and P. A. Kennedy, Chem. Phys. Letters, 1976, 43, 186. D. S. Leniart, H. D. Connor and J. H. Freed, J. Chem. Phys., 1975, 63, 165. (PAPER 7/793)

ISSN:0300-9238    

DOI:10.1039/F29787400067

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Phenomenological Coefficients and Frames of Reference for Transport Processes in Liquids and Membranes Part 1.-Resistance Coefficients, Friction Coefficients and Generalised Diffusivities in Isothermal Systems BY JOHNW. LORIMER Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7, Canada Received 1Ith May, 1977 Phenomenological descriptions of transport processes in isotropic, non-reacting systems are discussed in terms of resistance coefficients, friction coefficients, generalised diffusivities and conductance coefficients. These quantities are defined rigorously, and it is shown that, for a non- viscous system containing n species, there is a set of (n-1)2 independent coefficients whether or not the system is at mechanical equilibrium.If the coefficients are symmetric, then the number of independent coefficients is reduced to n(n-1)/2. It is also shown that resistance and friction coefficients and generalised diffusivities are invariant under transformations among frames of reference moving at different velocities. Conductance coefficients do not possess this invariance, but do preserve symmetry when they are defined properly. If any set of conductance or resistance coefficients is symmetrical, then the symmetry of any other set is implied. Conductance coefficients, resistance coefficients, friction coefficients and generalised diffusivities provide alternative phenomenological descriptions of the transport properties of solids and fluids. For small non-equilibrium deviations from an equilibrium state, the flux ji of a species i in a multi-species system can be written as a linear function of the affinities xj, with conductance coefficients defined formally as LiI = ljil/lxjlwith xk = 0 (k # j).Hooyman and de Groot '* and Kirkwood et aL3have shown how to define unique sets of conductance coefficients when either or both the fluxes and affinities are linearly dependent. If mities are written as linear functions of fluxes, resistance coefficients can be defined formally as Rij = lxil/ljjlwith jk = 0 (k # j). The matrix of resistance coefficients is not simply the inverse of the matrix of conductance coefficients because of linear dependencies among the fluxes and affinitie~.~ Friction coefficients 6-9 are related to resistance coefficients in a simple way, and appear to be the oldest forms of transport coefficients.10 They have been placed in the framework of irreversible thermodynamics in a number of papers 49 11-13 and their molecular significance 59 8p in electrolyte systems has been discussed.14 Generalised diffusivities are also related to resistance coefficients in a simple way, and have been called l5 Stefan-Maxwell diffusivities after workers in early kinetic theories of diffusion in gases [cJref.(10) and (16)]even though their introduction seems to be due to Curtiss and Hirschfelder." The definition of resistance coefficients and of the relations that hold among them have not been considered in as rigorous a manner as the definition and interrelations of conductance coefficients.Similar remarks hold for generalised diffusivities and friction coefficients. In descriptions of membrane phenomena, the use of friction coefficients began with Spiegler,l' who obtained equations analogous to those of Klemm * by assuming 75 TRANSPORT PROCESSES that relative motion of species i andj was resisted by a mutual friction that depended on species i and j only and that was proportional to their differences in velocity [cf. ref. (6), (7) and (19)]. Shortly afterwards, Kedem and Katchalsky 2o showed the equivalence between the approaches of Spiegler and Klemm. More recently? Staverman 21 has discussed applications to membrane phenomena in more detail. Generalised diffusivities have also been applied to the description of membrane processes.22 It has been claimed 11* l3 that, for systems in mechanical equilibrium, friction coefficients (and hence resistance coefficients and generalised diffusivities) are independent of the choice of the frame of reference with respect to which fluxes are measured. This claim is based on the form of the defining equation for friction coefficients rather than on the transformation properties of the coefficients under a change of reference velocity.In this paper, resistance coefficients, friction coefficients and generalised diffusivities are defined rigorously for isothermal, non-reacting systems and their interrelations and transformation properties are examined. In Part 2 of this series, non-isothermal transport coefficients will be considered in a similar manner, and in Part 3, special problems involving viscous, non-isothermal fluxes (especially in membrane systems) will be discussed.DISSIPATION FUNCTION The dissipation function 23* 24 for a non-elastic, non-reacting isothermal fluid containing n species i is given by where o is the rate of production of entropy per unit volume, Tis the thermodynamic temperature and the molar fluxjiis measured relative to the velocity u of the centre of mass : ji = ci(vt-u) (2) where v = Micivi/p i= 1 (3) ct is the molar concentration, Miis the molar mass of species i and the density is n The affinities are Xt = -(VP*),+& where (VP*)l-= (VPI)T,P+ ViVP is the gradient of molar chemical potential of i at constant temperature, Vt is the partial molar volume of i, p is the hydrostatic pressure and Ft is the external force per mole of i.The total pressure tensor is P = n+pl (7) where n is the viscous pressure tensor and Ithe unit tensor, and the axial vector Pa represents the antisymmetrical part of the total pressure tensor that gives rise to a rotation of angular velocity oin the fluid. The symbols :and -in eqn (1) represent J. W. LORIMER 77 the scalar product of two tensors and the transpose of a tensor, respectively. Schmitt and Craig 25 have shown recently that if transformed affinities 26 n xf = xi+Mi(Vp-C cjFj)/p (8)j=1 are used in the dissipation function, then cix: = 0 (9)i=l i.e., the affinities xfare linearly dependent.Relation (9) follows from eqn (4)-n (6) and (8) by use of the Gibbs-Duhem equation and the Euler relation ciVi = 1. i=l If new fluxes j; = ci(vi-ua) (10) are defined relative to an arbitrary reference velocity n where the wi are weighting factors whose sum is unity, then the dissipation function, eqn (l), is invariant under the transformation to the new set of fluxes and afiitie~.