ISSN:0300-9238    

DOI:10.1039/F298581FX001

出版商:RSC    

年代:1985

数据来源: RSC

ISSN:0300-9238    

DOI:10.1039/F298581BX003

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1985, 81, 11-19 Binary Systems of Acetone with Tetrachloroethylene, Trichloroethylene, Methylene Chloride, 1,2-Dichloroethane and Cyclohexane Part 3.-Dielectric Properties and Refractive Indices at 303.15 K BY JAGAN NATH"AND ATMAPRAKASH DIXIT Chemistry Department, Gorakhpur University, Gorakhpur 27300 1, India Received 22nd February, 1984 Measurements of dielectric constants, E, and refractive indices, n, have been made for binary liquid mixtures of acetone with tetrachloroethylene (CC12CC12), trichloroethylene (CHC1CC12), methylene chloride (CH2C1,), 1,2-dichloroethane (CH2ClCH2C1) and cyclo- hexane (c-C6H12) at 303.15 K. Values of the quantity AE, which refers to the deviations of experimental values of the dielectric constants of the mixtures from values arising from the law of the ideal-volume-fraction mixtures, have been calculated.The positive values of Ae for acetone +CH2ClCHzCl and acetone +CH2C12 and very slightly negative values of Ae for acetone +CHClCCl, are attributed to the formation of molecular complexes of acetone with CH2ClCH2C1, CH2Clz and CHClCCl,. The values of the equilibrium constant, K,, for the formation of 1 :1 complexes of acetone with CH2C1CH2C1, CH,C12 and CHC1CC12 have also been calculated, using the dielectric-constant data. The values of the apparent dipole moments paPpof acetone at various mole fractions in the solvents c-C6H,, and CC12CC12 have also been calculated. The values of p2aPpsuggest that there exists a specific interaction between acetone and CC12CC1,.Binary systems of acetone with tetrachloroethylene (CC12CC12), trichloroethylene (CHC1CC12), methylene chloride (CH2C12), 1,2-dichloroethane (CH2ClCH2Cl) and cyclohexane (c-C6HI2) are of considerable interest with regard to the existence of a specific interaction between the components. The specific interaction of acetone with these chloro-compounds can be visualized as being due to the presence of lone pairs of electrons on the oxygen atom in acetone which can act as an n-donor towards these chloro-compounds. CC12CC12 and CHC1CC12 can act as both u-and .rr-type acceptors toward acetone, whereas CH2C12 and CH2ClCH2C1 can act as u-acceptors toward, and be involved in the formation of a hydrogen bond with, acetone. The system acetone +c-C6HI2, in which only dispersion, dipolar and induction forces are believed to be present between the components, can be used as a reference system.Extensive studies of the interactions between components of such systems have not been made. Only recently' have measurements been made of ultrasonic velociiies, adiabatic compressibilities and excess volumes for binary mixtures of acetone with CC12CC12, CHC1CCl2, CH2C12, CH2ClCH2C1 and c-C6H 12 at different temperatures. Further, measurements of viscosities2 for these mixtures have also been carried out at 303.15 K. These studies have indicated the existence of a specific interaction of acetone with CH2C12, CH2ClCH2C1, CHC1CCl2 and CC12CC12. Dielectric-constant measurements of binary mixtures are also known3-' to give reliable information concerning the existence of a specific interaction between components.Further, values of the apparent dipole moments, paPp,of the various components in solution are also knownc8 to furnish evidence concerning the 11 DIELECTRIC PROPERTIES OF BINARY SYSTEMS existence of specific interactions. Values of paPpcan be obtained from measurements of the’dielectric constants of binary liquid mixtures. Hence, in order to obtain conclusive evidence about the existence of a specific interaction of acetone with CHC1CC12, CC12CC12, CH2C12 and CH2ClCH2Cl, we have made measurements of the dielectric constants for binary liquid mixtures of acetone with these chloro- compounds and c-C6HI2 at 303.15 K.Since the refractive indices of the various pure components and their binary mixtures were needed for calculations of paPp and the molar polarizations, these have also been measured at 303.15 K, and the results of these measurements are interpreted in this paper. EXPERIMENTAL MATERIALS The components acetone, tetrachloroethylene, trichloroethylene, methylene chloride, 1,2- dichloroethane and cyclohexane were purified and their purity was checked as described previously.’ METHODS Measurements of the dielectric constants (E) were made at a frequency of 1.8 MHz, with a dekameter (type DG3, Wissenschaftlich-Technische, Werkstatten, Germany), using one cell (MFL 1/S, no. 2078) for mixtures having dielectric constants <7.0 and another (MFL 2/S, Nr.2084) for mixtures having dielectric constants >7.0. The cells were thermostatted through the outer jacket using a water bath whose temperature was maintained at 303.15* 0.01 K. The two cells were first calibrated using liquids of known’ dielectric constant, and then measurements of the dielectric constants were made for the pure liquids and binary mixtures studied in the present programme. The precision of the dielectric-constant measure- ments is ca. 0.0004 units for dilute solutions of acetone in c-C6H,, and CCI2CCl2, and ca. 0.001 units for mixtures of acetone with CHCICCI,, CH2CI2 and CH2C1CH2C1 and for mixtures having higher concentrations of acetone in c-C6H, and CC12CC12. Refractive-index (n)measurements accurart to *0.0002 were carried out using a thermos- tatted Abbe refractometer at 303.15 *0.01 K.The values of the refractive indices were obtained for sodium-D light. RESULTS AND DISCUSSION The experimental values of E for tetrachloroethylene, trichloroethylene, methy- lene chloride, l ,2-dichloroethane, cyclohexane and acetone and for binary mixtures of acetone with CC12CC12, CHC1CCl2, CH2C12, CH2ClCH2C1 and c-C,H at 303.15 K are given in table 1. The values of E obtained for c-C6H12, CC12CC12, CHClCCl,, CH2ClCH2Cl and acetone are 2.0070, 2.2690, 3.340, 10.075 and 20.218, respectively, in excellent agreement with the literature value^^^^^'^ of 2.0070, 2.272, 3.337, 10.076 and 20.216, respectively. Values of n for acetone, c-C6HI2, CHC1CCl2, CH2ClCH2Cl, CC12CC12 and CH2C12 at 303.15 K were found to be 1.3540, 1.4202, 1.4710, 1.4390, 1SO00 and 1.41 70, respectively. The experimental values of the refractive indices, nI2,for the various mixtures at 303.15, as obtained in the present investigation, were fitted by a least-squares method to the equation n,2= a +bx, +cx: (1) where xI refers to the mole fraction of acetone and a, 6 and c are constants characteristic of a system.The values of the constants a, b and c, along with those of the standard deviations S(n) in the experimental values of n from the values J. NATH AND A. P. DIXIT Table 1. Dielectric constants for the various mixtures of acetone at 303.15 K" XI E XI E XI & acetone +CCI2CCl2 acetone +c-C6H12 acetone +CH2CI2 0.0000 2.2690 0.0000 2.0070 0.0000 8.621 0.0063 2.3203 0.0024 2.0238 0.2208 12.085 0.0 185 2.42 12 0.0028 2.0267 0.279 1 12.810 0.0308 2.5285 0.0284 2.2132 0.3023 13.184 0.0572 2.7608 0.0475 2.3630 0.3 102 13.280 0.0629 2.806 1 0.0756 2.5947 0.4363 14.809 0.0863 3.0221 0.1738 3.498 0.47 16 15.183 0.1 165 3.325 0.2567 4.383 0.5243 15.782 0.1995 4.250 0.3 132 5.027 0.5630 16.145 0.3272 5.852 0.3379 5.397 0.645 1 16.976 0.3470 6.1 13 0.3705 5.834 0.7030 17.617 0.3667 6.420 0.4322 6.66 1 0.7200 17.720 0.3892 6.759 0.4432 6.862 0.7352 17.803 0.4274 7.352 0.5649 9.006 0.7626 18.074 0.5278 9.045 0.6 105 9.905 0.9444 19.750 0.6678 1 1.775 0.6626 10.987 1.oooo 20.2 18 0.7665 14.11 I 0.7288 12.480 0.78 19 14.334 0.8266 14.922 0.8742 16.657 0.8587 15.835 0.8963 17.278 0.90 19 16.994 0.94 15 18.337 0.9667 18.958 1.oooo 20.2 18 1.oooo 20.2 18 acetone +CHClCCl, acetone +CH2ClCH2C1 0.0000 3.340 0.0000 10.075 0.3985 9.242 0.2366 13.179 0.44 10 9.938 0.3839 14.938 0.523 1 1 1.298 0.4466 15.686 0.6002 12.632 0.5846 17.134 0.6446 13.411 0.6669 17.908 0.6954 14.328 0.6728 18.013 0.7085 14.562 0.7670 18.763 0.7535 15.391 0.7892 18.952 0.8707 17.613 0.842 1 19.308 0.8944 18.081 1.oooo 20.2 18 0.9296 18.779 1.oooo 20.2 18 refers to the mole fraction of acetone.obtained from eqn (l), are given in table 2 for the various systems containing acetone.The values of the quantity A&,which refers to the deviations" of the dielectric constants of the mixtures from values arising from the law of ideal-volume-fraction mixtures, have been plotted against mole fraction of acetone, xl, in fig. 1. It is known that the dielectric constants of polar mixtures can be represented as linear functions of the volume fractions of the components." Comparison on a volume- fraction basis largely compensates for the 'dipole-dilution' effect, as discussed by DIELECTRIC PROPERTIES OF BINARY SYSTEMS Table 2. Values of the constants a, b and c of eqn (l), and the standard deviations 8(n),for the various systems of acetone at 303.15 K acetone +CH2C1CH2Cl 1.439 0 -0.081 175 -0.004 260 0.0003 acetone +CH2Cl, acetone +CHClCCl, acetone +CCl,CCl, 1.416 74 1.47I 48 1.499 81 -0.079 239 -0.09 1326 -0.100 285 0.016 037 -0.026 675 -0.045 788 0.0003 0.0004 0.0005 acetone +c-C6H1, 1.420 36 -0.066 28 -0.000 257 0.0002 1.2 O8 0 0 0 0 .E 0 0 v7v TVvv vv00.4 VV d" 0.0 -0.4 0 A Ah a A 0A -0.8 0 A0 0 -1.2 A 00 0 0 0 0 oo 1 I I 1 X1 Fig. 1. Plot of the values of A& against mole fraction of acetone, xl, at 303.15 K: 0, acetone +CH2C1CH2Cl; V, acetone +CH,Cl,; V, acetone +CHClCCl,; A, acetone + CC12CC12;0,acetone +c-C6H,,. Franks and Ives.l2 Fig. 1 shows that the values of AE are highly positive for the systems acetone +CH2ClCH2C1 and acetone +CH2C12, very slightly negative for acetone +CHC1CCl2 and highly negative for acetone +CC12CC12 and acetone + c-C6HI2.At x1 =0.5 the values of A&are in the sequence CH2C1CH2Cl >CH,C12 > J. NATH AND A. P. DIXIT CHC1CC12>CC12CC12>c-C6HI2. The negative values of AE for acetone+ c-C,H,, and acetone +CCl,CCl, can be attributed to decreases in the degree of alignment of the dipoles with changing composition of the mixture. Values of A& are found to be positive for systems in which molecular complexes are believed to be f~rmed,~ owing to the existence of a specific interaction between the components in the liquid state. The measurements of ultrasonic velocities, adiabatic compressibilities, excess volumes and viscosities for acetone +CC12CC12 have indicated'., the existence of a specific interaction between acetone and CCl,CCl, in the liquid state.The negative values of AE for acetone +CCl,CCl, may be explained by the predominance of contributions to AEarising from dispersion, induction and dipolar forces over those due to specific interactions. The highly positive values (see fig. 1) of AE for acetone+CH2ClCH2Cl and acetone +CH2C12, show that acetone forms strong intermolecular complexes with CH,ClCH,Cl and CH,Cl, in the liquid state, as is also found to be the case for the pure binary system pyridine +chloroform, in which a 1 : 1 complex is believed to be formed3 owing to hydrogen bonding between the two components. The very slightly negative values of AE for acetone +CHClCCl, indicate the existence of a specific interaction between acetone and CHClCCl, in the liquid state.To obtain further evidence supporting the fact that acetone forms molecular complexes with CH2C1CH2C1, CH2C12 and CHClCCl,, we have calculated values of the total molar polarisation, P, for acetone, CH2C1CH2Cl, CH,Cl, and CHClCCl, and for binary mixtures of acetone with CH2ClCH2Cl, CH2C19 and CHClCCl, at 303.15 K, using the Kirkwood-Frohlich equation13 (E -n2)(2s+n') vP= 9E where E, n and V refer to the dielectric constant, refractive index and molar volume, respectively. The values of the molar volumes needed to calculate P for pure liquids from eqn (2) were obtained from the density data available in literat~re,'~ whereas the molar volumes for the various mixtures were estimated from the molar volumes of the pure liquids and the measurements of excess volumes.' The values of n used to calculate P for mixtures were obtained from eqn (1).Values of the apparent molar polarisation, PD,of acetone at various concentrations in CH2ClCH2Cl, CH2C12 and CHC1CCl2 were calculated from the total molar polarisation in a manner similar to that described by Rastogi and Nath." The values of PDwere found to increase sharply with decreasing mole fraction of acetone in the case of the systems acetone + CH2ClCH2Cl, acetone +CH2C12 and acetone +CHClCCl,. The variation of PDwith the mole fraction of acetone, x,, in the case of the system acetone +CH2C1CH2Cl is shown in fig. 2. This behaviour of PDindicates that acetone forms strong complexes with CH2C1CH2C1, CH2C12 and CHClCCl, in the liquid state.Considering that a 1 :1 molecular complex (DA) is formed between acetone (D) and the chloro- compound (A), the molar polarisation, PDA,of DA was calculated by the method of Earp and Glasstone.16 The values of PDAfor the 1 :1 complexes of acetone with CH2C1CH2C1, CH2C12 and CHClCCl, were found to be 557.7, 499.1 and 380.1 cm3mol-', respectively. In addition, values of the equilibrium constant, Kf, for the formation of DA in the systems acetone +CH2C1CH2Cl, acetone +CH2C12 and acetone +CHClCCl, were calculated following the procedure described by Earp and Glasstone.I6 The results show that there is a significant variation in the values of Kf with the composition of the mixture in the case of the systems acetone+ CH2ClCH2C1, acetone +CH2C12 and acetone +CHClCCl,. Rivail and Thiebaut3 have also found that in the case of the pure binary system pyridine +chloroform, DIELECTRIC PROPERTIES OF BINARY SYSTEMS 3 Fig.2. Plot of the apparent molar polarisations of acetone, PD,against mole fraction of acetone, xl,for the system acetone +CH,ClCH,Cl at 303.15 K. the values of Kf estimated from dielectric-constant data exhibit a significant variation with the composition of the liquid. Rivail and Thiebaut3 have pointed out that a theory” based upon electrostatic interactions of the solute with the liquid predicts a linear variation of log Kf with the quantity where E and Em have the same significance as described by Rivail and Thieba~t.~ To calculate f(~) =from eqn (3), we have put E~ n2.Fig. 3, where the values of log Kf have been plotted against f(E), shows a linear variation of log Kf with f( E). In order to determine if the values of the apparent dipole moments, paPp,of acetone in CC12CC12 give information on the existence of a specific interaction between acetone and CC12CC12, we have calculated the values of paPpof acetone at various mole fractions in the non-polar solvents c-C6HI2 and CCl,CCl2, using the equation6”* J. NATH AND A. P. DIXIT -0.4 n # I hS -0.6 .I Y cd& W -0 E -0.8 v -2* Y -M -1 .o I I 0.13 0.15f(.) 0.14 Fig. 3. Plot of log [K,/(mole fraction)-’] against f(~)for the various systems at 303.15 K: 0,acetone+CH2ClCH2Cl; 0,acetone+CH2C12;A, acetone +CHC1CC12.where xA is the mole fraction of the polar solute, el is the dielectric constant of the non-polar solvent (c-C6HI2 and CCl,CCl,), E is the dielectric constant of the solution and E; is the internal dielectric constant of the solute. V,, V, and V, are the molar volumes of the non-polar solvent, the polar solute and the solution, respectively. N is Avogadro’s constant, k is Boltzmann’s constant, g is the Kirkwood correlation parameter” and ps,ois the moment of the isolated molecules. As has been mentioned by Stokes and Marsh,6 we have taken (gpi,o)1/2to be equal to the apparent dipole mome;t, paPp,of the solute. The values of V,, V2 and V, needed for calculations of (gps,o)1’2from eqn (4) were estimated from the densities of the pure liquids and the measurements of excess volumes.The value of 1.943 for E; was obtained from the refractive index of acetone as described by Stokes and Marsh.6 Fig. 4 shows that the values of p2ppfor acetone increase with increasing con- centration in the non-polar solvents (c-C,H,, and CC12CC12). Thiebaut et aL20 have also found that the values of the apparent dipole moments of the polar solutes chloroform, 1,1,1 -trichloroethane and diethyl ether, as determined from measure- ments of the dielectric constants of their solutions in the non-polar solvent cyclo- hexane, increase with increasing concentration in the non-polar solvent. Fig. 4 shows that the increases in the values of p:ppfor acetone with increasing concentra- tion are more pronounced in its solutions with CC12CC12 than those in solutions with c-C6HI2.Fig. 4 also shows that the abrupt increases in the values of p:ppfor acetone start at lower concentrations in solutions with CC12CC12 and that these are delayed in solutions with c-C,H,,. .This behaviour can be attributed to the more orderly alignment of the molecules of acetone in CC12CC1, than that in c-C6HI2, which may be due to the existence of a specific interaction between acetone and CC12CC12 in the liquid state. The specific interaction between acetone and CC12CC12 may be due to a charge-transfer interaction of C1 atoms in CC12CC12 with the lone-pair electrons of acetone.On the other hand the existence of a specific interaction of acetone with CH2C1CH2C1, CH2CI2 and CHClCCl, can be explained as being due to the formation of a hydrogen bond on account of the interaction of DIELECTRIC PROPERTIES OF BINARY SYSTEMS 1 I I -2.0 -1 .o 0.0 1.o log (c/mol dm-3) Fig. 4. Plot of pzppagainst the logarithm of solute concentration for acetone in (i) 0,CC12CC12 and (ii) 0,c-C~H,~. hydrogen atoms in CH2ClCH2Cl, CH2C12 and CHC1CC12 with the lone-pair electrons on the oxygen atom of acetone, as it is also known21722 that a complex is formed through hydrogen bonding between acetone and chloroform. There is, however, also a possibility of involving CH2ClCH2Cl, CH2C12 and CHC1CCl2 in the formation of charge-transfer complexes with acetone, owing to the interaction of C1 atoms in these chloro-compounds with the lone-pair electrons of acetone.CONCLUSIONS In conclusion, we note that the function A& shows that acetone forms inter- molecular complexes with CH2ClCH2Cl, CH2C12 and CHC1CC12 in the liquid state. The values of the equilibrium constants Kf for the formation of 1 : 1 complexes of acetone with CH2ClCH2Cl, CH2C12 and CHC1CC12, as estimated from the dielectric- constant data, are in accord with the theory of Barrio1 and Weisbecker,” which is based upon electrostatic interactions of the solute with the liquid. The plots of the values of p:ppin fig. 4 show that the molecules of acetone have a more orderly alignment in solutions with CC12CC12 than in solutions with c-C6HI2, a fact which has been attributed to the existence of a specific interaction between acetone and CC12CC12.The specific interaction between acetone and CC12CC12 may be explained as being due to the charge-transfer interaction of the chlorine atoms in CCI2CCl2with the lone-pair electrons of acetone. On the other hand, the presence of a specific interaction of acetone with CH2ClCH2Cl, CH2C12 and CHClCC12 can be explained via the formation of a hydrogen bond between a hydrogen atom of the chloro-compound and the lone-pair electrons on the oxygen atom of acetone. The possibility that CH2ClCH2Cl, CH2C12 and CHC1CCl2 may be involved in the formation of charge-transfer complexes with acetone has also been indicated.We are grateful to Prof. R. P. Rastogi, Head of the Chemistry Department, Gorakhpur University, for encouragement during the course of this investigation. Thanks are also due to the Indian University Grants Commission, New Delhi, for financial support. J. NATH AND A. P. DIXIT 19 I J. Nath and A. P. Dixit, J. Chem. Eng. Data, 1984, 29, 313. J. Nath and A. P. Dixit, J. Chem. Eng. Data, 1984, 29, 317. J. L. Rivail and J. M. Thiebaut, J. Chem. SOC.,Faraday Trans. 2, 1974, 70, 430. J. Nath and S. N. Dubey, J. Phys. Chem., 1980, 84, 2166. J. Nath and S. S. Das, Indian J. Pure Appl. Phys., 1981, 19, 343. R. H. Stokes and K. N. Marsh, J. Chem. Thermodyn., 1976, 8, 709.' C. Campbell, G. Brink and L. Glasser, J. Phys. Chem., 1975, 79, 660.C. Campbell, G. Brink and L. Glasser, J. Phys. Chem., 1976, 80, 686. N. A. Lange, Lange's Handbook of Chemistry (McGraw-Hill, New York, 1973). lo J. Nath and B. Narain, J. Chem. Eng. Data, 1982, 27,308. I' T. B. Hoover, J. Phys. Chem., 1969, 73, 57. l2 F. Franks and D. J. G. Ives, Quart. Rev., 1966, 20, 1. l3 C. Moreau and G. DouhCret, J. Chem. Therrnodyn., 1976, 8, 403. 14 J. Timmermans, Physico-chemical Constants of Pure Organic Compounds (Elsevier, Amsterdam, 1950). R. P. Rastogi and J. Nath, Indian J. Chem., 1967, 5, 249. 16 D. P. Earp and S. Glasstone, J. Chem. SOC.,1935, 1709. " J. Barriol and A. Weisbecker, C.R. Acad. Sci., Ser. C, 1967, 265, 1372. l8 H. Frohlich, Trans. Faraday SOC.,1948, 44,238. l9 J. G. Kirkwood, J. Chem. Phys., 1939, 7, 911. 2o J. M. Thiebaut, J. L. Rivail and J. Barriol, J. Chem. Soc., Faraday Trans. 2, 1972, 68, 1253. 21 M. D. Magee and S. Walker, J. Chem. Phys., 1969, 50, 1019. 22 G. C. Pimental and A. L. McClellan, The Hydrogen Bond (Freeman, San Francisco, 1960). (PAPER 4/305)