~~ The proof has not been given in detail, buf is straightforward : in eqn (l), substitute for xi in terms of xf from eqn (8) and for ji in terms of j; from eqn (2)and (10) and use eqn (3) and (9). The result is n hr @ = C jf*xf-n: vu+2Pa-0. (12)i= 1 Further discussion of this transformed affinity will be included in part 2.The equation of motion for the fluid is 23 If there is no total acceleration and no viscous force (i.e., the fluid is at mechanical equilibrium throughout), then xi = xi and Prigogine's theorem 27 holds : at mechanical equilibrium, the dissipation function for a non-viscous system is indepen- dent of the choice of reference velocity. Specific effects of viscous contributions will be discussed in Part 3 of this series. Here, eqn (9) is important because it states that, even when mechanical equilibrium does not hold, the dissipation function can be expressed in terms of linearly-dependent affinities x:. The fluxes j; are also linearly dependent, since from eqn (10) and (1l), 2 wij;/ci= 0. (14)i= 1 In writing phenomenological equations, the viscous and non-viscous terms in the dissipation function are of different tensorial character, and may be considered separately 28 if only the velocity of the centre of mass occurs in the dissipation function.Here, systems in which viscous flows are negligible are considered, so that n xi (15)@' = C jyi=1 n-1 = C (Sij +wicj/ciwn)j; xJ i,j=l n-1 = Aijj:.xJ. i,j= 1 TRANSPORT PROCESSES Eqn (17) can be regarded as the sum of products of fluxes$ and transformed affinities or, equally as well, as the sum of products of transformed fluxes and affinities xf: n-1 /n-1 \ In deriving these equations, eqn (9) and (14) have been used ; dij is the Kronecker delta. Eqn (18) and (19) give the non-viscous part of the dissipation function in terms of n-1 independent fluxes j; and n-1 independent affinities xf.PHENOMENOLOGICAL EQUATIONS Linear phenomenological equations may be written on the basis of eqn (18) and (19) in either of the forms n-1 or n-1 xf = c riaj(Akjj;) j,k= 1 n-1 The second line of either eqn (21) or (22) follows from the first line by noting that the order of summation overj and k can be reversed. The quantities 26 are conductance coefficients and the r$ are resistance coefficients, each measured in a frame of reference moving with velocity va. Since the fluxes and affinities are linearly independent, it may be assumed that the Onsager reciprocal relations hold :2 l:j = l;i i,j = 1, . . ., n-1 (23) r?. = rji i,j = 1,.. ., n-1 (24)1J but the following discussion does not depend on this assumption. The consequences of writing eqn (21) and (22) in the forms j: = C i = 1,. . ., n j=l and n xf = R:jj; i = 1,. . ., n (26)j=l will now be investigated. The relations among the coefficients ri"j and Rij can be established by at least three different methods. METHOD 1. Eliminatej," from eqn (26) by use of (14) to give n-1 xi = (R:j-wjcnR~n/cjwn)j,' i = 1,. . ., n. j= 1 J. W. LORIMER From eqn (9), (20) and (22), n-1 n-1 Comparison of eqn (22) and (28) with (27) gives n-1 and n-1 Rij-WjCnR,",lcjwn = -c Ci(r;j+(Wj/CjW,) n-C1 Ckrfl,]/Cn i= 1 k=l j = 1,. . ., n-1. (30) The identification between r& and may be completed by the arbitrary choice Rrj = rt i, j = 1,.. ., n-1. (31) Then, from eqn (29) and (30) n-1 Ri", = -cjr:j/cn (32)j=1 n-1-.. Rii = -cjr;i/cn (33)j=l n-1 RZ, = cicjr;j/cn2 (34)i,j=1 and ciR;j = C,R;~= 0 j = 1,. . ., n. (35)i= 1 i=1 Eqn (35) provide 2n- 1 relations among the n2 coefficients R;j, leaving (n-1)2 independent coefficients. If the reciprocal relations (24) hold, then the Rf'jmatrix is also symmetric and there are a further n(n-1)/2 relations among the coefficients, leaving n(n -1)/2 of them independent. These are the n -1 main diagonal coefficients R;i (i = 1, . . ., n -1) and the (n-l)(n-2)/2 cross-coefficientsRfj (i,j = 1, . . .,n-1). Relations (32) and (35) complete the definition of the coefficients Afj [cf.ref. (29) and (30)].Eqn (35) have been deduced previously by a different method '9 l1 for a system in mechanical equilibrium, from which viscous forces are absent. These restrictions are not made here. Method 1 has been used to relate the conductance coefficients L;j and Z& in a similar way.l* The relations corresponding to eqn (31) to (35) for the conductance coefficients can be obtained formally by replacing Rfj by L:j, &A;k by ZfkAtj and ct by wt/clin these equations. METHOD 2. From eqn (29) and (30), n-1 n-1 This equation can be simplified only if we set R?n = R;i, ri"j= r;i and choose Rin arbitrarily, which gives n(n-1)/2-1 independent, non-arbitrary, symmetric co- efficients R;j. Again, analogous relations for the conductance coefficients can be T obtained formally by the replacements given under method 1.Previously, the conductance analogue of eqn (36) was obtained only for symmetric Zb and L;j coefficients.l* Clearly, method 2 is inferior to method 1 because of the requirement of symmetry for the simplification of eqn (36), even though this equation may be considered to be more general if the reciprocal relations hold. METHOD 3. From eqn (22), choose n-1 Rb = 1 ?$&tjk i, j = 1, . . ., n-1. (37)k=l The coefficients and Rii can then be determined by imposing the conditions (35) and Rtn can be chosen arbitrarily. If the coefficients riaj are symmetric, eqii (37) shows that the coefficients Brj are symmetric only for the special case wi= 0 (i f n), wn = 0; i.e., the frame of reference is fixed on species n.Method 3 can be applied to conductance coefficients as well, with similar conclusions concerning symmetry. Method 1 remains the most general of the three methods for relating the ri"jto the R$, and the equations for practical use are (25) and (26)combined with the restrictions (35). It should be noted that either the resistance or conductance coefficients involving species n are arbitrary;l* their values depend on how the identification is made between eqn (21) and (25) or between eqn (22) and (26). The complete physical content of the phenomenological description is contained in eqn (21) and (22); eqn (25) and (26) are merely convenient, and not necessary. If methods 1 or 3 are used to define Coefficients for species rz, then relations (35) have their physical origin in the linear dependence of the affinities, eqn (9) (or in the Gibbs-Duhem equation for a system in mechanical equilibrium), while the corresponding relations for the conductance coefficients have their physical origin in the linear dependence of the fluxes, eqn (1 4).TRANSFORMATION PROPERTIES OF RESISTANCE AND CONDUCTANCE COEFFICIENTS Now consider fluxes j: relative to velocity va defined by eqn (10) and (11) and another set of fluxes relative to velocity vb: and the ui are weighting factors whose sum is unity. The two fluxes are related by j: = jp+ ci(vb-va). (40) Multiply this equation by wi/ciand sum over n,using eqn (14), to obtain n va -vb --C wij!/ci (41)i=l and thus n j; = C (6ij-ciwj/cj)j9 i = I, .. ., n. (42)j=l Now eliminatej; using the analogue of eqn (14) to obtain n-1 j; = C qjj; i = 1,. ..,n-l (43)j=1 where Ti, = S,,+cl(ujwn/un-wj)/cj i,j = 1,. . ., n-1. (44) J. W.LORIMER Eqn (43), (44) have been deduced previously '* by a more complicated method, while eqn (42) has been deduced by Haase 31 and is a generalisation of similar equations deduced by Kirkwood et aL3 Either eqn (42) or (43) gives the relation between fluxes in different frames of reference. Transformations of the phenomenological coefficients are easier to investigate if the relevant equations are written in matrix form. Thus, eqn (17) becomes N N @' =ja A x' = Aja x' (45) and analogous equations hold for reference frame b : N N @' = 'IjbBx' = Bjbx' where the matrix elements of B are [cf.eqn (20)] Bi j = 61j +uicjJc~u~ (47) Lower-case, boldface italic letters represent (n-1) x 1 column matrices ; upper-case boldface letters represent (n-1) x (n-1) square matrices ; italic letters represent scalar matrices; a tilde indicates a transposed matrix. Application of eqn (43) to (45) and (46) gives B = TA. (48) The phenomenological eqn (22) are u x' = RaAja= Rbgjb. (49) Application of eqn (43) and (48) to (49) gives R" = Rb R (50) that is, the resistance coefficients are independent of the choice of reference velocity. The reason for this independence is that both sets of coefficients Rfj and R!' must satisfy simultaneously the constraints of invariance of the dissipation function, eqn (45)-(46), and invariance of the affinities, eqn (49), on transformation to a new reference velocity.These constraints cause the A-and ja-matrices to transform in reciprocal ways. Transformations of the conductance matrix can be considered in a similar way. The phenomenological equations are ja = LaAx', jb = LbBx'. (51) Application of eqn (43) and (48) to (51) gives N ~a = ~~bor f ~b = 'JT-IL~T-~ (52) where T-l is the reciprocal of matrix T. Thus, the conductance coefficients are not independent of the choice of reference velocity because of the lack of the constraint of invariance of the affinities, eqn (49). However, the transformations (52) preserve the symmetry of the coefficients; if La = za,then eqn (52) shows that Lb = Ebas well.Kirkwood et aL3have considered transformations based essentially on eqn (43), and have derived transformed phenomenological coefficients that did not preserve symmetry. They neglected, however, to require that the transformation be consistent with an invariant dissipation function. Since this invariance is a necessary condition for use of fluxes relative to arbitrary reference velocities, their conclusions concerning loss of symmetry on transformation are in error. TRANSPORT PROCESSES RELATIONS BETWEEN THE RESISTANCE AND CONDUCTANCE MATRICES Eqn (49) and (51) give XL~A= R-I or ARG= (~")-1. (53) In the special case wi = 0 (i # n), w, = 1 (velocities measured relative to species n), the matrix A becomes the unit matrix, and the (n-1) x (n-1) matrices Laand R are reciprocal.The n x n matrices in eqn (25) and (26) with elements L;j and RLjare not reciprocal ; there are 2n-1 linear relations among each set and the determinants of each matrix then vanish. The sets of eqn (53) are also equal to R~ = A =L~AR;~I (54) where I is the unit matrix, or, on using eqn (20) and (31)-(35), to Rik(L:j-CjL;,,/Cn) = L~~(Rkj-WjCnRkn/CjW,)= 6ij i,j = 1, .. ., It. (55) k=l k= 1 Eqn (53) shows that the matrix R is symmetric if the matrix Lais symmetric, and conversely. If it isestablished by experiment or by molecular theory that a particular matrix of coefficients L:j is symmetric in a frame of reference a, then all other sets of coefficients LFj (in other reference frames b) and the set of coefficients Rgj are also symmetric.The transformations among frames of reference and between resistance and conductance coefficients are particular cases of general transformations discussed by Meixner 329 and by Coleman and Tr~esdell.~~ FRICTION COEFFICIENTS AND GENERALISED DIFFUSIVITIES Eqn (26) may be written n xf = C Rijcj(vj-va) j=l = Rijcj(vj-vi) i = 1,. . ., n j=l where the second equality follows from eqn (35). Klemm incorporated a factor of c (total concentration) in the dissipation function and defined friction coefficients as R{ = -R1jc,while Lamm defined friction coefficients as 41 = -cicjRlj. While both these sets of friction coefficients have the same symmetry as the resistance coefficients, the definition of friction coefficients as f..= cjRij i,j = 1, .. ., n-1 (57)15 18* 2o If the conductance coefficients are symmetric, is more consistent hist~rically.~~ fijlcj =fjrlci. (58) As well, eqn (35) and (57) give n fii= -C fij i= I, ..., n (59)j=i=1 a relation that gives the missing friction coefficient in eqn (57). That eqn (56) does not contain the reference velocity has been the basis of previous claims that Rijandf,, are independent of the choice of reference velocity.ll* l3 However, the affinities are necessarily independent of the choice of reference velocity; thus, eqn (56) is only a statement of the invariance of the sum of the products of n-1 friction J.W. LORIMER coefficients and the corresponding differences in velocities and says nothing directly about the transformation properties of the individual coefficients. The correct procedure for establishing the invariance of the coefficients is contained in the transformation eqn (44) and (49). Generalised diffusivities 34 gi are essentially reciprocal friction coefficients :9 and if the Rij coefficients are symmetric, gij= 9ji. Generalised diffusivities were defined originally for systems of perfect gases.lS The extension used here for more It is seen that friction coefficients and general systems originated with Ne~man.~~ generalised diffusivities are related directly to resistance coefficients, and, through their definitions, obey relations (31)-(35).Thanks are due to Dr. J. B. Craig who kindly made ref. (25) available prior to publication. G. J. Hooyman and S. R. de Groot, Physica, 1955,21,73. S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam, 1962), p. 38; chap. VI, sect. 2, 3, 5. J. G. Kirkwood, R. L. Baldwin, P. J. Dunlop, L. J. Gosting and G. Kegeles, J. Chem. Phys., 1960,33, 1505. L. Onsager, Ann. N. Y. Acad. Sci., 1945, 46, 241. R. J. Bearman and J. G. Kirkwood, J. Chem. Phys., 1958,28, 136. J. Stefan, Wiener Sitzungsber., 1871, 63, 63. Lord Rayleigh (J. W. Strutt), Proc. London Math. Soc., 1874, 4, 357. A. Klemm, 2.Naturforsch., 1953, 63, 80. 0. Lamm, J. Phys. Chem., 1947, 51, 1063 ; 1957, 61,948. lo C. Truesdell, Rational Thermodynamics (McGraw-Hill, New York, 1969), Lecture 7.l1 R. W. Laity, J. Phys. Chem., 1959, 63, 80. l2 R. W. Laity, J. Chem. Phys., 1959, 30, 682. l3 D. G. Miller, J. Phys. Chem., 1966, 70, 2639. l4 M. J. Pikal, J. Phys. Chem., 1971,75, 3124. l5 R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena (John Wiley, New York, London, 1960), p. 570. l6 J. H. Jeans, The DynamicaE Theory of Gases (Dover, 1954 ; reprint of 4th edn, 1925), sect. 419. C. F. Curtiss and J. 0. Hirschfelder, J. Chem. Phys., 1949, 17, 550. l8 K. S. Spiegler, Trans. Faraday SOC.,1958, 54, 1408. l9 M. J. Johnson and E. 0. Hulburt, Phys. Rev., 1950,79, 802. 2o 0.Kedem and A. Katchalsky, J. Gen. Physiol., 1961, 45, 143. 21 A. J. Staverman,J.Electroanalyt. Chem., 1972, 37, 233. 22 E. M. Scattergood and E. N. Lightfoot, Trans. Faraday Soc., 1968, 64, 1135. 23 S. R. de Groot and P. Mazur, ref. (2), pp. 14-15, 304-310. 24 D. C. Mickulecky and S. R. Caplan, J. Phys. Chem., 1966,70, 3049. 25 A. Schmitt and J. B. Craig, J. Phys. Chem., 1977, 81, 1338. 26 J. 0. Hirschfelder, C. F. Curtiss and R. B. Bird, The Molecular Theory of Gases and Liquids (John Wiley, New York, 2nd printing, 1964), p. 714. 27 I. Prigogine, Etude Thermodynamique des PhtnomBnes Irrtversibles (Dunod, Paris, 1947), pp. 100-102. 28 S. R. de Groot and P. Mazur, ref. (2), chap. IV, Sect. 2. 29 E. Helfand and J. G. Kirkwood, J. Chem. Phys., 1960, 32, 857 ; eqn (3.6). 30 R. E. Howard and A. B. Lidiard, J. Chem. Phys., 1965,43, 4158. 31 R. Haase, Thermodynamik der Irreversiblen Prozesse (Steinkopff Verlag, Darmstadt, 1963) ; Thermodynamics of Irreversible Processes (Addison-Wesley, Reading, Mass., 1969), Sect. 4.3. 32 J. Meixner, Ann. Phys., 1943, 43, 244. 33 B. D. Coleman and C. Truesdell, J. Chem. Phys., 1960, 33, 28. 34 J. Newman, Adv. Electrochem. Electrochem. Eng., 1967 5, 87. (PAPER 71823)