ISSN:0300-9238    

DOI:10.1039/F29858100011

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1985, 81, 21-41 Mixing Coefficients for d Orbitals in Copper(11) Complexes of Lower Symmetry BY NORMANFOGEL Department of Chemistry, University of Oklahoma, Norman, Oklahoma 73019, U.S.A. Received 26th March, 1984 A study has been made of the effect of D,, symmetry on the wavefunctions of copper(I1) complexes. The wavefunction becomes an extended orbital with the form qb,, = a,1(x2-y')') + b,((z2)+)+c,l(xy)+)+d,((xz)-) +e,l(yz)-). Mixing coefficients (a-e) have been calculated in two different ways from spectral and e.s.r. data and are compared. The relative potential and equations have been derived and can reproduce spectral and e.s.r. g values within experimental error. The mixing coefficients a, and b, for the ground state [2Ag,l(X2-Y2)] are related to the radii in the xy plane, and the b2 mixing coefficient [2Ag,1(Z2))]for the excited state is related to the radius in the z direction by a simple linear equation of the form r = K (r)(+*I qb) +r,.The ratio of the radii in the xy plane has been calculated. The radius in the z direction has been calculated by assuming that the radius is proportional to the energy of the 1(Z2))excited state, giving the equation r,/A =0.01384(2b2)2E,,,2,,+0.68. A most likely electronic configuration can be determined if the symmetry is approximately correct, although the symmetry used in this paper is probably too high for many compounds. In the study of the magnetic and spectral properties of copper(I1) compounds, the use of a lower symmetry to explain the physical properties is a necessity.In many cases the highest possible symmetry is used,'-' a symmetry which does not even approximately correspond to the correct symmetry. In other cases, a more correct symmetry is used, but approximations are made that negate some of the advantages of the lower symmetry.- For copper( 11) compounds, the usual symmetry used is the elongated tetragonal (D4h), although almost every copper( 11) compound has a lower symmetry. In cubic (0,)and tetragonal (D4h) symmetries, the-d-orbital set is usually treated as separate simple orbitals. Spin-orbit coupling mixes some of these simple orbitals into extended orbitals which are linear combinations of the simple orbitals. However, not all of the simple orbitals will mix into the extended orbitals under spin-orbit coupling. The usual ground state for tetragonally elongated copper( 11) compounds is 2Blg,which has the same symmetry as the dxzVy2 orbital, and the usual lowest excited state has the same symmetry as the dz2(2AIg)orbital.These two states (or the orbitals) will not be mixed by either symmetry or spin-orbit coupling. Experi- mentally, the point groups of most copper(I1) compounds are of a lower symmetry where these two states (and orbitals) will mix. In the lower symmetry the other states (and orbitals) will also mix in, either from symmetry, spin-orbit coupling or both. This mixing will affect the calculation of spectral and magnetic properties'** of these compounds.Since a reasonable amount of experimental information is already present in the this investigation was undertaken to check if, by using a lower symmetry, the amounts of different d orbitals mixed into the wavefunction (mixing coefficients) could be calculated and if the result was consistent with the physical properties of the compounds. This is difficult since most of the symmetries 21 EFFECT OF SYMMETRY ON WAVEFUNCTIONS used are only approximate, which sometimes leads to errors in the assigned optical transitions. It is possible to assign an electronic structure from the spectral calcula- tions and some interesting relationships of the mixing coefficients to ionic radii in the molecule are observed. The crystal-field simplification, including spin-orbit coupling, of ligand-field theory is used in these calculations because many of the original interpretations used this semi-empiricle theory.In addition, it is relatively simple to extend the tetragonal (D4h)potential to the lower potential used here (D2h)by standard methods.’ The alternative approach, the angular-overlap model, offers some theoretical advantages in terms of classifying the bond types, but must use the same number of pararneteq2 so there is no calculational advantage. It is also more convenient to use crystal-field theory since the programs were already in use in the analysis of the optical spectrum. It is relatively easy to transfer the results to a more complete molecular-orbital model if desired.CALCULATIONS POTENTIAL The full potential for D2hsymmetry is used in this study. This lower symmetry mixes the excited state with A, (dz2)symmetry into the ground state with A, (dX2-,,2) symmetry. This is equivalent to mixing the two simple orbitals. The potential developed below for this symmetry is adequate for D2and C,, point groups, since these also mix the same orbitals because of symmetry. Any additional parameters in these symmetries should be small and can be ignored. From group theoretical consideration^^^" the form of the potential for D2hsymmetry is VDZh=pe +q( Y:+ YY2)+rys: +s( Y:+ Yi’)+t( Y::+ YT4). (1) The values of the coefficients (p, q etc.) can be evaluated for the ionic model by standard methods. l4 Subtracting the cubic and tetragonal potentials, this can be reduced to where VLZhis the specific residual potential for this symmetry, r2 and r3 are the average extensions of the d orbitals along the x and y axes, the Y: are the spherical harmonics and the C,, contain the charges of the central ion and the ligands.Following standard practice,’.2 the charges and radial extensions are symbolized and the potential can be written vL2hCs[E+(J6/2)(Y;+ YY2)]= +Ct[3ys: +JG(Y:+ YT’) +(J70/2)( Y:: + YT4)] where Cs and Ct now contain the C2,C4,r:, r’, and some constants. Operators can now be substituted for the spherical harmonics by standard methods.14 The sign conventions used were those in Ballhausen,’ which makes the d,2 lower than the dX2+. This is consistent with the sign convention for other crystal-field symbols.Some constant number factors were moved into the Cs and Ct symbolism to keep the numerical coefficients minimal. The results for the d orbitals are ((x2-y2)1Vl(x2-y’)) =6Dq +~Ds-Dt +CS+19Ct (44 N.FOGEL Fig. 1. (a) Splitting of d orbitals in different symmetries. (b) Splitting of 2Dstate in copper( 11) ions in different symmetries. ((.Z’)/V((X’))=~D~-~DS-6Dt-Cs+9Ct (4b) ((XY)~V~(XY)) -= -4Dq +~DsDt + CS-16Ct (44 ((xz)~V((XZ))=-4Dq -DS+4Dt +(5/2)C~+4Ct (44 (( YZ)(Vl( YZ)) = -4Dq -DS+4Dt -(7/2)C~16Ct (44-(4f1 The last term is the only off-diagonal element in this basis set. All other integrals are zero in this symmetry using a basis set of d orbitals. The splitting diagram for the d orbitals and the term diagram are in fig.1. The signs invert the d9 case with a ’A, ground state. Residual potentials for CZvsymmetry will take the same form as eqn (3) if odd spherical harmonics are ignored. These odd harmonics will only interact with orbitals that have odd orbital momentum (p or f),which are not included in this restricted basis set. The odd harmonics interact with orbitals that are energetically distant, therefore the interaction should be small. Ignoring these orbitals also reduces the number of parameters to be considered. WAVEFUNCTIONS In the lower symmetries considered here, those orbitals with the same symmetry will mix, so the symmetry correct wavefunctions for the (X2-Y’)and (2’)extended orbitals are (in Dirac notation) IW’-Y’N = [I(X’-Y’)> -bl(z2))1/JN, (5)W’))= [l(z’)> +bl(X2-Y’))IIJN, (6) where the Niare the normalization constants and b is the mixing coefficient.These two extended orbitals are orthogonal. The parameter 6 is EFFECT OF SYMMETRY ON WAVEFUNCTIONS from perturbation theory. However, from eqn (4) this is equal to A = ((x2-y’) I V’I(z’)) = -h(2Cs-5Ct) (8) and if A, Cs or Ct can be determined, along with the energy differences, the mixing coefficient can be calculated for these orbitals. The remaining three orbitals are symmetry correct and remain simple. The complete wavefunction must include the effect of spin-orbit coupling. The operator is =V” = Vs.o.A kiZisi (i = x, y, z) (9) where A is the spin-orbit coupling constant (-828 cm-1),3 ki is the orbital reduction factor, Zi is the orbital angular momentum operator and si is the spin momentum operator.The spin-orbit operator is applied to each of the five symmetry-correct orbitals using perturbation theory to form the new extended orbitals. The three simple orbitals have the spin-orbit operator (V”) applied first, then the lower- symmetry operator ( Vl,,,) to form the new complete extended wavefunctions. The resultant wavefunctions are A ik A/(x2-y2)+)--1(z’)+)-A I(Xy)+)-- kf,A I(xz)-)J“ EZ2 EXY 2 remembering that this orbital represents the ground state [copper( 11) considered as a hole], Ex2-y2= 0, so all the E,, used here are the differences (E,,-Ex2-y2) unless indicated otherwise, and i = ,/?.All of the symbols used in eqn (10) have been identified before except N1= 1 + (A/ EZ2)’+ ( kzA/ E,,)’ + (A ’/4) (f:k; + g: k:) 1 fl =-(1 +*).-EX, Ez2 1 J~A g, = -( 1 -%).EYZ where N.FOGEL The remaining orbitals are where and f4=-(l-(1 EX, Ex, -Ezz) g4 = (Ex,-1 Ez2)(J+. The final extended orbital is where A phase factor of (-l), or ( -i)" has been assigned so that the mixing coefficients of the I(x2-y2))+)will be real and positive for the ground state. This agrees with the standard convention used in the diagonalization program. All of the wavefunc- tions have the standard form I@) = u,(x2 -y')+ +b,( z2)++c,(xy)' +d,(xz)-+e,( yz)-(15) where a-e are the mixing coefficients, n is from 1 to 5, indicating the orbital to be considered, 1 for eqn (lo), (X2-Y'), 2 for eqn (1 1) and so on in regular order.EFFECT OF SYMMETRY ON WAVEFUNCTIONS It is clear from eqn ( lo)-( 14) that the mixing coefficients for the extended orbitals can be calculated if the ki, A and the energies of the optical transitions (En)are known. These can be compared with mixing coefficients derived from the spectrum using an appropriate theory5y6 to construct a matrix to be diagonalized. However, if the spectrum is to be analysed, eight parameters are necessary (including spin-orbit coupling), but the visible spectrum only gives data for five parameters, therefore more information is necessary to determine the other parameters. This information is contained in the electron spin resonance (e.s.r.) spe~trum.~ The equations that define the measured g values derived from the e.s.r.spectra are g, =2((X2-Y2)+*lvy(x’-Y’)’) (164 gx=2((X2-Y2)+*lvy(x’-Y’)-) (16b) gy= 2i((x2-Y’)+*(V;I(X’ -Y’)-> (164 where Vy= (kiZi +g,si), g, = 2.0023, and the other symbols have been defined above. When eqn (lo), its conjugate and negative forms are substituted into eqn (16), the results are gz=N,[g:--’g:[(”)”($)2]2 4k2h EXY EXY where g: = g,/2 and the En represent the energy differences from the ground state, as discussed above. Ordinarily these equations are simplified and the approximate equations are used in interpreting the In this study the exact equations are used in a non-linear Marquardt fitting program” that uses the experimentally determined e.s.r.g values and the known spectrum to determine the best values for the parameters. COMPUTATIONS Different computer programs were used to fit the e.s.r. and visible spectrum of compounds of copper( 11). Compounds with approximately the correct symmetry have been used where possible. The results for the selected compounds are displayed in tables 1-3. The experimental data used are all from the literature4-” and are assumed to be accurate for the purposes of this investigation. In only one case, the barium copper(I1) formate hydrate, were any changes necessary and these were in the values of the energy levels in the visible spectrum. For this compound the Table 1. Electronic configurations and the values of selected parameters electronic configuration/ lo3cm-' compound, 2Af-2Bl, 2B2g 2B3g A A symmetry and ref.(2 ) (XU) (XZ) ( YZ) (esr.) (spectra) kz k, = ky fit gz gx gy ~ copper hydrogen 16.0 11.5 13.6 13.7" 80.0 126 79.823 0.791 06 0.887 40 2.64 x lop6 2.353 2.085 2.089 maleate hydrate, 11.5 16.0 13.6 13.7h 59.2285 59.2361 0.932 86 0.874 02 5.77 x lo5 c2h'2 11.5 13.6 13.7 16.0 -187.026 -187.027 0.859 43 0.9 15 79 3.50 x 10' 13.6 11.5 13.7 16.0 -221.635 -221.633 0.790 42 0.923 46 5.72 x lo5e 13.7 11.5 13.6 16.0 -237.783 -237.783 0.790 47 0.921 89 5.72 x 10' Ba,Cu( HC02)6.4H20, 8.40 10.6 12.95" 13.65 h,c 288.6I0 288.204 0.792 36 0.912 70 9.47 x lop6 2.383 2.078 2.109 c2J 13.65" 8.40 10.6 12.95" 158.62 158.62 0.706 68 0.870 63 2.03 x 10' 8.40 13.65" 10.6 12.95" 112.388 112.388 0.900 14 0.846 11 5.86 x 10-~ copper ethoxy- 10.1 11.6 13.6 14.5b*c 368.359 367.326 0.805 72 0.835 80 4.16 X lo-' 2.364 2.061 2.087 acetate hydrate, 10.1 11.6 14.5 13.6 554.388 554.920 0.806 33 0.838 46 1.88 x 10-~ c2,6 14.5 10.1 11.6 13.6 313.730 313.740 0.752 67 0.798 65 1.13 x 10' 10.1 14.5 11.6 13.6 23 5.699 235.699 0.901 39 0.780 40 6.28 x lo-, n-mpHCuCl,, 17.0 12.5 14.05 14.45kc -69.401 1 -69.513 0.648 24 0.619 22 1.34 X lo-' 2.221 2.040 2.040 DZhlO'1 1 12.5 14.05 14.45 17.0 -294.866 -294.864 0.686 86 0.644 04 3.03 x lo4 12.5 17.0 14.04 14.45 -5 1.1826 -5 1.199 0.755 79 0.607 36 1.03 x lo5 copter lactate 9.75 11.6 12.55 13.6',' 766.707 766.341 0.771 02 0.849 22 3.08 X lo-' 2.330 2.057 2.1 14 hydrate 13.6 9.75 11.6 12.55 1044.88 1044.90 0.707 74 0.825 71 9.90 x c2," 9.75 13.6 11.6 12.55 775.1 14 776.24 0.835 71 0.811 11 6.27 x lo3 e " These values are altered slightly from the original reference, from 15.5 x lo3 to 13.65 x lo3 cm-' and from 13.1 x lo3 to 12.95 X lo3 cm-', see text.'Electronic configuration assigned in reference. Electronic configuration calculated in this work. Final electronic configuration calculated in this work same as initial electronic configuration. Final electronic configuration different from initial electronic configuration, see text. Table 2. Mixing coefficients for the ground states electronic configuration/ lo3 cm-' mixing coefficients compound 2Ag(z2)2B,,(xY) 2B2g(xa 2B3g(YZ) a, bl C1 dl el source copper hydrogen 16.0 11.5 13.6 13.7" 0.997 83 -0.005 14 0.054 64i 0.026 08 -0.025 44i a maleate hydrate 0.997 65 -0.004 98 0.056 82i 0.027 18 -0.026 52i b 11.5 16.0 13.6 13.7d 0.998 18 -0.005 06 0.047 36i 0.026 37 -0.025 86i a 0.996 25 -0.005 16 0.048 04i 0.027 89 -0.025 15i b Ba2Cu( HC02),.4H20 8.40 10.6 12.95'* 3.651 0.996 90 -0.033 85 0.059 15i 0.029 57 -0.025 80i a 0.997 30 -0.034 27 0.061 73i 0.030 92 -0.026 04i b copper ethoxyacetate 10.1 11.6 13.6 14.5"*' 0.997 23 -0.035 93 0.055 44i 0.026 11 -0.022 07i a hydrate 0.997 08 -0.036 26 0.057 34i 0.026 97 -0.022 30i b 10.1 11.6 14.5 13.6" 0.996 41 -0.054 16 0.055 43i 0.025 83 -0.022 30i a 0.996 24 -0.054 74 0.057 34i 0.026 12 -0.023 Oli b n-mpHCuC1, 17.0 12.5 14.05 14.45c3d 0.998 81 0.004 12 0.041 93i 0.017 57 -0.017 45i a 0.998 75 0.004 08 0.042 89i 0.0 18 09 -0.017 84i b copper lactate 9.75 11.6 12.55 13.6 0.994 9 1 -0.077 45 0.051 5% 0.031 59 -0.022 14i a hydrate 0.994 68 -0.078 22 0.054 74i 0.031 66 -0.022 2 1i b a Calculated from spectral data by diagonalization of matrix.Electron spin resonance data, eqn (lo)-( 14). Assignment of electronic configuration from this work. Assignment of electronic configuration in original paper. " Alternate assignment which could not be distinguished from other assignment shown here, see text. Table 3. Mixing coefficients ( un-en)afor excited states of selected compounds wave-compound function n an bn source 2 0.033 35 0.983 08 0.009 94i -0.128 50 -0.125 79i b C0.033 73 0.981 84 0.0 10 06i -0.138 43 -0.124 83i 3 0.048 06 0.009 89 -0.967 87i 0.179 14 -0.169 47i h 0.060 54 0.007 94 -0.978 19i 0.157 28 -0.121 19i C 4 0.051 25 -0.030 77 -0.242 08i -0.789 62 0.560 6% b 0.023 08 -0.126 18 -0.142 89i -0.888 65 0.416 45i C 5 0.01 1 65 0.177 07 -0.031 89i 0.571 86 0.800 29i b 0.027 10 0.1 12 78 -0.1 10 73i 0.468 60 0.893 78i C copper ethoxyacetate 2 0.035 28 0.978 35 0.019 