ISSN:0300-9238    

DOI:10.1039/F29787400075

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Phenomenological Coefficients and Frames of Reference for Transport Processes in Liquids and Membranes Part 2.-Resistance and Conductance Coefficients iii Non-isothermal Systems BY JOHNW. LORIMER Department of Chemistry, University of Western Ontario, London, Ontario N6A 5B7, Canada Received 1 1 th May, 1977 Phenomenological descriptions of transport processes in isotropic, non-reacting, non-isothermal systems are discussed in terms of thermal conductance and thermal resistance coefficients. It is shown that, for a non-viscous system containing n species, there is a set of 2n-1 independent thermal coefficients whether or not the system is at mechanical equilibrium. If the thermal coefficients are symmetric, then the number of independent coefficients is reduced to n.It is also shown that thermal resistance coefficients are related simply to heats of transfer, and both quantities are independent of the choice of reference velocity. If any set of thermal coefficients is symmetric, the symmetry of all other sets of thermal and isothermal coefficients is implied. The converse, however, is not necessarily true. In Part 1,l transport processes in isothermal systems that are not necessarily in mechanical equilibrium were described in terms of independent resistance coefficients or other related coefficients, and it was shown that these resistance coefficients are iudependent of the choice of the reference velocity with respect to which fluxes are measured. This paper extends the treatment of Part 1 to non-isothermal systems.The dissipation function for an isotropic, non-reacting system containing rz speciesis 29 3 All symbols have been defined in Part 1 except for the following affinities : X; = -TV(pi/T)+ Fi = -Vpi+piVTJT+Fi = -(V,Ui)T+ HiVT/T+Fi X, = -VT/T. The quantity -V.j, = Pdddt defines the negative divergence of the heat flux as the rate of absorption of heat q per unit mass and Hi is the partial molar enthalpy of species i. It is customary' to define a new heat flux, called here the "reduced heat flux ", by n so that the dissipation function becomes n Q = C ji xi+ji xu-fi: VV+~P"u) i=1 84 J. W. LORIMER and where xt is defined in Part 1, eqn (5). The reduced heat flux, eqn (5), is the " pure heat flux " due to transport of heat and matte^,^ and not to enthalpy changes accompanying the addition of matter to the volume element under consideration.For systems at mechanical equilibrium and with no viscous forces, eqn (6) is then invariant under a transformation to new fluxes&' measured with respect to an arbitrary velocity va [see Part 1, discussion following eqn (13)], providedji is similarly invariant. Haase has, in fact, proved this invariance by considering the form of the energy equation for the system under the conditions of mechanical equilibrium and absence of viscous forces. The extension to systems not at mechanical equilibrium will be discussed below. De Groot and Mazur have defined quantities called " absolute " heat fluxes, but have not considered the generalisation of eqn (5) to include fluxes relative to an arbitrary reference velocity.This generalisation will also be discussed below. The phenomenological equations corresponding to the dissipation function (6) now contain terms in Zkxu(conductance formulation)6 or in rkj; (resistance formula- ti~n),~analogous to the formulations discussed in Part 1 for isothermal systems. Heats of transfer 2* 4-6 are then defined in terms of the 2;and Zi"j conductance coefficients. Complete formulations in terms of thermal resistance coefficients r;u have not been given, and the transformations properties of the Z;, ri., or heats of transfer have not been examined. Consideration of these problems completes the subject-matter of this paper.REDUCED HEAT FLUX AND THE DISSIPATION FUNCTION In terms of the molar energy, Urn,of the system, the negative divergence of the heat flux jgis -V jq = c dU,/dt+pV v+n: Vv-2Pa*a-5ji Fi. (7)i= 1 This equation may be written in terms of enthalpies rather than energies by using the relations : w, = cn C,Hi/C = U,+pV* (8)i= 1 where H, and Vm are the molar enthalpy and volume, respectively; the general relation for any quantity a : pda/dt = a(pa)/at+V (pav) and ac,/at = -v (c,v,). The resulting equation is n n -V j,+ civi Fi+v (Vp-c,F,)-dplat-i=l i=l fi:vn+2Pa*m-i C,aHJat-i civievw, i= 1 i= 1 = -V cjiHi = -V c civi(Hi-Hm). (11)i a If viscous forces are absent and the system is in mechanical equilibrium [Part 1, eqn (13) and following discussion], then the third term on the left-hand side of eqn (11)is zero and no terms in the velocity of the centre of mass, v, remain.Thus, the heat TRANSPORT PROCESSES flux is independent of the choice of reference velocity. From eqn (9,Haase's equation for the divergence of the reduced heat flux is found by replacingj, by j,'and setting the right-hand side of eqn (1 1) equal to zero. Under the same conditions, both heat fluxes jgand$ are independent of the choice of reference velocity. A more general definition of the reduced heat flux is n ji =j4-(Hi-H,)j;. (12)i=l Unlike definition (5), the enthalpy term in eqn (12) is independent of the choice of reference velocity, since eqn (8) gives It follows from eqn (8) and (13) that definitions (5) and (12) are identical. Eqn (12) is the analogue of the definition of reduced heat flux in a discontinuous system where fluxes are referred to the walls of the experimental apparatus.Eqn (12) also shows that if either one of the heat fluxes is independent of the choice of reference velocity, then the other one is as well. De Groot and Mazur defined an absolute heat flux nji = j,+cH,u = ji+ C c~H~~~ i= 1 that is "absolute " in the sense that the second equality in eqn (14) contains only absolute fluxesji = cpl. Use of this heat flux does not lead to adissipation function that contains only fluxes relative to an arbitrary reference velocity or only absolute fluxes.The reduced heat flux, eqn (12), is the more useful definition. In Part 1, a transformed affinity xi was used to show that the dissipation function was independent of the choice of reference velocity for systems not in mechanical equilibrium. This affinity can be considered to be made up of two parts. One part is the isothermal gradient of chemical potential -(V&. The other part involves the external force per mole : n Fi = Fi +Mi(Vp-C ciFi)/p (15)j= 1 For non-viscous systems, the second term on the right so that xf= -(V~~)~+li;.is, using Part 1, eqn (13), M,du/dt, the total acceleration per mole of i. Since ji =j:+ ci(ua-u) (16) wherejl is the flux relative to the centre of mass and j; is the flux relative to velocity ua, substitute for ji and Fi in eqn (1 1).The result is n -V j; = -1jr Ff-va vp-ap/dt + i= 1 where the relations 2 Mij:/p = v-va i= 1 and cn CiFf = vp i= 1 J. W. LORIMER have been used. It follows that the affinity xf can be considered as a transformation of the external force only, and that, if the transformed external force F[is used in eqn (1 l), the reduced heat fluxfor a non-viscous system is independent of the choice of reference velocity. Finally, the dissipation function, eqn (6), can be written where j; is defined by either eqn (5) or (12) and @ is independent of the choice of reference velocity in a non-viscous system that may or may not be in mechanical equilibrium. PHENOMENOLOGICAL EQUATIONS The phenomenological equations can now be written in terms of the set of n independent fluxesj; and& and the n independent affinities xi and xu.As in Part 1, eqn (21) and (22), n-1 j: = (l:kAkj)X~+l~,,X,, I! = 1, . . ., n-1 (21)j,k= 1 n-1 and n-1 xf = C (rfl,Ajk)jja+rruji i = I, . . ., n-1 (23)j,k= 1 n-1 These equations are to be compared with n j! = C LI~,xJ+L;‘,x,, i = I, . . ., n j= 1 n ji = C LijxJ+~~,,x,, j=1 and n xf = R:jjjj”+R;ujl i = 1, . . ., n (27)j= 1 n xu = C R:jjj”+R:,,j;. (28)j=l Eqn (31)-(35) of Part 1 still hold for the isothermal resistance coefficients R; and analogous equations hold for the isothermal conductance coefficients Lb. Method 1 of Part 1, when applied to the pairs of eqn (21) and (25), (22) and (26), (23) and (27), (24) and (28) gives readily l:,, = L;,,, l:i = L$, l:,, = L&, i = 1, .. ., n-1 (29) and r;,, = Ri”,, rii = R:i, ri,, = Ri,,, ciR;,, = ciRti = G. i= 1 i= 1 TRANSPORT PROCESSES Eqn (30) and (32) define L,,, Lin,Ri,, and REn. There are thus 212-1 independent thermal coeficients Li",,L$, L:, or Rfu,Rit, R:,. If the coefficients are symmetric, there are a further n-1 relations among them, leaving n independent thermal coefficients. The coefficients L:,, Lii and A:,, R:i clearly transform in the same way as the corresponding isothermal coefficients L:j and R:j* Thus, the analogue of Part 1, eqn (52) holds for the Lf, and L:,, while the R;, are independent of the choice of reference velocity.Neither of the coefficients L,, and R,, depends on the reference velocity, since xuand ji do not. The set of n2 independent coefficients [or n(n+ 1)/2 coefficients if the Onsager reciprocal relations hold] Rij, Rji, Ri,,RUi and A,,, (i,j = 1, . ..,n-1) are, therefore, independent of the choice of reference velocity in systems that may or may not be in mechanical equilibrium. As in Part 1, the complete physical content of the phenomenological description is contained in eqn (21)-(24) ; eqn (25)-(28) are unnecessary but convenient. Definitions (29)-(32) of coefficients for species y1 are arbitrary, but have their physical source in the linear dependence of fluxes and affinities. RELATIONS AMONG THE THERMAL RESISTANCE AND CONDUCTANCE COEFFICIENTS These can be derived from eqn (21)-(24).The details are given in the Appendix. From eqn (A.10), (A.18), (A.24) and eqn (29), n-1 Ri, = c RijAkjL&,R,, i = 1, ...,n-1 (33)j,k= 1 n-1 RUU = 1lLull* (35) These 2n- 1 equations, with eqn (53), Part 1, determine the 2n- 1 thermal resistance coefficients in terms of the conductance coefficients. Eqn (33) and (34) include the normalisation relations (A. 17), (A.22) : n-1 n-1 C L,"iAijRj,= C RUiAjiL;,= 1. i,j= 1 i,j=l Eqn (A.21), (A.22) give : Lfu = n-c1 L;jAjkRkuLuu j,k= 1 n = C L;jRj,,L,, i= 1, ..., n (37)j=l where eqn (20), (35) from Part 1 (conductance analogue) and (32) have been used. Similarly, eqn (A.17), (A.21), (A.22) and the procedure leading to eqn (37) give n L;~= 2 L;,R,,~L,, i = I, ..., n j=1 while eqn (33), (34) and (36) give R~,= R~~L;,,R,,, i = I,..., n (39)j=1 RUi= RjiLijR,, i = 1, ..., n j=1fL$Riu= fRUiL;,,= 1. i= 1 i= 1 J. W. LORIMER Eqii (37), (38) and (41) give 2(n-1) independent equations for the 2(n-1) indepen-dent coefficients L;uand L:i, while eqn (39)-(41) give 2(n-1) independent equations for the 2(n-1) independent coefficients Ri, and Rui. These sets of equations are equivalent to the sets (33), (34) and (36). HEATS OF TRANSFER Heats of transfer q: and q:' can be defined in a general way by the relations fl-1 If: = C 2:jAjkq: i = 1,. . ., n-1 or I, = LaAq* j,k= 1 and n-1 N = C 2,"iAjkq:' i = 1, . . ., n-1 or I: = LaAq*' (43)j,k= 1 Agar and Breck's nomenclature is de Groot and Mazur use the term " reduced heat of transfer ".The existence of two different sets of heats of transfer has not been recognized explicitly in previous work. The customary definitions of heats of transfer are equivalent to n L:~,= C L:jqy i = I,. . ., n (44)j= 1 and n = C L;iq;' i = I,. . ., n. (45)L,"~ j= 1 These definitions are identical to the more general definitions (42) and (43) if relations (29) and (30) and Part 1, eqn (31)-(35) (conductance analogues) are used to define conductance coefficients for species n and q:, 4:' are defined by i= 1 i= 1 Eqn (42) and (43), when compared with eqn (A.21) and (A.25), give q: = riu/ruu i = 1, . . ., n-1 (47) 4;' = ruJrUu i = I, .. ., n-1. (48) These relations between heats of transfer and thermal resistance coefficients do not appear to have been suspected. They lead at once to the conclusion that the heats of transfer are independent of the choice of reference velocity and have the same symmetry properties as the thermal resistance coefficients. The heats of transfer for species n are subject to the same arbitrariness as the thermal resistance coefficients. Howard and Lidiard lo [see also ref. (7)] have stressed this arbitrariness, and have claimed that there is no physical basis for eqn (46). However, relations (47) and (48), with (32), show that this basis is, in fact, the linear dependence of the affinities (or the Gibbs-Duhem equation for systems in mechanical equilibrium).The flux equations with heats of transfer are n-1 jf = 1 l$Akj(xi+qj*xu) i = 1, . . ., n-1 (49)j,k= 1 n-1 n-1 n-1 TRANSPORT PROCESSES If the coefficients for species n are used, equations similar to the customary equations are found : n ji. = C L,",(xj.+qj*x,) i = I,. . ., n j=1 n n Alternatively, the affinity equations xi = n-C1 rikAjkj;+q)ruuji i = 1,. . ., n-1 (55)j,k= 1 or n xf = C Rijj;+q)Ru,jl, i = 1,. . ., n (57)j=1 contain only coefficients that are independent of the choice of reference velocity. Finally, eqn (A.26) shows that, if either set of coefficients riuor Zk is symmetric, then all isothermal coefficients are also symmetric. However, symmetry of the isothermal coefficients does not imply symmetry of the thermal coefficients. Completesymmetry in this case requires as well the symmetry of one of the sets riuorZ;.APPENDIX RELATIONS AMONG THERMAL RESISTANCE AND CONDUCTANCE COEFFICIENTS As in Part 1, eqn (49) and (5l), the phenomenological equations in matrix form are ja= LaAx'+IUxu (A4 j; = i~Ax'+luux, (A4ly x' = RAja+ruji (A.3)Iv xu = F:Aja+ruuj; (A.4) where I,, Zi are the (n-1) x 1 column matrices with elements l;, Zti,respectively, and r,, v: are the (n-1) x 1 column matrices with elements riU,r,,, respectively. Substitu-tion of eqn (A.3), (A.4) in (A.l), (A.2) gives N j" = LaARAja+L"Ar,,ji+ l,(Fgj")+ l,,r,,,ji (A.5) N jh = i:ARAja + i:Arujk + lut,(F;ija) + luuruuji. (A4 Substitution of (A.l), (A.2) in (A.3), (A.4) gives X' + r,,(i:~x') + r,~u,x, (A.7)= RZL~AX' + RLz,~, xu = F:kaAx' + F;iZUxu+ ruu(i:Ax')-tr,,,,luuxu. (A.8) J.W. LORIMER 91 For consistency with the isothermal case, Part 1, eqn (53) is used : ha^ = L~A& = I (A.9) where I is the unit matrix. As well, at x' = 0,ja = 0, eqn (A.2) and (A.4) give I,,r,,, = 1. (A. 10) Elimination of the bracketed matrix products from eqn (A.5), (A.6) and (A.7), (A.8) and use of eqn (A.9), (A.10) gives rv I,,LaAr, ji +1,jk = 1,i:ARA.j" +E,i:Ar,, jl, (A.ll) and N N N r,,RAl,x, +rux,,= r,,F:AL"Ax' +v,r"/Al,x,. (A.12) Eqn (A.l), (A.2), with (A.9), give a different equation for ji : N N ji = i:ARAj" +(E,, -i:ARAl,)x, (A.13) which, by use of eqn (A.l l), gives N (I,,E"Ar, -l,i;Ar,)jk +I,( I,, -i:ARAl,)x, = 0.(A. 14 At xu = 0, this equation and eqn (A.9) give N r, = RAl,,( i:Aru)/lu,, (A.15) and at ji = 0, NI,,, = ILARAZu = ih(La)-' Zu, (A. 16) Eqn (A.15), (A.16) are seen to be identical when it is noted that i:Aru is a scalar matrix. Eqn (A.15) are then n-2 linear equations for the n-1 elements Y,,, since this scalar matrix permits only the ratios ri,,/r(,,-ljuto be determined. A complete set of n-1 linear equations for the n-1 elements rfucan be found by imposing the condition i:Aru = 1 (A.17) so that r, = &izu/luu. (A.18) Eqn (A.3), (A.4), with (A.9), give a different equation for xu : H N xu = v;ALaAx' +(ruu-P;AZaAr,) ji (A.19) which, by use of eqn (A.12), gives N N N (r,,JUZu-rur";AZ,,)x,,+r,,(r,,,-r":AZaAru)ji = 0.(A.20) At ji = 0, this equation and eqn (A.9) give ryI, = LaAr,,(r"~AZ,)/r,,. (A.21) At xu = 0 the same result is obtained. Eqn (A.21) is identical with eqn (A.18) if the condition N r":AZu = 1 (A.22) is imposed. The two conditions (A.17}, (A.22) give ILAr,, = iuAr: (A.23) so that, if l: = Z, (Le., I,, = Ilu),then rui = riu,and conversely. From (A.17), (A.18), N N FLAZ, = iLAr, = i~AR(AZu)/lu,, which gives after transposing -I1 r: = RAIJluu. (A.24) 92 TRANSPORT PROCESSES Similarly, NN 1; = L"Ar;/r,,. (A.25) Eqn (A. lo), (A. 18), (A.24) constitute 2n -1 equations that determine the 2n -1 elements riu,rut and r,, in terms of the conductance coefficients. Eqn (A.18), (A.24)give N u-ZuIt(r,,-r:) = RAI, -RAIL.(A.26) u Two cases arise. (a) If either r, = r: or I, = Z:, then eqn (A.26) gives R = R; i.e., the symmetry of either r, or 2, implies the symmetry of R and, from part 1, the symmetry of all other non-isothermal coefficients. (b) If R = E, then it is not necessary that I, or r, be symmetric. Conditions (A.9), (A.lO), (A.17), (A.22) may be deduced in a different way by setting all six possible pairs formed from x', xu,jaand ji equal to zero in eqn (A.1) to (A.4). Eqn (AS) to (A.8) may also be deduced in a different way. Define the partitioned matrices '1.-& (A.27) Eqn (A.l) to (A.4) become J = AX, x = PJ (A.28) from which J = APJ, X = PAX. (A.29) Eqn (A.29)are identical to (A.5) to (A.8).However, it does not follow that AP = I, since if this condition is imposed, the non-conformable products I,,F:i and r1&A appear in the resulting equations. The correct method of solution again starts with eqn (A.9). Further discussion of the relations between the sets of thermal conductaiice and resistance coefficients appears in eqn (36)-(41). J. W. Lorimer, J.C.S. Faraday 11, 1977, 73, 75. S. R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam, 1962)) pp. 14-15, 26, 304-310. D. C. Mickulecky and S. R. Caplan, J. Phys. Chem., 1966,70,3049. H. J. V. Tyrrell, Diffusion and Heat Flow in Liquids (Butterworth, London, 1961)) pp. 15, 32. R. Hasse, Thermodynamik der Irreversiblen Prozesse (Steinkopff Verlag, Darmstadt, 1963) ; Thermodynamics of Irreversible Processes (Addison-Wesley, Reading, Mass., 1969), sect. 4.7. S. R. de Groot and P. Mazur, ref. (2), pp. 283,481, 495-6; chap. XI.'R. J. Bearman and J. G. Kirkwood, J. Chem. Phys., 1958,28,136. S. R. de Groot, Thermodynamicsof Irreversible Processes (North-Holland, Amsterdam, 1952), p. 70. J. N. Agar and W. G. Breck, Trans. Faraday Soc., 1957,53,167. lo R. E. Howard and A. B. Lidiard, J. Chem. Phys., 1965, 43,4158. (PAPER 7/824)