43i -0.153 99 -0.132 29i b hydrate 0.035 61 0.976 35 0.015 84i -0.163 66 -0.135 79i C 3 0.045 76 0.025 90 -0.964 88i 0.196 27 -0.166 50i b 0.056 19 0.013 80 -0.976 98i 0.169 03 -0.1 16 57i C 4 0.042 84 -0.079 32 -0.248 17i -0.874 63 0.406 59i b 0.019 02 -0.154 12 -0.158 19i -0.913 46 0.338 88i C r 5 0.018 99 0.185 93 -0.062 99i 0.414 86 0.888 25i b z1 0.025 24 0.127 69 -0.1 10 24 0.342 43 0.923 93i C ni3 n-mpHCuC1, 2 0.003 68 -0.968 48 0.000 57i -0.167 09 -0.184 70i b r 0.003 98 -0.974 53 -0.000 48i -0.147 03 -0.169 29i C 3 0.034 68 -0.002 19 -0.969 25i 0.185 14 -0.158 32i b 0.041 98 0.000 65 -0.977 54i 0.161 68 -0.128 51i C 4 0.031 33 0.087 75 -0.229 29i -0.901 41 0.355 25i b 0.014 29 0.123 22 -0.135 Oli -0.8 16 36 0.670 93i C 5 0.013 82 -0.233 07 -0.078 86i 0.353 49 0.902 39i b 0.015 18 -0.141 49 -0.107 42i 0.548 22 0.817 lli C 2 0.076 62 0.962 12 0.032 1 li -0.21 1 82 -0.150 16i 0.075 70 0.962 72 0.026 12i -0.230 50 -0.159 19i 3 0.024 78 0.093 67 -0.789 02i 0.527 40 -0.299 86i 0.052 98 0.021 96 -0.923 94i 0.356 29 -0.169 24i 4 0.048 13 -0.134 18 -0.590 55i -0.781 67 0.141 15i 0.013 05 -0.185 91 -0.327 9i -0.886 10 0.269 37i 5 0.036 69 0.203 81 -0.158 16i 0.254 9 1 0.93 1 19i 0.032 40 0.152 22 -0.163 84i 0.283 33 0.932 02i a Mixing coefficients from the configuration indicated as most probable in this paper.Spectral data. E.s.r. data. EFFECT OF SYMMETRY ON WAVEFUNCTIONS Ex, and the Ey, were moved away from a hypothetical centre by 0.15 x lo3 cm-' (i.e. from a reported 13.1 X lo3 to 12.95 x lo3 cm-' and from 13.5 x lo3 to 13.65 x lo3 cm-', respectively). This put the fitting error into an acceptable range without altering any conclusions. This seems justifiable since the reported values are from peak positions and there is considerable overlap of the two transitions, even in the polarized spectrum.The three g values from the e.s.r. ~pectrum~-~,~-" are solved simultaneously by a computer program to yield the best values of A and the ki for a particular electronic configuration. Since there are only three g values but four parameters (A, k,, k,, and k,), it was assumed, following Hitchman and Waite,8.9 that for the purposes of this calculation k, = k,,. Hitchman and Waite indicate that when the ratio kx/kyis varied over a reasonable range the error is small. In this study a reasonable variation is allowed, and it is found that if the variation was five percent or less, the percentage change in k, and the average of k, and ky from the simplest approximation (kx= k,) was <0.4% for the ki and < 15% for the A value. With a variation of ten percent, the error was 1.6% in the ki and 25% in A, but the calculated values of the e.s.r.g began to deviate from the experimental values. For the compounds reported here, the variations from the values calculated with k, = ky seem small enough to be ignored in this preliminary study. One indication of this is that the calculated g values are the same as the experimental values to much less than experimental error (see below). Another factor that favoured this simple approach was that the errors in the calculated values (e.s.r. fit) remained approximately constant as the ratio was varied, so at least initially there is no preferential choice for the ratio. This is something to be investigated later. This approach will create some errors in the mixing coefficients, especially those of the I(xz)-) and I( yz)-) (3-5% in the ground state) simple orbitals and of the I(XZ)-)and I( YZ)-)extended orbitals.These are the orbitals and mixing coefficients that show larger errors. The error is probably smaller than expected since where a mixing coefficient overestimates the contribution of a simple orbital, the paired mixing coefficient will underestimate the effect for the other simple orbital, and vice versa. These contributions tend to balance out, although not completely. Since other errors are ignored, this is probably not the major error in these calculations. It does allow the calculations to be carried out and the correlations to be checked. The calculated best values are then used in the crystal-field program that fits the visible spectrum, where only the parameters Dq, Ds and Dt are allowed to vary.Since A = --$(2Cs -5Ct), a value of A from the e.s.r. data fixes the ratio of Cs to Ct. The spectral fitting program is solved by iteration, using the best value of A and the ki for a particular configuration. Varying the value of Ct automatically varies Cs through the relation to A. Five energy levels are expected in the spectral data, four excited states and the ground state, arbitrarily considered to be zero. Only four parameters need to be fitted. The function that indicates the total difference between the final calculated values and the experimental values is called the fit function (abbreviated as fit in table 1).The fitting program automatically tries to minimize the fit function by varying the original input parameters.16 For the best values of the input parameters, the fit function is very small. The spectral matrix to be diagonalized is in table 4. The fit function for the e.s.r. program is (gi,calcgi,obs)-for each experimental g value, while for the spectral program it is C: (Ecalc-Eobs)2,the squaring to eliminate signs so that the total difference is minimized. The E are the En defined before. The total difference must be minimized because the matrix for all the observed spectral values must be diagonalized to obtain a better answer. In fitting the e.s.r. N. FOGEL Table 4. Matrix for spectral properties in D2hsymmetry I(XY 1+> ((X’-Y’)+] 6Dq +2Ds -Dt + CS + 19Ct -h(ZCs -5Ct) -ik,A 0.5 kyA 0.5ikxA ((z2)+1 -&(2Cs-5Ct) 6Dq-2D~ -6Dt-Cs 0 0.5&k,,~ 0.5ihkxA +9Ct ~~ -4Dq+2D~ ((XY)+l ikzA 0 -Dt+Cs -0.5 kxA 0.5 kyA -16Ct -4Dq -DS 0.5kyA 0.5hk,,A 0.5ikxA +4Dt -0.5 ikzA +2.5Cs +4Ct -4Dq -DS -0.5i kxA -0.5ihkxA 0.5kyA -0.5ikzA +4Dt -3.5cs -16Ct data, differences are in the range 10-6-10-8 for each g value, much better than experimental data.There is no need to diagonalize a matrix so each g value can be handled separately. For the spectral data, the acceptable accuracy was in the range l OP4-l 0-5, again much better than the experimental accuracy. DISCUSSION For compounds where the effective symmetry is approximately one of those for which the potential is correct, it is possible to fit the spectrum and e.s.r.values to much better than experimental error. Usually only one of the possible electronic configurations gives an acceptable value for the fit function. Other possible assign- ments for the electronic configuration give much larger errors in the fitting of the spectrum. In table 1 the most plausible configurations considered are displayed below the appropriate orbital designations. The reported values for the e.s.r. g values are also in table 1, as are the calculated values for the A, kiand the values of the fit function for the spectral values. The values of fit reported in table 1 are for the spectral values only, because the e.s.r. fit function was always excellent and never indicated erroneous configurations.The criteria used to determine the best electronic configuration are two-fold; first the smallest value for fit and secondly the electronic configuration associated with the lowest-fit value must be the same as the input configuration. The program that fits the spectrum16 will arbitrarily shift the configuration in an attempt to minimize the fitting function. It was found that when input and final configurations are different, it was not possible to get a good value for fit for that initial configuration. Therefore, for a particular electronic configuration, it is not enough to have a good value for the fitting function, the initial and final configurations also must agree. This is observed for copper(I1) EFFECT OF SYMMETRY ON WAVEFUNCTIONS lactate hydrate, where the program reassigned the orbital energies several different times.These are included in table 1 to show the possible fit values, but are also footnoted to indicate the changes in assigned electronic configurations. In the other compounds reported it is easily seen that only one configuration gives the best value for fit. A best value for fit can lead to a reassignment of the correct orbital energies. This was observed with copper( 11) hydrogen malleate hydrate where the initial assignment by Hathaway and Billingt2 made the reasonable assignment that the I(XY)') was the highest excited state. On the basis of our calculations the highest excited state is assigned to the l(Z')>'),as with several other compounds where the g, value is the same as the gy value, or else very close.In all of the other compounds reported here, our most plausible configuration is in agreement with the literature assignments. This type of calculation allows the selection of the most probable orbital energies. Since the extended orbitals are a mixture of all the d orbitals, the designation used for the orbital is that of the major component (tables 2 and 3). In many cases, the d,, and dYzorbitals are so mixed in the extended orbital that the calculations will not distinguish between them and they interchange position. This behaviour is observed in some of the calculations. The only difference in the energy of these two orbitals comes from Cs, Ct and the spin-orbit coupling para- meters.In most cases these differences are not great in the diagonalization program for the spectrum and it is necessary to observe carefully how these orbitals are assigned to the extended orbital. In most cases it is possible to determine which of the two is the major component of the extended orbital. Where the two criteria above will not distinguish, the I( YZ)-)is assigned to be the higher in energy [e.g. copper( 11) ethoxyacetate hydrate, table 13. Each electronic configuration has a best value for the A parameter (assuming k, = k,,), as well as the orbital reduction factors (kj).The value of the A parameter can become an identifying factor for a particular electronic configuration.Note that there is a slight possibility that the best value of A may be the same for two different configurations. The values of the orbital reduction factors often correct for erroneous symmetries and calculational or observational error^,^ so the values reported here are for the best fitting of the experimental g values when the A value is the best the program can produce. The orbital reduction as normally used cannot be considered to arise only from the overlap of the two orbitals to form a molecular orbital. It is possible because of these factors that the value of the orbital reduction factor can become greater than one, although this was not acceptable in these calculations. It is worth correcting as many factors as possible if it is desired to know the true effect of the ligands on the central metal ion. One problem in this study is that most of the compounds studied do not have the same symmetry as the potential used, although the data are treated as though it had this symmetry.The most common seems to be c2h (or lower), where the d,,, is mixed into both the dx2-,,2 and the dz2by symmetry as well as by spin-orbit coupling. This is a more difficult problem than reduction to It is interesting, although expected, that an important error in the calculation of the mixing coefficients using eqn (lo)-( 14) is in the mixing coefficients of the dxyin the extended orbitals. The \(XU)') extended orbital also shows a maximum error in energy calculations. The mixing coefficients that are considered best in this study came from the diagonaliz- ation program during the fitting of the visible spectrum.These can be compared with the mixing coefficients calculated from eqn (lo)-( 14). These allow a simplified method of calculating the mixing coefficients, but ignore the more complete configur- ation interaction that occurs in the diagonalization of matrix for the optical spectrum. N. FOGEL In the simplified calculation it is expected that the differences will be larger for wavefunctions of the higher excited states, but the peaking in the I(XY)’) extended orbital for many compounds is interesting and indicative of the error caused by ignoring the symmetry mixing of the dxyorbital into the extended orbital. A way of checking how well the wavefunctions predict is to use them in calculating the properties of molecules.One such property is the spectrum which is used to derive the wavefunctions. If the mixing coefficients can be used to reproduce the spectrum, using a simpler approximation, it will be a useful test. The spectrum is the difference in the energies of the excited and ground state, but all the energies can be calculated from the mixing coefficients, since where $ is the form in eqn (15). The equation for the energy is E = Dq[6( a2+b’) -4(c2+d2+e’)] +Ds[2( a2-b’ +c’) -(d’ +e’)] +Dt[4(d’+e2)-(a2+6b’+c2)]+ Cs[a2+c2+2.5d2-(b’+3.5e2) -4habI +Ct[19a2+9b2+4d2-16(c2 +e’) + lOhab] +kzh(2ac+de) -k,h(ae +3be +dc) +kyh(ae-3be +ce) (18) where a, b, c etc.are the appropriate mixing coefficients (tables 2 and 3), Dq, Ds, Dt, Cs and Ct are the crystal-field parameters taken from the best fit of the spectral data and the other parameters are those from the e.s.r. data calculated for that electronic configuration. The results are good and some sample results are in table 5. The error between the observed and calculated spectrum runs from very low to a maximum of ca. 7%, depending on the transition. The difference between the ground and first excited state is usually in excellent agreement with the observed value. The energy of the transitions are calculated using two sets of mixing coefficients for the same level, one from the matrix diagonalization for fitting the spectral data and the other from eqn (10)-(14).The agreement between the two sets of mixing coefficients varies, being excellent for the first two states (2-5%) but getting worse for higher excited states (200-300%). Note that even when the percentage error of the mixing coefficients is a maximum, the worst difference between the calculated and observed spectrum is only 7% (table 5). This indicates that even poor wavefunctions can be used successfully in many applications. Several have indicated that wavefunctions are related to the ionic size in the molecule. To test this, the ratio ry/rxfor the copper(I1) ion in different molecules has been calculated. The ionic radius is found by subtracting the radius of the coordinating atom in the ligand as reported in the 1iterat~re.l~ From quantum mechanics (4=($*lrl$) but r is a constant, not an operator, therefore r = ri(+*l+) or = if dealing with real wavefunctions.If + is put into polar coordinate form r’ ~+=-[uJ15/nsin2 8cos2~+b~15/3~(3cos~8-l)+cJ15/~sin’ 8sin24 4 +2d-sin 8 cos 8 cos 4 +2eJ15/n sin 8 cos 8 sin 41. (20) w Table 5. Observed and calculated energies of selected compounds P ~~ ~~ error (% ) spectra/ cm-' energy9 mixing compound orbital symmetry eqn (29)/ cm-' calculated observed spectra' coefficientd -9499.04 " 0 (-9500.0)" 0 0.0 10 7.68-9498.96' 0.01 1 2 17.006" 97 16.0 0.359750.0 29.7821 6.778' 97 15.74 0.35 2298.2" 1.65791*8 11 600.0 283.42258.4' 11 757.4 1.36 3 190.73 " l2 689'8 12 550.0 1.1 1 260.13221.31' 12 720.3 1.36 3 8 00.74" 2.21l3 299'8 13 600.0 5 1.833753.84' 13 252.8 2.55 -11599.8" 0 (-11600.0)' 0 0.002 8.46-1 1599.7' 0.003 5366.9" 0.196l6 966*7 17 000.0 213.3537 1.5 ' 16 971.2 0.170 1088.8" 1.51688.6 12 500.0 183.11036.6' 12 636.3 1.09 2.372783.1 " l4 382.9 14 050.0 234.23446.7' 15 046.4 7.09 2702.1" 1.02l4 301'8 14 450.0 149.92370.9' 13 970.5 3.32 " Calculated from mixing coefficients from spectral data.'Calculated from mixing coefficients from e.s.r. data. Eobs -Ecalc/Eobs 100. From data in table 2 and 3, qn(spectra) -qn(e.s,r.)/ qn(spectra) loo, 4 = 4, b, c, d, e, Sum of errors in hybrid orbital. 