ISSN:0300-9238    

DOI:10.1039/F29787400084

出版商:RSC    

年代:1978

数据来源: RSC

摘要:

Kinetic Study of Ground State Silicon Atoms, Si[3~~(~P,ll,by Atomic Absorption Spectroscopy *BY DAVIDHUSAIN AND PETERE. NORRIS Department of Physical Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EP Received 23rd May, 1977 A direct detailed kinetic study of ground state silicon atoms, S~[~P'(~PJ)] is described. The silicon atoms were generated by repetitive pulsed irradiation of SiCL and monitored photoelectrically using resonance line absorption coupled with signal averaging. Particular attention is directed towards investigation of the kinetic behaviour of the close lying individual spin orbit states by optical isolation of the appropriate resonance transitions during the time-resolved measurements. Thus we have employed for si(33P0), X = 251.43 nm (43P1+-33P0), for Si(33P1), = 250.69 nm (d3P2-+33P1) and, for Si(33P2), X = 251.61 nm (43P2+-33P2).Kinetic measurements were also made on a group of lines centred on A = 251.6 nm. Detailed kinetic studies with a range of SiC14+ He mixtures clearly showed the maintenance of a Boltzmann equilibrium between the spin orbit states during the measurements on the decays of the transient atoms. We report absolute second- order rate constants for the chemical reaction of Si(33P~)with O2 and N20, namely, ko2 = 2.7k 0.3 x 10-lo and ~N~O= 1.9kO.2x 10-lo cm3 molecule-' s-I (300 K). These rate data are com-pared with analogous data for C(23P~), Ge(43P~),Sn(53Po)and Pb(63Po), and are discussed within the context of the nature of the appropriate potential surfaces resulting from symmetry arguments based on the weak spin orbit coupling approximation for light atom-molecule collisions and (J,Q) coupling for heavy atom-molecule coliisions.A major experimental objective in the investigation of the relationship between electronic structure and atomic reactivity is the kinetic study of all atomic states arising from the gross overall electronic structure of the ground state configuration, and the characterisation of such collisional behaviour for all atoms within a given group of the Periodic Tab1e.l This latter purpose is stimulated by our interest in the role of increasing spin orbit coupling in atomic reactions. The framework for dis- cussion is to employ, for light atom-molecule collisions on the one hand, the weak spin orbit coupling approximation to describe the nature of the appropriate potential energy surfaces,2* and, on the other, for heavy atom-molecule collisions, (J, a)~oupling.~Within this overall programme, the atomic reactivity of the various states arising from the np2 configuration of group IV elements (3P0,1,2,ID2, 'So) has been the object of extensive investigation.Thus, direct detailed kinetic studies of the atomic states CQ3PJ, 2102, 21S0),4-12Ge(43Po,1,2, 41S0),13-15 Sn(53P0,1,2, 51D2,SISO)16-21 andPb(63P,,l,2, 6102,61So) 22-30 have beencarried out, principally by resonance line absorption following generation of the transient atoms by pulsed irradiation. To the best of our knowledge, no direct kinetic experiments on atomic silicon have, as yet, been reported.Gaspar et aL31*32 have described nuclear recoil experiments involving the generation of silicon atoms and end-product analyses using gas chromatography. As it happens, limited previous efforts in this research group on direct monitoring of Si atoms were unsuccessful mainly because we used " single-shot mode " experiments rather than " repetitive-mode " experiments coupled with signal averaging. We have recently described an experimental system 93 GROUND STATE SILICON ATOMS for the study of P(34S+),generated by flash phot~lysis,~~ involving repetitive pulsed irradiation on a flow system, " pre-trigger " gating of a high gain photomultiplier, and attenuation of atomic resonance radiation coupled with a new signal averaging 35 This system is now applied to the kinetic study of the ground state silicon atom, Si[3p2(3PJ)].Of course, monitoring of steady concentrations of silicon atoms, such as in a flame, is a routine analytical proced~re.~~ The application of the time-resolved technique to Si(33PJ) is described here in detail. The resulting kinetic data are compared, where appropriate, with those for other atoms within group IV and within the context of symmetry arguments on the nature of the potential surfaces using the weak spin orbit coupling approximation. EXPERIMENTAL The silicon atoms were generated photochemically by the repetitive, pulsed irradiation (E= 45 J, 0.2 Hz) of SiC14 in a coaxial lamp and vessel as~embly,~~-~~ with a common wall of high purity quartz (Spectrosil).Hence, the lower wavelength limit of photolysis is :PS~CI~;1 = -165 nm. Excess helium buffer gas was always employed (p~~: : N 50 000: 1)to ensure no significant rise in temperature above ambient conditions. The experimental arrangement is that of a flow system, kinetically equivalent to a static sy~tem.~~'~~ The silicon atoms were monitored in absorption in the ultraviolet using resonance radiation derived from a microwave-powered flow lamp (psic14= 1.3 N m-', ptotalwith He = 133 N m-' ; incident power = 75 W) employing an Evenson The absorption signals were detected by means of a high gain photomultiplier (E.M.I. 9816 QB) mounted on the exit slit of a grating monochromator (Czery-Turner mount, McPherson Corporation, U.S.A.) always employed, in fact, under vacuum (N Torr, 1 Torr = 133 N m-2).This further ensures no electrical discharge from the photocathode of the photomultiplier but is not an absolutely necessary requirement, The circuitry required for repetitive " pre-gating " of the photomultiplier has already been given in We have also described the new data handling system for dealing with the photoelectric 35 This essentially comprises a fast transient recorder (Data Laboratories DL 920), employed in the " A/B " mode in order to permit averaging of the unattenuated signal I. for each decay, interfaced to a signal averager (Data Laboratories, DL4000). This is, in turn, interfaced to a paper tape punch (Datadynamics 1183) whose output is given in ASCII code for direct input into the University's IBM 370 computer. In general, individual decays for Si(3 "J) always represent the average of at least 32 individual experiments.Again, as we employ the numerical data smoothing procedure of Savitsky and G01ay.~~ MATERIALS SiCI4 (liquid, B.D.H.) was transferred from a sealed ampoule in a nitrogen box and thoroughly degassed before use. All other materials (He, Kr, 02,N20) were prepared essentially as described in previous paper^.^^-^^ RESULTS AND DISCUSSION Fig. 1 shows a portion of the atomic spectrum of silicon, derived from the micro- wave-powered flow lamp, indicating transitions connecting with the ~P~(~P,, levels. Considerations of slit widths and photomultiplier amplification show the microwave lamp to be an intense source of resonance radiation.Detailed considera- tions of line strengths 39 for the observed transitions between the individual spin orbit levels (fig. l), together with the effects of the degeneracies for the upper levels, and an assumption of an effective lamp temperature in the range of 500-1000 KY4Olead one to the conclusion that this source is reversed, as seen from the measured intensity ratios (fig. I). D. HUSAIN AND P. E. NORRIS As the present experiments include kinetic measurements on individual spin orbit levels in the ~P~(~P,) state as well as investigations in which the spin orbit components are not resolved, we employ transitions within the R = 251 nm group (fig.1) as the atomic lines are more widely separated. At the slit widths required for kinetic measurements, it is not possible in the present arrangement to resolve the lines for individual spin orbit components in the il = 221 nm group (fig. 1). The close spacing within this latter group arises, of course, from the small spin orbit splitting in the 3~l(~Dj)state (Jo-.l = 77 cm-l, J2-.3 = 28 ~m-l).~l Emission from the lamp from S~[~S(~P,)] to the ground state is found, in practice, to be more intense (by about a factor of two) than that from Si[3d(3Dj)]. This is in accord with the magnitudes of the appropriate transition probabilities [e.g.,line (A) : gA = 0.3 x FIG.1.-Portion of the spectrum of atomic silicon, derived from a microwave-powered flow lamp, indicating transitions connecting with the 3~~(~P.r) ground state.psiC14 = 1.3 N mp2,pt0talwith He = 133 N m-2 ; incident power = 75 W ; slit widths = 0.02 mm ; scan speed = 2 nm min-' ; p.m. (E.M.I. 9816QB) voltage : 1.9 kV (221 nrn group), 1.7 kV (251 nm group). A 220.87 33D1 +33Po G 250.69 43P2 +33P, B 221.16 33D2 +33P1 H 25 1.43 43P1 -+ 33Po C 221.24 33D1 -+ 33P1 I 251.61 43P2 -+ 33P2 D 221.I5 33D3 +33P2 J 25 1.92 d3P1 --f 33P1 E 221.88 33D2 -+ 33P2 K 252.41 43Po +33P1 F 221.96 33D1 33P2 L 252.85 d3P1-+ 33P2 lo8 s-l ; line (H) : gA = 5.7 x lo8 s-~].