'Calculated from the fitting of the spectra. fDq = -1207.24 cm-', Ds = -1418.8044 cm-', Dt = -761.2824 cm-', Cs = -204.266 cm-', Ct = 6.783, other data in table 1.Dq = -121 1.108 cm-', Ds = -2639.5868 cm-', Dt = -1220.9386 cm-I, Cs = -10.247 cm-', Ct = -232.682 cm-', other data in table 1. Copper lactate hydrate is one of the better cases calculated, the n-mpHCuC1, one of the poorest. N. FOGEL In D2hsymmetry = 90", 8 = 0" in the y direction and 90" in the x direction, and the only parts of the wavefunction that survive in the xy plane are the I(x2-y2))') and the 1(z2)+).Put in terms of the mixing coefficients, the ratio of ionic radii of copper(rr) ions in the xy plane is where a, and bl are the mixing coefficients for the ground state [(X2-Y')'). This ignores, in this approximation, the extension of the l(2')') orbital in the plane, which should be small, and all orbitals but the ground state.The calculated ratios are compared with the ratio from the experimental radii and the results are in table 6. The results are surprisingly good for this simple a treatment. The ratio is calculated since the complete normalization factor for the wavefunction is not known. In the ratio only the relative extension of the lobes becomes important. In most of the compounds studied the mixing coefficient of the [(z')') (i.e. b,) has a negative sign and the relative extension along the y axis is greater than that along the x axis. This agrees with the convention used in the experimental data where the longer radius in the xy plane is the y direction from the e.s.r. data. An interesting exception is that in n-mpHCuC1, the b, mixing coefficient is positive, and in this molecule the radius along the x axis is longer following the convention above.In this molecule, however, g, = gy and the axes can be defined for the convenience of study. If the radius along the y axis is always defined as the longer radius in the xy plane, there will be certain consequences which must be observed in the normaliz- ation constant. Attempts to improve the agreement between the calculated and experimental values of the ratios by considering the actual angles in the molecules did not show any improvement in this approximation and are not included. Since eqn (2 1) ignores the residual radius in the I( 2')') and other excited states as well as the radius of inner-core electrons, these were treated as an additive constant and the radius for each direction in the xy plane was assumed to be a linear function.The linear equation can be written where K(ri) contains the normalization constant and the basic radius for that extended orbital and r, contains all the residuals. The ratio now takes the form if K(ri) is the same for the two directions. Eqn (23) reduces to eqn (21) if ro/K ( ri)= 0. The results are in table 6. The value of ro/ K ( ri)used for the reported calculations is 2.22 from the fitting of the copper ethoxyacetate hydrate. This led to a minimum overall error and improved the fit. The individual values needed to produce an exact fit for each compound ranged from 2.28 to -0.83. The two compounds (copper hydrogen malleate hydrate and n-mpHCuC1,) where the l(2')') was the highest excited state both had negative values for this parameter (-0.66 and -0.83, respectively).Two values were close (copper lactate hydrate, 2.28 and copper ethoxyacetate hydrate, 2.22) and one was smaller (barium copper formate hydrate, 0.16). It is not clear what the physical significance of rO/K(ri)is, so it is used only as a fitting function. Table 6. Calculations of the ratio of radii of copper(I1) ions in the horizontal plane ratio a1 bl compound ry "/A r, "18, exptl eqn (21) eqn (23) ratio spectrab ratio spectrah ref. copper hydrogen 0.739' 0.713' 1.036 1.012 1.004 0.9999 0.99818 -0.0153 -0.00514 12maleate hydrate Ba2Cu(HC02)6-4H20 0.80' 0.75' 1.07 1.08 1.024 0.9996 0.99690 -0.028 1 -0.03385 7 copper ethoxyacetate 0.77' 0.75' 1.03 1.09 1.03 0.9999 0.99723 -0.01 1 1 -0.03593 6hydrate n-mpHCuC1,' 0.578d 0.61 Id 0.946 0.990 0.997 0.9997 0.9988 1 0.0240 0.00412 10, 11 copper lactate 0.75" 0.7 1' 1.06 1.20 1.06 0.9997 0.99491 -0.0236 -0.07745 6hydrate a Values calculated from rCu-ligand -rligand oxygen radius = 1.22 A, chloride ion radius = 1.67 A." Values from table 2.Coordinating atom is oxygen. Coordinating atom is chloride ion. 'Analysis suggests r, is longest radius, see text. N. FOGEL If the ratio of distances is known experimentally and ro/ K ( ri) is known and a true constant, the ratio of a, to bl can be evaluated. This was attempted with both the simple eqn (21) and the linear form, eqn (23).To get values for a, and 61, other information is necessary, but reasonable values can be calculated by using the normalization condition, a: + 6: = 1, which ignores the mixing coefficients of the other three orbitals. The values calculated from eqn (21) are in table 6, with the spectral values from table 2 for comparison. This approximation works better with a,,the major component, than with 6,. Attempts to use eqn (23) for these calculations showed no improvement if ro/ K ( ri)has the constant value used above, and therefore the values are not included in table 6. The analysis above indicates that the difference in the radii in the xy plane can be assigned mainly to the behaviour of the lobes of the l(X2-Y’)’) ground state due to the mixing-in of the dzzorbital.The coordinating-ligand atom can be assigned a constant extension if it is the same on both axes. This implies that the overlap is the same, or almost the same, along these two axes if the coordinating atom is the same and in the same environment. The corrolary is that if the orbital reduction is due only to the overlap of the ligand atom on the central ion, the orbital reduction factor should be the same, or reasonably close, on both the x and y axes. This is the simplest approximation which is used in this paper, as well as in others. This is only the first approximation since there can be other causes for a reduction in orbital angular momentum, but may be a better approximation than is usually acknowledged.The successful treatment of the ionic distances in the xy plane encouraged a linear treatment along the z axis as well. The distance along the z axis would not be a major function of the ground state, since this extension along the z axis only occurs through the mixing of the dz2 into the ground state, which is usually small. The excited state [(Z’)+)has a strong z component and is expected to be the major orbital overlapping the ligands along that axis. The overlap should be proportional to the radius and the radius proportional to the energy of the orbital, therefore we expect a simple relationship between the radius and the energy of the 1(2’)+).A plot of the ionic radius of the copper(1r) ion in the z direction against the energy (in lo3 cm-’) of the transition assigned to I (2’)’)state produces a reasonable straight line, with some scatter. A simple treatment from eqn (22) produces r, = K (2b2)2E,(z2)+)+ r, (24) where b2is the mixing coefficient for the simple I(z2)+) orbital mixed into the l(Z’>’) extended orbital.The constant is due to several factors and is not necessarily the same as ro in eqn (22). The factors involved can be the radius of the d orbitals at zero energy, collapsing into the ground state, the extension of the ground state, the core electrons, as well as other factors. Treating this as a constant is probably a simplification, but the error is probably smaller than the other errors in this treatment. K will contain the normalization factors as well as the constants that relate distance to the energy of the orbital.No attempt is made, at this time, to calculate a value for either constant from theoretical considerations, since they are evaluated experimentally. Fifteen compounds of copper( 11) that are approximately tetragonally distorted are included. A plot of the ionic radius against the assigned energy of the (2’)’ state is in table 7 and fig. 2. Although there is considerable scatter, the results indicate that for the compounds studied the average error is surprisingly small, only 3.5%. The equation for the best straight line is rz/A= 0.O5O8E1(,2)+)+ 0.68. (25) Table 7. Energy of l(Z2)') orbital as a function of radius of copper(rr) ion along the z axis radius of Cu2'/A b2 El(Z2)+)l radius of compound lo3 cm-' Cu-ligand/A environment exptl/Aa eqn (25) exptl spectra ref.0 //Ba2Cu(HC02)6-4H20 8.40 2.18 H-C-O--0.97 1.1 1 0.7897 0.983 08 7 copper ethoxyacetate 10.1 2.38 C2H50--1.17 1.19 0.936 1 0.978 35 6 hydrate copper lactate hydrate 9.75 2.30" H20-1.08 1.18 0.8608 0.962 12 6 0 //2.90 R-C-O--1.68 copper hydrogen 16.0 2.682 H20-1.462 1.492 0.9396 0.903 05 6 maleate hydrate copper formate hydrate 9.20 2.36 H20-1.14 1.15 0.9503 0.983 99 7 copper tutton salt 6.40 2.23 H20 -1.01 1.005 0.965 1 0.973 15 9 (SO4) 2.6 H 20 7.14 2.278 H20-1.058 1.042 0.9779 0.979 21 9K~CU 0 //11.5 2.54 R-C-O--1.32 1.26 1.0026 0.924 11 6 13.4 2.65 O2N-1.33 1.36 0.9360 -4 14.3 3.OO NCS-1.30 1.41 0.8850 0.926 34 4 14.0 2.59' H2O-1.37 1.39 0.9435 -4 3.37 H20-2.15 Rb,Cu( S0,)2.6H20 7.63 2.307 H2O-1.087 1.067 0.98 16 -9 S04),-6H20 8.00 2.3 17 HZO-1.097 1.086 0.9703 -9TI~CU(K2Cu( ZrF,),.6H20 7.80 2.327 H20-1.107 1.076 0.9944 -9 CU(N H 3) 4( CuC12) 2.H20 16.1 2.73 H20-1.51 1SO 0.9650 -4 a Oxygen radius = 1.22 A, nitrogen radius = 1.32 A, sulphur radius = 1.70 Experimental values calculated from ionic radii and eqn (26), those under spectra from table 3. Only closer ligand is considered, no averaging of distances. N. FOGEL 0.9r I I I 1 I 1 I 6 8 10 12 14 16 10 El(z2)+,/1~3cm-' Fig. 2. Plot of radius in z direction against energy of 1(Z2)'):(a) eqn (36) and (b)eqn (37) :::I with b2= 1.00. 1.5 5 a 1.4 4 21.3 c____1.2 1.1 1.o 0.9I I I I I I 1 I 24 28 32 36 40 44 48 52 (2b2)2q(z2,+, Fig.3. Plot of radius in the z direction against (2b,)2Elcz2,+,:1, copper tutton salt; 2, K,CU(SO~)~.~H~O;3, copper formate hydrate; 4, copper ethoxyacetate hydrate; 5, copper hydrogen maleate hydrate. The results using this equation are compared with the experimental values in table 7. There are several compounds that fall far outside the average error. From eqn (24)it is clear that the slope of the line is K(2b,)2, and b, can be evaluated. From a study of five compounds (fig. 3) where b, is known from spectral data and the ligand radius was unambiguous, the following equation is found: r, = 0.01384(2b2)2E1,z21+,+0.68 (26) with a correlation of 0.968.The maximum normalized value of b2 as a mixing coefficient can only be one [ie. l(2')') is a pure dZ2 orbital]. The value of b2 that EFFECT OF SYMMETRY ON WAVEFUNCTIONS reproduces the slope in eqn (25) is 0.958. It is clear that in most compounds the value of 6, is not exactly 0.958 but will vary around this value (tables 3 and 7). Those compounds not directly on the straight line defined by eqn (25) can be explained as having different values for b2. In many of the compounds in table 7, the spectral parameters have not been analysed, so for these compounds the value of b2 reported is that calculated from eqn (26). For those cases where b2 is known, the value is included in table 7 for comparison. In only one compound is the b2 value slightly greater than that calculated from eqn (26), and in this case [copper(rI) glycollate hydrate] the oxygen atom in the ligand on the z axis is a carboxyl oxygen which can form a resonance hybrid.A larger error in the opposite direction is found in compounds where the oxygen atom is part of the formate ion and again the probability of resonance is present. Another problem is one of compounds with two different ligands on the axis, or two ligands at different distances. In these cases it is observed that the radius in the direction of the ligand that is closest to the central ion usually correlates well with the linear equation. If the distance of the molecular radius is > 3 A, the correlation is excellent; under 3 A there seems to be some averaging process.There is no correlation if the ligand on the z axis is a metal ion from another complex ion, although for one copper(r1) complex the distance seems to be roughly double the calculated distance for the radius of the central ion. The difficulty in these cases is in determining the radius for the complex that is acting as the ligand. These types of compounds are excluded from this study. In the molecule there is also a correlation with the molecular distance between the central ion and the ligand if the coordinating atom in the ligand is constant (ie. oxygen in these studies). This has not been explored further. The constant (r,,) along the z axis should be related to the size of the bare ion and it is gratifying that 0.68 A is approximately the ionic radius assigned to the copper(I1) ion (0.71-0.89 A)." CONCLUSIONS This study and many others4-12 show that the fitting of the physical properties can be used to decide which is the best electronic configuration.That the symmetry assumed in this study is still too high is indicated by the differences calculated for the mixing coefficients for the dxYorbital in the different extended orbitals. In D2h and the other symmetries used here, this orbital is mixed into the others only by spin-orbit coupling. The values of the mixing coefficients can change considerably if the mixing also occurs because of the symmetry. Many of the errors that are produced by simplifications are partly compensated for by the orbital reduction factor, so that it is dangerous to use this factor as an indication of the covalence of the bond, except as a rough appro~imation.~.~ The mixing coefficients can be used to estimate spectral and magnetic parameters.Mixing coefficients may be estimated from the ionic radii in different directions. It has been shown that the ionic radius on the z axis is related to the value of the normalized b2 mixing coefficient (the amount of the dz2 orbital) of the l(2')') extended orbital. In the same way the a, (mixing coefficient for the dX2-,,2) and the 6, (for the d,~)for the ground state [(X2-Y')') are related simply to the ratio of the ionic radii in the xy plane. The ratio of a, to bl can be calculated, but not the values of these parameters, although they can be approximated.The treatment here is a more complete treatment of a lower symmetry than is usually attempted, but there are still some drawbacks. Only the d-orbital basis set is used, and the inclusion of higher excited states may be important." The symmetry N. FOGEL 41 used is the simplest low symmetry, not necessarily the correct symmetry for the compounds studied. If the correct symmetry is used, especially the site symmetry, most transition-metal complexes have a very low symmetry. The treatment here is an indication of what can be done in future studies. I thank Dr Harmon Abrahamson for useful discussions and acknowledge a grant for computing from the University of Oklahoma. C. J. Ballhausen, Introduction to Ligand-$eld Theory (McGraw-Hill, New York, 1962).M. Gerloch and R. C. Slade, Ligand-jield Parameters (Cambridge University Press, Cambridge, 1973). F. E. Mabbs and D. J. Machin, Magnetism and Transition Metal Complexes (Chapman and Hall, London, 1973). A. A. G. Tomlinson, B. J. Hathaway, D. E. Billing and P. Nichols, J. Chem. SOC.A, 1969, 65. M. A. Hitchman, J. Chem. Soc. A, 1970, 4. K. Dawson, M. A. Hitchman, C. K. Prout and F. J. C. Ressotti, J. Chem. SOC.,Dalton Trans., 1972, 1509.' D. E. Billings and B. J. Hathaway, J. Chem. SOC.A, 1968, 1516. M. A. Hitchman and T. D. Waite, Znorg. Chem., 1976, 15, 2150. T. D. Waite and M. A. Hitchman, Inorg. Chem., 1976, 15, 2155. 10 R. L. Harlow, W. J. Wells 111, G. W. Watt and S. N. Simonsen, Znorg. Chem., 1974, 13, 2106.'I M. A. Hitchman and P. J. Cassidy, Znorg. Chem., 1979, 18, 1745.'* B. J. Hathaway and D. E. Billings, Coord. Chem. Rev., 1970, 5, 143. l3 F. A. Cotton, Chemical Applications of Group Theory (Wiley, New York, 2nd edn., 1971). l4 H. Watanabe, Operator Methods in Ligand-field Theory (Prentice-Hall, Englewood Cliffs, N.J., 1966).15 S. D. Christian and E. E. Tucker, Am. Lab., 1982, 14, 36. 16 H. W. Joy and N. Fogel, J. Phys. Chem., 1975, 79, 345. 17 R. D. Shannon, Acta Crystallogr., Sect. A, 1976, 32, 751. 18 M. A. Hitchman and J. B. Bremner, Znorg. Chim. Acta, 1978, 27, L61. (PAPER 4/493)