~~The apparent relative intensities of the two groups of lines as shown in fig. 1 result from a higher photomultiplier voltage used, purely for the purpose of visual display, to increase the intensity of the lower wavelength group.Routine atomic absorption spectroscopic analysis (AAS) generally employs the il = 251.6 nm line. It must be emphasised that such measure- ments, which use chopping of radiation and phase sensitive detection with " lock-in " amplification, do not require high incident intensities. This procedure may be contrasted with time-resolved measurements of the present kind where a high incident intensity is required to achieve signal-to-noise ratios sufficiently high to overcome the scattered light intensity. Standard AAS measurements sometimes also employ the il = 288.16 nm transition [4s(lPf)+-3p2(lO2), gA = 1.5 x lo9 s-l] 42 which monitors the low Boltzmann population in the 'D2 state [3p2(102)-3p2(3P0) = 6299 ~m-~].~~ At equilibrium -1 % of the atomic silicon concentration is in the lD2 state for a flame temperature of2000 K.GROUND STATE SILICON ATOMS KINETICS OF INDIVIDUAL SPIN ORBIT LEVELS, si(33po,1,2) We first investigate the kinetics of the individual spin orbit levels of the 3p2 ground state (3P0: 0; 3P1= 77.15 ; 3P2 = 233.31 cm-l) 41 by employing the transitions at A = 251.43 nm (43p1 f-33p0,line H, fig. l),A = 250.69 nm (43p2+ 33P,, line G) and A = 251.61 nm (43P2 +-33p2,line I). Optical isolation of these lines involves the use of relatively small slit widths (0.02 mm), with a corresponding reduc-tion in the signal-to-noise ratio, especially when employing the weakest line (H).Fig. 2(a)-2(c) show the digitised time-variation of the transmitted light intensity for 1 0.16 -0.16 -...............0.15 -! :0.15 -.................................. ........... ............. .....".......... ! :::::':; .:" " ........................................... ......................... .................................. .............. (b)................ 0.14 -...............(a) 0.14-................. .................. .......... ............... ....................... .................. ............ .. ...... ... 0.13 y.~:;:::;':.:+::.... ........... ............... ...0.1 2 ...7: . lii I I I 1:: 1 1 i t/ms FIG.2.-Digitised time-variation of the transmitted light intensity indicating the decay of resonance absorptionby ground state silicon atoms following pulsed irradiation.(a)si(33Po),X = 251.43 tun (d3P1 + 33P0) (6) Si(33Pr), h = 250.69 nm (43Pz-+ 33P1) (c) si(33Pz), A = 251.61 nm ('13P2 +33P2) (d) Si(33P~), = 251.6nm (43P~+-33P~).PSiC14IN m-2 : (a), (b),(c) 0.063 and (d),0.22 ; Ptotal with He = 2.8 kN m-2 ; E = 45 J ; repetition rate = 0.2 Hz ; no. of experimentsfor averaging = 32. lines H, G and I, indicating the decay of resonance absorption due to Si(S3P0), si(33p1) and si(33p2),respectively. Fig. 2(d)shows the decay of resonance absorp-tion at wider slit widths (0.150 mm), centred at the strongest line, 2 = 251.6 nm (I) and including resonance absorption at H and J.We employ the empirical, modified Beer-Lambert law :43 It, = I. exp [-E(cZ)Y] (9 or It, = I. exp [-(EcZ)~] (ii) (where the symbols have their usual significance) 43 in order to relate the extent of resonance absorption to atomic concentration. We have given in previous papers 33-35 extensive and detailed discussion for the justification of these empirical expressions when used in logarithmic form {Le., In [ln (IO/Itr)]against In c), in terms of " curve-of-growth " calculations.33*44-46 Fig. 3(a)-3(d) show the computerised output for the first-order kinetic decays, i.e., In [In (Zo/Itr)] against time, for the data 97D. HUSAIN AND P. E. NORRIS ...... ...-1s=*:%.;.;***... ...... .+. ....(a) -2.0 -................. .........I....... ... ...... .......,. .. .. -2.5 -.: ,.......-..,.....,. .. ....L 1 0.4 0.6 0.a 1.0 I.Z 1.4 9 :.. i t/ms FIG.3.-Typical pseudo first-order plots for the decay of ground state silicon atoms obtainedibymonitoring atomic resonance light absorption. (a) Si(3P0)y A = 251.43 nm (43P1+-33Po)(6) Si(3P1), h = 250.69 nm (43Pzc-3jP1) (c) Si(3P2), A = 251.61 nm (43P2c 33Pz) (d) Si(3P~)yA = 251.6 nm (43P~+33P~).psicld/N m-2; (a), (b), (c) 0.063 and (d), 0.22; ptotzl with He = 2.8 kN m-' ; E = 45 J ; repetition rate = 0.2 Hz ; no. of experiments for averaging = 32. n v -3 I I UI 0-? In [SiC14/molecules~rn-~] FIG.4.Beer-Lambert plots for light absorption by the individual spin orbit states Si(33P0),Si(3jPJ and Si(33Pz): (a), 'Po : A = 251.43 nm (43P1c-33P0); (b), 3P1: h = 250.69 nm (43Pz + 3jP1) ; (C), "Pz: = 251.61 M.1 (43Pz+ s3Pz); Ptotd with He = 2.8 kN 11-4 98 GROUND STATE SILICON ATOMS presented in fig.2(a)-2(d). Si[3p2(lD2)] and Si[3p2(1So)][3p2(1So)-3p2(3P0)= 15 394 cm-l 41] were also both, in fact, detected in absorption at 1= 288.16 nm and 1= 390.53 nm [4s(lPp) + 3p2(lS0), gA = 0.86~lo8 s-I 42]. The yields of ID2 state were particularly low. Using conditions under which Si(33PJ) could be moni- tored for -2 ms, Si(31D2) and Si(3lSO) could be monitored for -0.5 and 1 ms, respectively. The sensible linearity in the first-order plots for Si(3"P,> (fig. 3) justifies the neglect of any effects of quenching into the 3PJ state from these higher lying singlet states and, indeed, could indicate chemical reaction of Si(3lDJ and In[SiCl4/rnolecules ~rn--~] FIG.5.-Beer-Lambed plot for light absorption by Si(33P~) at X = 251.6 nm (Si[4~(3~P'')]f-S~[~P~(~P'J)]).Ptotal with He = 2.8 kN Si(31S0) with SiC14 rather than physical relaxation into the ground state.The slopes of the first-order plots for Si(33PJ) (fig. 3) following eqn (i) or (ii) are given by -yk'. k' is the overall first-order decay coefficient for the silicon atom in the state being monitored by resonance absorption and y is the effectively constant value 33-35 that may be applied for the particular transition (H, G,I) or group of transitions (I +H +J) being employed. The values of y are determined by the empirical procedure described by Donovan et aZ.,43 namely, in this case, the assumption of a linear relationship between [Si(t = O)] for the particular atomic state and psic4 (initial). This assumption, coupled with D.HUSAIN AND P. E. NORRIS expressions (i) or (ii), leads, for an effectively constant value of y, to a linear relation- ship between In [In (IO/Itr)](t0) [determined by extrapolation of plots of the type = shown in fig. 3(a)-3(d)to t = 01 and In [SiC14] (initial). The slope of such a plot is y. Beer-Lambert plots for all the transitions employed in fig. 2 and 3 are shown in fig.4 and 5 which yield the following y-values : line transition Y H (A = 251.43 nm) G (A = 250.69 nm)I (A = 251.61 nm) I+H+J (A = N 251.6nm) 43P1 +-33P() 43P2f-33P2 43f + 33P,,,,, 43P2+-33P1 0.98k0.15 0.68 f0.1 0.61kO.1 0.51f0.04 Two points arising from these observations may be noted.First, the numerical value of y decreases as the concentration of Si atoms in the particular 33PJspin orbit level increases. [At the effective Boltzmann temperature involved here, the degeneracy determines the order in the concentrations in the spin orbit levels for Si(33PJ).] This is entirely in accord with observations on the extent of absorption on individual spin orbit levels in O(23PJ)(where the levels are inverted) reported by Bemand and Cl~ne.~~ Secondly, the apparent extent of absorption is less when employing the group (I + H +J) than that for lines for transitions between individual spin orbit levels.Both of these considerations are qualitatively in agreement with curve-of-growth calculations for the A = 177.50 nm transition of P(34S3).33 Especi-3t 3-ns 2-0 1 2 3 lo-’ 3[SiC~4~molecules~rn-~l FIG.6.-Plots of pseudo first-order rate coefficients (yk’) for the decay of (a), Si(3’PO),(b), Si(33P1), (c), Si(33P2)in the presence of SiC14. GROUND STATE SILICON ATOMS ally important, in our view, is the fact that when the curve-of-growth generates an apparent y-value of less than unity, the effect of increasing atomic concentration in the spectroscopic source is to decrease y further. Furthermore, the second point is only true in the region of the curve-of-growth where y is less than unity.33 These general considerations thus support the use of empirically determined values of y which are less than unity.k' is expressed in the form : k' = K+kR[R] (iii) where K is taken to be a constant in a series of experiments in which the concentration of the reactant gas, R, is varied. k, is the appropriate absolute second-order rate constant for the reaction of Si with R. Fig. 6 and 7 show the variation of yk' with b lo-' 3[SiC14]/molecule~rn-~ FIG.7.-Plots of pseudo first-order rate coefficients (yk') for the decay of Si(33P~)in the presence of SiCl.,. [SiCl,] for all the resonance transitions employed. The slopes of the plots in these two figures, coupled with the values of y above, give ksic14 for the various spin orbit levels. These yield : ksi~/cm3molecule-1 s-1, at 300 K (line H) Si(33P0) (line G) Si(33P,) (line I) Si(33P2) (lines I +H +J) Si(33P,) 7.3 1.4 x 10-l1 6.9+ 1.3 x 10-l1 7.2+ 1.3 x 10-l1 7.5k0.8x 10-l1 Thus the (apparent-see later) second-order rate constants as measured for the individual spin orbit levels are seen to be equal within experimental error.It can easily be shown from a simple kinetic analysis that such an observed equality will D. HUSAIN AND P. E. NORRIS 101 arise if all the levels in the 3PJmanifold were in a Boltzmann equilibrium, even if the rate constants for the individual spin orbit levels were, in fact, different. The observed (apparent) second-order rate constant applied to any of the 3PJlevels then becomes a function of the two equilibrium constants combining the three states and the three rate constants for reaction of each Si(33PJ) level with SiC14.A Boltzmann distribution in Si(33PJ) during the time scales over which removal of Si is investigated here would certainly be expected in terms of the small spin orbit energies to be trans- ferred on collision,41 and on the basis of empirical observations for similar processes observed hitherto for different 48 Hence, from the kinetic viewpoint, we may monitor any of the individual spin orbit levels or group of levels. We follow the latter course by employing the conditions described in fig. 2(d) and 3(d),basically on account of the reduced noise levels. It is clearly a good assumption to expect no significant differences in the atomic reactivities for these spin orbit levels, differing so little in energy.Indeed, were one to attempt considerations of (J, 0) coupling on the collision to discuss any possible differences in the reactivity for the individual J levels,l the time and space variable magnetic field generated on collision would destroy the identity of the separate close lying spin orbit levels via the appropriate rnJ values. Hence, all kinetic data are attributed to the overall S~[~P~(~P~)] state. With SiC14 we presume that Cl atom abstraction takes place on collision with the ground state atom. 4OO 1 2 3 10-13[N20,02]/moleculecmq3 FIG.8.-Plots of pseudo first-order rate coefficients (yk') for the decay of Si(33P~)in the presence of (a) O2and (b)N20. GROUND STATE SILICON ATOMS As examples of the study of added reactant gases, kinetic measurements were made on the removal of Si(33PJ) by N20 nd 02.The optical conditions given in fig. 2(d) and 3(d)(I+H +J, fig. 1) were employed. The resulting variation of yk’ with [N20] and [O,]/(fig. S), coupled with the above value of y for the group of lines I+H+ J (A = 0.51 +_0.04), yield the following absolute rate constants : l~[Si(3~P,)+N20] = 1.9 k0.2 x cm3 molecule-’ s-l (300 K) = 2.7k0.3 x cm3 molecule-’ s-l (300 K). Si+O2 The high reactivity if Si(33PJ) with O2 is in accord with the thermochemistry of the atom transfer reactions and with correlations based on the weak spin orbit coupling approximation : AH/eV 41* 49 sio(XlX+) +0(23~,) -2.813 2’A’+ IA/’\ SiO(XIZ+)+O(2lD2) -0.846.Reaction to yield 0(2’S0) would be endothermic (AH = +1.387 eV) 419 49 and there are no direct channels leading to these products. This consideration of Si +O2may be readily compared with that for C(23PJ)+02 for which reaction is also rapid (k[C(23PJ)+02] = 2.610.3 x 10-l’ cm3 molecule-’ s-l, 300 K) l1 and for which there are analogous, exothermic, symmetry-allowed routes : AH/eV 41. 49. 50 CO(X~X+)+o(23pJ) -6.00 7 3A‘+ fA”/ c(23~,)+02(x3Zg)/ \ 21A’f 1A”\ L CO(XIE+)+O(2’02) -4.02. Ogryzlo et aLS1have measured the vibrational distribution of CO from the reaction of C+O, in a flow system and concluded that, on this basis, the route to O(2l0,) was preferred.To our knowledge no similar measurements have been reported on the infra-red emission from SiOIXIC+(v” = n)] from Si+ 02. Rate measurements on the individual spin orbit levels of Ge(43PJ) with 02(J = 0,O cm-l ;J = 1,557 cm-l ; J = 2, 1410 cm-l ;41 1 cm-l = 1.239 81 x eV) 52 indicate that reaction is also rapid [I~(~P,) = 1.2k0.1 x 10-lo; Ic(~P,) = 1.3kO.l x 10-lo; k(3P2)= 1.5f0.3 x 10-lo cm3 molecule-l s-l, 300 K].13 Reaction of Ge(43f‘o) with O2to yield ground state products is exothermic (AH = -1.66 eV) 49* 52 but is endothermic (AH = +0.34 eV) 41* 49* 52 for the production of 0(2lD2). Brown and Husain l3 have presented the correlation diagram connecting the states of Ge +O2and GeO +0 in D. HUSAIN AND P. E. NORRIS 103 (J, 0)coupling in order to consider the routes available to the individual spin orbit levels.Whilst this extreme coupling case for heavy atoms has been found to be of very limited use in considering the behaviour of Ge(43PJ),139 l4 we may note that, on such a basis, Ge(43P0, 1) + 02[X3Z9(0-, l)} correlate with GeOIXIX+(O+, l)]+ OQ3PJ), but Ge(43P,) + 0, do 110t.l~ With the heavy atoms, Sn and Pb, where the spin orbit splitting is relatively large,41 reaction between Sn and Pb+O, must, of course, be considered in terms of (J, Q) coupling. Foo et have monitored Sn(53P0, ,,) in time-resolved atomic resonance absorption experiments but did not report a rate for SJI(~~PO)for which chemical reaction to SnO[X'Z+(O+)] + O(z3P~) would be ex other mi^,^^ although these authors presented a diagram in (J, a)couplingindicating correlations between the ground state of Sn + 0, and SnO + 0.Wiesenfeld and Yuen 53 later reported a rate constant of k = 3.5 x cm3 molecule-l s-l (300 K) for this process. Reaction between Pb(63Po) and 0, is endothermic 'OD 52 and time-resolved resonance absorption measurements on Pb(63P0)show the removal of the atom to be characterised by kinetics which are overall third-order.26 Si+ N20 Thermochemistry alone is not a particularly useful guide in considering reactions of atoms with N20. On energetic grounds, this molecule appears to be an attractive proposition for the production of energised diatomic oxides resulting from atomic reactions, in terms of the low bond dissociation energy of N20[D(N2 -0) = 1 A77 eVJS4 On the other hand, there are many reactions of atoms in specific electronic states for which the processes are highly exothermic and symmetry-allowed, whether on the basis of the weak spin orbit coupling approximation or (J, 0)coupling but which are characterised by low rates.l* 559 56 These primarily result from energy barriers arising from the 1 &electron, closed shell linear structure of this molecule.The rapid reaction rate of Si(33PJ) with N,O is certainly in accord with the high exothermicity of the process : Si(33PJ)+ N,O(XII=+) -+ SiO(XII=+)+ N2(X1Z+)AH = -6.252 eV 499 54 but is seen to be spin forbidden. Without a knowledge of all the states of SiO up to this reaction ex~therrnicity,~~ one cannot construct a correlation diagram with any confidence. The high exothermicity leads one to presume that there are electronically excited states of the products SiO+N, for which potential surfaces are directly available.Consideration of the alternative atom transfer route : 3A'+ 3A" Si(33PJ) +N20(X1Z+) -+ SiN(X2X+) +NO(X211) depends upon the choice of the value of the bond dissociation energy for SiN[D(N-NO) = 4.9303 eV].54 Gaydon's compilation 52 gives D(SiN) = 4.5k0.4 eV. The more recent compilation of Vedeneyev et aZ.,57based on a comparison of bond energies for other group IV diatomic nitrides, gives D(SiN) = 5.20k0.43 eV. The former bond energy of SIN yields a reaction endothermicity of 0.43 eV and the latter, a reaction exothermicity of 0.27 eV, the process being symmetry-allowed on the basis of the weak spin orbit coupling approximation using C, symmetry in the collision, which is not, of course, the " least symmetrical complex ".2 In comparison, the reaction of C(23PJ)with N20is rapid (k[C(23PJ)+N20) = 1.3k 0.3 x 10-1 cm3 molecule-l s-l (300 K)>.l The basic thermochemistry to yield either CO+N, or CN+NO indicates high reaction exothermicities (AH = -9.41 104 GROUND STATE SILICON ATOMS 529eV 52* s4 and -2.82 eV,so* 54 respectively).Husain and Kirsch have listed the symmetry-allowed, exothermic pathways to excited states of both pairs of pro- ducts. The same considerations of thermochemistry and correlation apply to Ge +N20 as described for Si +N20.Reaction to yield GeO(XIZ+) +N2(X1Z;) is highly exothermic (AH = -5.1 eV),49*52* 54 and whilst spin forbidden, there are presumably direct pathways to excited state products 49 which are energetically accessible. The second-order rate constants of the three spin orbit levels [k(3P0)= 5.8f0.8~ k(3P1)= 5.3kO.l x and k(3P2)= 9.5f0.7~ cm3 molecule-' s-l (300 K)]14indicate that no large energy barriers are involved in the removal processes. Wiesenfeld and Yuen 53 have measured the temperature dependence of the re- action : SII(~~P,)+N20(X1Z+)-+ SnO[X'Z+(O+)] +N2[X1E,'(0+)]AH = -3.72 eV 529 54 and report k = 5.0k1.0 x 10-13 exp (-0.17f0.009 eV/RT) cm3 molecule-' s-l. The product states of SnO in this reaction have not been determined. These authors 53 employ correlations based on the weak spin orbit coupling approximation which indicate direct pathways leading exothermically to electronically excited states of SnO(d3C+,A311)+N2(X1C:), and similar but more complex correlations will arise in (J, 0)coupling.The observed energy barrier is in accord with the con- siderations given above of the electronic structure of N20; the low Arrhenius A factor supports effects arising from non-adiabatic transitions following surface crossings. Presumably, similar considerations will apply to the exothermic reaction of Pb(63Po)+N20(AH = -2.193 eV) 529 54 for which only a lower rate limit at room temperature has been reported [k(Pb(63P,) +N20) < 1.8 x cm3 mole- cule-I s-'].~ s We thank the S.R.C.and the Pye Unicam Company for the award of a CASE Studentship to one of us (P. E. N.), during the tenure of which this work was carried out. We also thank the S.R.C. for an equipment grant used for the purchase of the data handling system. 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ISSN:0300-9238    

DOI:10.1039/F29787400093

出版商:RSC    

年代:1978

数据来源: RSC