ISSN:0300-9238    

DOI:10.1039/F29858100021

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. Soc., Faraday Trans. 2, 1985, 81, 43-61 Spino dal Decomposition and the Liquid-Vapour Equilibrium in Charged Colloidal Dispersions BY JEAN-MARCVICTOR*AND JEAN-PIERREHANSEN Laboratoire de Physique ThCorique des Liquides, C.N.R.S., UniversitC Pierre et Marie Curie, 4 Place Jussieu, 75230 Paris Cedex 05, France Received 2nd April, 1984 The possibility of observing a phase separation between a high-density (‘liquid’) phase and a low-density (‘vapour’) phase in charge-stabilized aqueous dispersions of monodisperse spherical colloidal particles which are assumed to interact via the standard DLVO potential has been examined. The free energy of the system is calculated by thermodynamic perturbation theory. The contribution of the repulsive part of the potential is reduced to that of an effective hard-sphere fluid, while the van der Waals attraction is treated in the ‘high-temperature approximation’. The critical-point coordinates and spinodal and coexistence curves have been determined as functions of the potential parameters and show that a reversible ‘liquid- gas’ transition should be observable for physically reasonable values of these parameters.In a well defined range of salt concentrations it is shown that this reversible transition should be clearly distinct from irreversible coagulation, which will always occur beyond a critical salt concentration. Charge-stabilized colloidal dispersions are known to crystallize in sufficiently dialized aqueous solutions under the action of the electrostatic repulsion between the electric double layers surrounding the colloidal particles.’ The colloidal crystal melts when the Coulomb barrier is effectively screened by the addition of salt.At high salt concentrations attractive van der Waals forces take over and eventually lead to irreversible coagulation.2 Before that stage is reached, however, the balance between the short-range Coulomb repulsion and the longer-range van der Waals attraction develops into a so-called ‘secondary minimum’ in the standard DLVO potential between particles ;over a certain range of electrolyte concentrations this secondary minimum is separated from the deep primary minimum by a Coulomb barrier which effectively prevents coagulation as long as it appreciably exceeds the thermal energy kT.Some authors have examined the possibility of observing3 a ‘weak’ (reversible) coagulation into this secondary minimum, and the analogy with a liquid-vapour equilibrium has been put for~ard.~ In fact it is well known that under the conjugated action of a short-range repulsion and a long-range attraction between molecules a gas will condense into a liquid below a critical temperature, which is of the order of the depth of the potential welL5 Since the DLVO potential between colloidal particles in charge-stabilized suspensions exhibits precisely these characteristic features under favourable physical conditions, it is quite natural to expect a ‘liquid-gas’ separation in these suspensions. In this paper we present a statistical-mechanics calculation of the ‘liquid-gas’ coexistence and spinodal curves for monodisperse aqueous suspensions of charged colloidal spheres.In an earlier repod we specified the physical conditions (particle diameter, surface potential and salt concentration) under which the critical temperature of this ‘liquid-gas’ transition is expected to fall into a physically realistic range ( T > 273 K). The details of this calculation as well as the resulting phase diagrams and spinodal curves are 43 LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS reported here. A simpler, mean-field type calculation of such phase diagrams has recently been proposed by Grimson,’ but his results apply only to weakly charged dispersions. A related mean-field calculation, which includes the bending energy of the spherical globules, has been applied to uncharged microemulsions and also predicts a separation between low- and high-density globular phases induced by the van der Waals attraction between globules.8 In the case under consideration here we show that mean-field theory is not sufficiently accurate to predict quantita- tively reliable results.THERMODYNAMIC PERTURBATION THEORY We consider a suspension of n charged colloidal spheres of diameter a, per unit volume. The spheres are surrounded by counter-ions and stray ions which form electric double layers around them. The total DLVO potential energy of interaction between two colloidal particles is the sum of an electrostatic repulsion between double layers and the van der Waals attraction: v(r)= vl(r) +v2(r).(1) The electrostatic term can be calculated approximately from Poisson-Boltzmann theory:279 where E = eOeris the dielectric constant of the solvent (water), !Pois the surface potential of the colloidal particles and k,’ is the Debye screening length of the ions, which are assumed to be monovalent. The above expression for q(r) is expected to be valid when kD(r-ao)>> 1, i.e. in the strong-screening For not too large surface potentials (typically !Po< 25 mV), the hyperbolic tangent in eqn (2) can be linearized and vl( r) reduces to where J = mO~pO*:is the electrostatic coupling constant, x = r/o0 is the reduced pair distance and K = kDa0is the reduced Debye wavenumber.The van der Waals attraction between spherical particles is of the form2 -Ah(x) v2( r)= ~ 12 where A is Hamaker’s constant and 1 1 h(x) =-x2-1 +7+21n(lx 1 =-; x >> 1 3x6 J-M. VICTOR AND J-P. HANSEN 45 The total potential eqn (1) is written in the convenient form u(x)= Jq(x)= J 12J and its extrema [u'(x)= 01 are solutions of the equation exP(Y)-~Y2(l+--)(l+~)(1+~)2=0Y+l (74 where y = K(X- 1) and 24J y=-.AK For sufficiently large values of J and sufficiently weak screening (i.e. for large values of y), the Coulomb repulsion completely masks the van der Waals attraction. As K increases, because of the addition of salt, the Coulomb barrier is gradually reduced and the secondary minimum in u(x)deepens.In the strong-screening regime ( K >> 1), eqn (7a)reduces approximately to exP(Y)-?Y2=0 (76) which implies that cp(x) exhibits a positive maximum at x = xM> 1 ( y = yM)as soon as y > e, and a subsequent secondary minimum at x, > xM( ym> yM) (see fig. 1). Coagulation will be prevented as long as the Coulomb barrier u(x,) is substantially larger than the thermal energy kT." In the following we shall assume, arbitrarily, that the dispersion is charge-stabilized when q(xM)> 10T/ TJ,where T' = J/k, but the results are rather insensitive to the precise value of the assumed potential barrier. Whenever the preceding condition is fulfilled, we replace u(x) by the hard-sphere potential u(x)= afor x < xM,thus introducing an effective h.s.diameter; this is justified, because the high Coulomb barrier will prevent particles from getting as close as xM. In order to calculate the osmotic properties of this model of a charge-stabilized colloid suspension, we now make use of thermodynamic perturbation theory, as developed for simple liquid^.^ This is a reasonable procedure because of the strong analogy between colloidal suspensions and simple atomic liquids, which is illustrated by the liquid-like structure of these suspensions revealed by light-scattering experi- ments." Following Weeks et aZ.'*we separate the effective potential in fig. 1 into its repulsive [ -u'(x)> 01 and attractive [ -u'(x) < 01 parts: v(x)= uo(x)+ w(x) (9) where and w(x)= 0 ;x<s = V(xrn> ;S<x<x, = u(x) ;x>x, where S will be defined in eqn (14).A system of particles interacting uia the purely LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS 20 h” 10--. A 0 -10 Fig. 1. DLVO potential for T, = 60 000 K, TA= 1800 K and K = 230. S = 1 +3.1 / K for T = T, = 331 K. (---) u(x), (---.) uo(x)and (-) w(x). (a)True h.s. diameter, (b)effective h.s. diameter xM and (c)Barker-Henderson diameter S. Note that w(x)is not defined for x < S. repulsive potential vo(x) constitutes the ‘reference system’, while the attractive component w(x) will be looked upon as a perturbation. In a first stage, the properties of the reference system are related to those of an ‘equivalent’ fluid made up of hard spheres of diameter (T which is larger than the effective diameter coxM,because of the steep repulsion between colloidal particles separated by xM<x <x,.In the following we shall denote by rlo = 7rnai/6 the true packing fraction of the colloidal particles and by 7 = q0S3>rlo the corresponding packing fraction of the equivalent hard-sphere fluid, where S is the dimensionless ratio a/a0. This ratio can be determined from a functional Taylor expansion of the Helmholtz free energy in powers of the difference between the Boltzmann factors associated with the reference system and the equivalent hard-sphere fl~id.~,’~ Trunca-tion of the expansion after the first order leads to the following implicit relation for S: Jo* B(X)X~dx =0 where the ‘blip-function’ B(x) is defined by5.” J-M.VICTOR AND J-P. HANSEN where us(x)= +a ;x<s =o ;x>s is the potential of the equivalent hard-sphere fluid, y, (x) = exp [pus(x)]g, (x), g, (x) being the pair distribution function of the hard-sphere fluid for the packing fraction q and /3 = l/kT; note that by convention x = r/uoand not r/u. B(x) is non-zero only over the narrow range xM < x < x,, while the function y,(x) and its derivatives are continuous at contact (x = S). A systematic expansion for S can hence be obtained by expanding x2y,(x) in powers of (x-S).’~ The leading term in this expansion is the density-independent Barker- HendersonS7’’ diameter: (Ts=-=xM+ Je., *m{l-exp[-~uo(x)]}dx (70 XM =l+J+S($)K where terms of order higher than 1/~are negligible in practice, since in the strong-screening regime, relevant for the present investigation, K >> 1 (typically K blo2).Terms beyond the Barker-Henderson diameter in the expansion of S are of order l/~~ or smaller and can hence be safely neglected. This contrasts with the situation in simple liquids where corrections to the Barker-Henderson diameter are generally not negligible;5 the difference can be ascribed to the particular form of the repulsion uo(x),which is much steeper in colloid suspensions when K >> 1 than typical interatomic potentials, as illustrated in fig. 2. Once the equivalent hard-sphere diameter has been determined from eqn (14) for given values of the temperature and of K (i.e.of the salt concentration), the thermodynamic properties of the reference system are identified with those of the hard-sphere fluid of packing fraction r] = qOS3.In particular the Helmholtz free energy per particle of the reference system is given by where u = 7m3/6 is the equivalent hard-sphere volume, h is the thermal de Broglie wavelength and we have used the Percus-Yevick (PY) compressibility equation of state to determine the excess (non-ideal) part of the free energy, which is practically exact in the density range relevant to the present study (77 = 0.2) ;’note that the first two terms on the r.h.s. of eqn ( 15) depend on both density and temperature (uiaqo and S),while the third one depends only on temperature. In a second stage we use thermodynamic perturbation theory to account for the attractive part w(x) of the potential’ between colloidal particles. To first order in this perturbation the free energy per particle is given by16 Eqn ( 16) constitutes the so-called ‘high-temperature approximation’ (h.t.a.) and represents an upper bound to the exact free energy of the system and is known to be very accurate for dense atomic fluids;’ we shall briefly reconsider the validity of LIQUI D-VAPO UR EQUI LI BRIUM IN COLLOIDAL DISPERSIONS A --5 -h" -ly-..h 8 v s--1 2 the h.t.a. for colloidal systems later. The h.t.a. is equivalent to the assumption that the spatial correlations are not affected by the attractive perturbation. We have calculated the integral in eqn (16) with the PY expression for the hard-sphere pair distribution function g,,(x),which is known analyti~ally.'~ Details of the calculation are given in the Appendix; the final result is Iy(71)='y0+w71+"27?2 (174 where the numerical coefficients a, are given in the Appendix, TA=A/kand ym= K(X, -1) ; note that 8depends on the potential parameters K, J and A,but not - - DLVO -5- - - J-M. VICTOR AND J-P.HANSEN on q. The mean-field approximation (m.f.a.) neglects spatial correlations of the reference system in the calculation of the perturbation fl, i.e. it amounts to setting g,(x) = 1 (ideal-gas limit) in the integral of eqn (16). An ambiguity arises in the m.f.a.since the perturbation potential w(x)has no physical meaning inside the hard core, i.e. for x <S. We have adopted the convention that w(x)=0 in that range, which is equivalent to choosing g,(x) = 8(x -S), where 8 denotes the Heaviside step function. With this convention, correlations in the reference system are included at the level of the second virial coefficient for the calculation of the contribution of the perturbation potential w(x) to the total free energy. Under these conditions fl simplifies to where a,.f. is the constant (4+5 In 2)/3 =2.49. The equation of state in the h.t.a. follows by straightforward differentiation of eqn (1 5)-( 17): where the prime denotes differentiation with respect to q and vo=7&'6 is the volume of a colloidal particle.The corresponding m.f.a. expression is The chemical potential df the colloidal particles follows directly, in either the h.t. or m.f. approximations, from the thermodynamic identity: PPO0PP=Pf+--770 By differentiating once more with respect to qowe obtain the isothermal compressi- bility xr.In the h.t.a.: a(nkTxJl =-(PPVO) (3 770 The corresponding m.f.a. expression is much simpler: It is known that the isothermal compressibility is related to the long- w avel engt h limit of the static structure factor S(k),or equivalently of the Fourier-transform of LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS the Ornstein-Zernike direct correlation function, c( k), via5 It is interesting to recall the connections between the h.t.a.and the m.f.a. on one hand and various approximations which can be made for the direct correlation function c(r). The functional Taylor expansion, familiar from the theory of inhomogeneous fluids, leads to the so-called ‘mean-density’ approximation (m.d.a.):‘8719 where c,(r) is the direct correlation function of the reference system, i.e. of a hard-sphere fluid with packing fraction T. The corresponding compressibility is easily calculated by taking the Fourier transform of eqn (25) and the k-+0 limit [see eqn (24)]. The resulting expression for xT is then identical to eqn (22); thus the h.t.a. and the m.d.a. are equivalent, at least at the level of the isothermal compressibility. Neglecting the density dependence of g,( r) in eqn (25) leads to the so-called ‘extended random-phase approximation’ (e.r.p.a.): 19320 Making the further approximation g,( r)= 1 leads to the usual ‘random-phase approximation’ (r.p.a.) which yields a compressibility identical to that obtained in the m.f.a.[eqn (23)]. It is precisely this approximation which was used by Grimson’ to determine the phase diagram of weakly charged dispersions. However, the r.p.a., like the m.f.a., raises the problem of the choice of w(x)inside the core; this problem is automatically settled by eqn (25) and (26) since g,(x) = 0 if x < S. Grimson chose w(x)= u(x,) for all x < S and this choice favours phase separation since it leads to a large negative contribution to the free energy and a correspondingly high critical temperature.As indicated earlier our m.f.a. calculations are based on the assumption w(x)= 0, x < S, which is more consistent with the m.d.a. (or equivalently the h.t.a.) or the e.r.p.a. CRITICAL POINT AND SPINODAL DECOMPOSITION Since the expressions for the total free energy and its derivatives contain both positive (repulsive) and negative (attractive) contributions, we expect a phaseseparation below a critical temperature. The spinodal curve, which encloses the portion of the density-temperature plane where the system is thermodynamically unstable, is the locus of points where the compressibility diverges. According to eqn (24) the (7,T) points which lie on the spinodal curve are solutions of the equation 1 -C(0) = 0.(27) The h.t.a. expression eqn (22) leads immediately to the equation J-M. VICTOR AND J-P. HANSEN where which yields the desired 7-T or qO-T relationship. In practice r] = 70S3,which depends on temperature via S [eqn (14)], exceeds qoonly by a few percent, so that a reasonably accurate spinodal curve can be calculated more simply by replacing 7by q0in eqn (28). The results presented below are based, however, on calculations which distinguish between q0 and 7. The top of the spinodal curve is the critical point above which no phase separation occurs. The coordinates (q,,T,) of the critical point are determined by solving the coupled equations 1-E(0) = 0 (304 a -[1 -qo)] =0. a70 In the h.t.a., eqn (30a) leads back to eqn (28), while eqn (30b) can be cast in the form where 7(5 +27-8q2+5 73/2) q2(7-77 -4T2+4q3) Q(7)= 47)+ 1 +77 a’(77)+ 2( 1 +77) QlY7) (32) Eqn (3 1) and (32) are solved numerically for fixed values of the potential parameters J (or TJ),A (or TA)and K.Values obtained for some typical values of the potential parameters are listed in table 1. Note that eqn (31) and (32) are solved for a fixed value of the reduced screening wavenumber K. The corresponding monovalent salt concentration then follows directly, for a given value of the diameter o0,once the critical temperature T, has been determined. The spinodal curve is determined next by solving eqn (28) for several fixed temperatures 273 < T/K < T,;the physical roots of the equation yield qvsand qls, the packing fractions corresponding to the low-density (‘vapour’) and high-density (‘liquid’) branches of the spinodal curve.The calculations can be performed at fixed values of K, which is practically equivalent to working at constant salt concentration c, since any variation in temperature is cancelled by the corresponding variation of the dielectric constant E which varies essentially as I/ T. Two typical spinodal curves are shown in fig. 3 and 4. From an experimental point of view it is more convenient to work at constant temperature and observe the phase separation induced by increasing the salt con- centration c and hence K. Eqn (28) must then be solved for fixed values of TA, TJ and T and the corresponding critical point is obtained by solving simultaneously eqn (28) and (32).This yields critical values 7,and K, (and hence a critical salt concentration c,). Eqn (28) is then solved for K > K,, yielding q,, and qIsand in this way an ‘inverted’ spinoda17 is derived, as illustrated in fig. 5 and 6. Anysuspension ‘quenched’ into a thermodynamic state inside the spinodal curve will undergo spinodal decomposition. < $ 0 C 7 m Table 1. Critical temperatures and packing fractions predicted for some typical values of the diameter, u0,surface potential, To,and salt 05concentration, c. Two values of Hamaker's constant, A, are included. c: W Iuohm *o/mV TJK AJ T,/K ~,/moldm-~ K TJK T,",f,/K n,/ cmP3 7 770, 2 I0.6 25 6.0 X lo4 2.5 X lop2' 1800 1.3 x 230 33I 134 0.259 2.3 x 10I2 I .o 20 6.0 X lo4 5.0 X 3600 1.7 x 10-~ 135 323 184 0.177 3.4 X 10" z 01.o 25 105 2.5 x 1800 6.7 X 270 340 136 0.263 5.0 x 10" 0 1.o 25 I 05 5.0 x lop2' 3600 2.4 X lop3 160 337 189 0.181 3.5 x 10" z 0 J-M.VICTOR AND J-P. HANSEN 3 40 &--. h 3 00 270 0.1 0.2 0.377 Fig. 3. Spinodal (---) and coexistence (-) curves T(7)for TJ= 70 000 K, TA=4000 K and K = 135. 330 300 I I/ \ \ \I \270 0 0.1 0.2 0.3 77 Fig. 4. Spinodal (---) and coexistence (-) curves T(7)for TJ= 60 000 K, TA= 1800 K and K = 230. The calculations are of course much easier in the mean-field approximation. Eqn (30a,b) now lead to the relations LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS COAGU LAT I0NI -0 0.1 0-2 0.3 7) Fig.5. Spinodal (---) and coexistence (-) curves K( 7)for T, =70 000 K, TA =4000 K and T = 304 K. COAGULATION 250 1I \ // 0 0.1 0.2 0.3 0.4 0.5 77 Fig. 6. Spinodal (---) and coexistence (-) curves K (7)for T, = 60 000 K, TA = 1800 K and T=310K. 4(1+2d~-~~(6-+ d3(1-TO -ay,.f.)[(l T q-4qO)1 (33b) which replace eqn (29) and (31). Dividing eqn (330) by (33b) and neglecting the small difference between q0 and q = qOS3we find that qCis the positive root of the quadratic equation 1-77-6q2=0 (34) i.e. qr.f.=0.1287, a universal value.' The critical temperature is The m.f.a. critical temperatures are much lower than the h.t.a. values, as can be seen from table 1; note also that the q?.f.is always lower than the corresponding h.t.a. critical packing fractions. J-M. VICTOR AND J-P. HANSEN The spinodal curve in the m.f.a. is obtained by solving eqn (33a) for T < T:.f.. Ne lecting once more the small difference between qoand q,we find that qEf.and qz‘. are the roots of the equation so that the spinodal is a universal curve in the m.f.a. Working at constant temperature an ‘inverted’ spinodal can be calculated from eqn (33a) by varying the salt concentration and hence K and t(K) according to eqn (17b). The resulting spinodal curve is not universal since it is given by where K, is the solution of eqn (35) with Ty.f*replaced by the fixed temperature T. LIQUID-VAPOUR COEXISTENCE The coexistence curve of the low-density (‘vapour’) and high-density (‘liquid’) phases below T, is calculated in the usual way by equating the osmotic pressures and chemical potentials of both phases: PP(qv) uo = PP(71) uo (384 PPhV) = PP(T1).(386) This set of coupled equations is solved numerically for qvand ql,given a temperature T< T,. In the h.t.a., PPvo and Pp are given by eqn (15), (17), (19) and (21). The calculation may be done at constant K, which is again equivalent to working at constant salt concentration in a way quite analogous to the determination of the spinodal curves. Typical examples of T(q)coexistence curves are shown in fig. 3 and 4. As usual the area between the coexistence and spinodal curves corresponds to metastable thermodynamic states (supercooled ‘liquid’ or ‘vapour’).Again the experimentally more relevant situation is to work at constant temperature and vary the salt concentration. The corresponding ‘inverted’ K ( q) coexistence curves are shown in fig. 5 and 6. Of course when K increases because of the addition of salt, the height of the Coulomb barrier decreases, as is easily understood from the analytic form of the DLVO potential.6 When this height drops below some critical value, corresponding to K = K,,,, coagulation sets in (coagulation is to be understood in the sense of irreversible coagulation) ; above we adopted the energy criterion u(x,) =: 10kT,which determines K,,, for given T, T, and TA. The ‘inverted’ coexistence curves are hence bounded above by K = K,,,, higher values of K corresponding to the onset of coagulation; the boundary is of course not very sharp since its location depends on the precise value adopted for the critical energy ratio u(x,)/kT. As can be seen from the coexistence curves in fig.3-6, the packing fraction of the ‘liquid’ phase can become quite large (qB 0.2). In fact for q > 0.5 the colloid suspension is expected to crystallize like the underlying ‘equivalent’ hard-sphere fluid. 35 This raises the interesting possibility of observing a solid-liquid-vapour triple point in colloid suspensions, just as in ordinary atomic or molecular liquids, provided the temperature of this triple point falls into the physically accessible range T> 273 K. The calculation of the coexistence curve is again much simpler in the m.f.a.Substituting eqn (15), (18), (20) and (21) into the two coexistence relations eqn LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS (38) and dividing the two resulting equations, we obtain first a symmetric relation between ql and 7,: where and we have once more neglected the small difference between 7 and vo. On the other hand the equality of chemical potentials eqn (38b) together with eqn (35) yields a second equation Eqn (39) yields a universal relation q,(7,)and eqn (40) leads then to a universal coexistence curve 7(T/ T,) where 7 is q1 or 7,. The m.f.a. results are only of qualitative relevance, since the coexistence curves are shifted to considerably lower temperatures and densities compared with the predictions of the more accurate h.t.a. PHASE EQUILIBRIUM VERSUS COAGULATION As stressed earlier, the critical coordinates (v,, T,) determined from eqn (28) and (31) in the h.t.a.for a given set of potential parameters TA,TJ and K may very well be unphysical. More precisely the critical temperature T,may either fall below the freezing point of the solvent (T,< 273 K) or, on the contrary, be too high compared with the repulsive potential barrier [1OkT,> u(XM)], thus leading to irreversible coagulation. For that reason it is instructive to consider the inverse problem, Le. to ask which values of the potential parameters TA,T' and K will lead to a prescribed, physically acceptable value of T, while maintaining stability against coagulation.We proceed by fixing T, at a value To ( = 300 K say) and also T' The minimum barrier condition which we have adopted reads: where yM is the solution of eqn (7b), and for the present purpose the approximate form of the DLVO potential, valid for x -1<< 1, is sufficiently accurate. Eqn (41) determines the minimum value of y below which coagulation sets in. Eqn (28) and (31) for T, and 7,depend parametrically on TA,7" and y. In the present context they are solved to yield TAand 7,for fixed values of To, TJ and y = y( To/TJ).A similar solution can be obtained from eqn (33) in the m.f.a. The resulting TA(TJ) relationship is shown in fig. 7 for To= 273 and 300 K. The corresponding curve separates the ( TA,TJ)plane into a lower region where only coagulation occurs at the prescribed temperature To,and an upper region where a reversible 'liquid-gas' separation takes place with a critical temperature T,> To.The predictions of mean field theory are seen to differ considerably from the more accurate h.t.a. to which we shall henceforth restrict ourselves. For TA< 1000 K, the latter predicts values J-M. VICTOR AND J-P. HANSEN --LIQUID-GAS TRANSITION !4 \ c--COAG U LAT ION I I I I I I I I I I o 103 TAI K 1OL Fig. 7. Separation of the ( TA,T,) plane in the mean-field (upper curves) and m.d.a. (lower curves) theories. The stability curves are drawn for two temperatures: (-) To= 273 K and (---) To= 300 K. The regions labelled ‘liquid-gas transition’ and ‘coagulation’ refer only to the lower pair of curves (m.d.a.) ; the corresponding regions in the mean-field approximation lie similarly above and below the upper pair of curves.The behaviour of the stability curve for TA < 1000 K appears to be unphysical since the corresponding critical packing fractions become too high ( > 0.5). GO 0 350 &2 300 270 Fig. 8. Variation of T, and 7,with K for fixed values of TA and Tp (1) T, = 70 000 K and TA= 4000 K; K,~” = 126 and K,~, = 144. (2) T, = 60 000 K and TA = 1800 K; K~~~ = 209 and aK,~, = 259. For K > K,,,, the suspension coagulates. For K~~~ < K < K,,, and To< Tc(~), liquid and a gas phase coexist in the suspension. LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS of 7,) 0.5, indicating that for too weak van der Waals attractions the ‘liquid’ phase no longer exists, since, as we have already indicated, the ordered (‘solid’) phase is expected to be stable beyond an effective packing fraction 7 -0.5.Given a ( TA,T,) point above the stability curve obtained for a given temperature To,the corre-sponding critical temperature T, under the minimum barrier condition eqn (41) will be higher than the temperature To for which the stability curve has been drawn. Eqn (41) determines the minimum value of y and hence the maximum value of the screening wavenumber, K,,,, ie. the highest concentration of added salt beyond which the suspension is expected to coagulate. For lower concentrations, such that K < K,~,, T, will decrease because the increasing range of the Coulomb repulsion reduces the depth of the secondary minimum in the DLVO potential.The situation is illustrated in fig. 8, which shows the variation of T, and 7,with K, in the range 273 < T,< ,Fax = T,(K,~~),as calculated from the h.t. approximation for a given set ( TA,TJ). A very important point illustrated in fig. 7 is the existence of a minimum value of T, (ca. 40 000 K) below which no ‘liquid-gas’ coexistence can be observed at room temperature. Since T,Eao*& this implies that for a fixed surface potential q,the colloidal particles must exceed a minimum size a. for the phase transition to be observable. Explicitly, under room-temperature conditions ( E = 78): As an illustration, if To= 25 mV, a:’” =r 4000 A.Note that for significantly higher surface potentials, details of the present calculation would have to be modified, since the (T-dependent) eqn (2) of the double-layer potential would have to be used, rather than the simplified eqn (3). This is not a serious limitation, however, since in the strong screening regime zeta potentials rarely exceed 30-40 mV. DISCUSSION We have shown, on the basis of a reasonably realistic statistical-mechanics calculation, that a monodisperse suspension of spherical colloidal particles in water should be expected to undergo a reversible ‘liquid-vapour’ separation below a salinity-dependent critical temperature. The transition is expected to take place for physically reasonable values of the particle diameter a,, surface potential Toand Hamaker constant A, and should hence be observable in the laboratory.In fact this ‘liquid-gas’ transition may already have been detected by Kotera et aL,3 who observed a reversible aggregation of charged colloidal particles into dense droplets, which can be redispersed by mere stirring; they refer to their findings as a ‘reversible coagulation’ into the secondary minimum of the DLVO potential. Their experimental conditions (particles ca. 1 pm in diameter, a zeta potential ca. 25 mV and salt concentrations ca. lo-* mol dm-3) are compatible with the range of physical parameters for which our theory predicts a ‘liquid-vapour’ transition. There is clearly a case for more experimental work in that direction. Particular precautions will be needed in controlling the salt concentration, since the ‘liquid-vapour’ transition is predicted to occur only over a relatively narrow range of concentrations, just prior to coagulation.However, the distinction between irreversible coagulation and reversible phase separation should be clearly visible. We also recall that according to our analysis the colloidal spheres must be sufficiently large (typical diameters ca. 1 pm) for the transition to occur above the freezing point of water. J-M. VICTOR AND J-P.HANSEN The liquid-vapour coexistence and spinodal curves which we have calculated for charge-stabilized colloidal suspensions bear a close resemblance to the corre- sponding diagrams for simple atomic or molecular systems (i.e.argon). There is, however, one clear difference: the critical packing fraction is significantly larger in the colloidal system (77, ==: 0.2) than in most atomic systems ( 77, = 0.19, and turns out to be sensitive to the values of the potential parameters, particularly of TA(see table 1). Moreover, the critical ratio Pc/(n,kTc) is of the order of 0.5 in the colloidal suspension compared with the nearly universal value 0.3 in simple liquids. We attribute these differences to the different qualitative behaviour of the DLVO poten- tial compared with a typical interatomic potential such as the Lennard-Jones (LJ) potential (see fig. 2). Compared with the LJ attraction the DLVO potential well is much deeper and narrower.Whereas the critical thermal energy kT is of the order of the depth of the LJ potential (i.e. T: = kT/cL,==:l), it represents only ca. 15% of the depth of the secondary minimum in the DLVO potential. Furthermore, the DLVO potential is practically zero (compared to its depth) at x = 1.5, whereas the LJ potential is still only ca.; of its maximum depth at that distance. These qualitative features of the DLVO potential raise some interesting theoreti- cal problems which we are planning to investigate. First, the h.t.a. is known to be an excellent approximation in the dense (‘liquid’) phase, but fluctuation corrections to the free energy beyond the h.t.a. become important in the low-density (‘vapour’) phase.’ It is difficult to know a priori whether the peculiar shape of the DLVO potential (in particular the narrow shape of the secondary minimum) will enhance or reduce the importance of these higher-order terms.This point can be clarified, e.g. on the basis of the standard integral equations of the theory of liquid^,^ which are well adapted to the low-density phase. The phase separation may also be sensitive to the unavoidable polydispersity of colloid suspensions. Even if experi- mental precautions can keep the polydispersity at the low-percentage level, the difference in diameters will be comparable to the width of the attractive ‘skin’ of the DLVO potential and may lead to sizeable effects which ye are planning to investigate along the lines of ref. (21) and (22).Finally, the correlation length is known to diverge at the critical point, but in view of the large size of the colloidal particles, the correlation length may become rapidly of the order of the size of the colloidal sample as the critical point is approached, which may lead to observable deviations from the universal scaling laws in the critical region, because of the suppression of very-long-wavelength density fluctuations. APPENDIX Here we give the details of the calculation of the h.t.a. contribution to the free energyf,, leading to eqn (17). We first note that the distribution function go refers to a system of effective hard spheres of diameter So, so that the distribution at a distance r is given by g, (r/SU)and However, g, (x) hardly varies with distances of the order S -1 = O(l/~),so that an accurate expression for fi is given by LIQUID-VAPOUR EQUILIBRIUM IN COLLOIDAL DISPERSIONS Then using eqn (1 1) and (16) we obtain Neglecting terms of order 1/~we obtain, using eqn (8): The last integral I3 is more involved because of the singularity of h (x) for x = 1.More precisely 1h(x) =-2(x -I) +2 In (x -1) +H(x) where 1H(x)= -~ +-+2 In (x + 1) -4 In x 2(x+1) x2 is the regular part of h(x) in the vicinity of 1. Putting G,(x) = xg,(x) we obtain the second term is rewritten as G,(x)(:+l(x- 1) In (x- l)+- 1 2(x -1) Putting X(X)=XH(X)+$ +~( X-I)In(x-1) we obtain = J1+J2+J3+J4. 644) As the integrands of J, and J2 are regular for x = 1 we obtain a good approximation to J, and J2by replacing x, by 1.On the other hand, J3 = -G, ( 1)[$ In (x, -1) +21= G, (1) In (K /y ,) -21. (A5) The last term J4 can be estimated by taking g,(x) = 1 because of its small value (cu. lop2). Then we obtain I3 in the following way: '3 = -g, ( 1)[In (K /Ym) -(7)I (A6) J-M. VICTOR AND J-P. HANSEN with [G,(x)-G,(l)]ln(~-1)dx I X x-1 I*1 G”(x)-G’(1)dx-2 i Gq(x)X(x)dx-2[2 g,(x)h(x)x’dx). a(q)is a slowly varying function; we thus computed a parabolic fit to the numerical results obtained for several values of 7: 477)= ‘Yo+a,rl+ad with a. = 2.49 a, = 0.86 a2= 1.21. Note that The final result reads: with For a review see P. Pieranski, Contemp. Phys., 1983, 24, 25.2 E. J. W. Verwey and J. Th. G. Overbeck, Theory of Stability of Lyophobic Colloids (Elsevier, Amsterdam, 1948).3 A. Kotera, K. Furusawa and K. Kudo, Kolloid 2. 2. Polym., 1970, 240, 837. 4 J. A. Long, D. W. J. Osmond and B. Vincent, J. Colloid Interface Sci., 1972, 42, 545. 5 J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic Press, London, 1976).6 J. M. Victor and J. P. Hansen, J. Phys. Lett. (Paris), 1984, 45, L207. 7 M. J. Grimson, J. Chem. Soc., Faraday Trans. 2, 1983, 79, 817. H S. A. Safran and L. A. Turkevich, Phys. Rev. Lett., 1983, 50, 1930. 9 G. M. Bell, S. Levine and L. N. McCartney, J. Colloid Interface Sci., 1970, 33, 335. 10 G. R. Wiese and T. W. Healy, Trans. Faraday SOC.,1970, 66, 490.I1 J. C. Brown, P. W. Pusey, J. W. Goodwin and R. H. Ottewill, J. Phys. A, 1975, 8, 664; W. Hartl, H. Versmold, U. Wittig and V. Marohn, Mol. Phys., 1983, 50, 815. 12 J. D. Weeks, D. Chandler and H. C. Andersen, J. Chem. Phys., 1971, 54, 5237. 13 H. C. Andersen, J. D. Weeks and D. Chandler, Phys. Rev. A, 1971, 4, 1597. 14 L. Verlet and J. J. Weis, Mol. Phys., 1972, 24, 1013. 15 J. A. Barker and D. Henderson, J. Chem. Phys., 1967, 47, 4714. 16 R. Zwanzig, J. Chem. Phys., 1954, 22, 1420. 17 W. R. Smith and D. Henderson, Mol. Phys., 1970, 19, 411. 18 R. L. Henderson and N. W. Ashcroft, Phys. Rer. A, 1976, 13, 859. 19 R. Evans and W. Schirmacher, J. Phys. C, 1978, 11, 2437. 20 R. E. Jacobs and H. C. Andersen, J. Phys. Chem., 1975, 10, 73. ‘I P. Van Beurten and A. Vrij, J. Chem. Phys., 1981, 74, 2744. 22 L. Blum and G. Stell, J. Chem. Phys., 1980, 71, 42; J. J. Salacuse and G. Stell, J. Chem. Phys., 1982, 77, 3714. (PAPER 41543)

ISSN:0300-9238    

DOI:10.1039/F29858100043

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. Soc., Faraday Trans. 2, 1985, 81, 63-75 Experimental Sulphur-33 Nuclear Mag net ic Resonance Spectroscopy BY PETERS. BELTON" Agricultural and Food Research Council, Food Research Institute, Colney Lane, Nor wich NR4 7UA AND I. JANECox AND R. K. HARRIS* School of Chemical Sciences, University of East Anglia, Norwich NR4 7TJ Received 13th April, 1984 The potential of sulphur-33 n.m.r. spectroscopy for chemical applications is explored. Data are presented for a range of compounds. Spectra have been obtained using both high-power and high-resolution systems. Moderately high field strengths were used (7.05 and 4.70T), together with a special pulse sequence to reduce the effects of the dead-time. Lines as wide as 5 kHz can be readily observed. A scalar sulphur-hydrogen coupling is reported for the first time.Sulphur-33 is the only naturally occurring isotope of sulphur with a non-zero spin (I =;). Since it has a moderate quadrupole moment (-5.5 X m2), a low natural abundance (0.76%) and a low magnetogyric ratio (2.055 x lo7 rad T-' s-')' it is clearly an intrinsically insensitive nucleus. However, despite these drawbacks, a number of studies using sulphur n.m.r. spectroscopy has been reported.2 The early work used continuous-wave spectrometers and concentrated on broad-line ~tudies.~More recent investigations, using Fourier-transform instruments (generally operating at modest field strengths in the range 1.88-5.87 T, corresponding to frequencies from 6.1 to 19.2 MHz), have shown that a few types of sulphur environ- ment give narrow lines and can be readily Such species include the sulphate ion, the tetrathiomolybdate ion, sulphones and sulphonic acids.The variation in chemical shift with substituent for the sulphones, for example, has been considered in detail.4" However, the short free-induction decays associated with the broader peaks are often obscured by the ringing typical at the low frequencies used. The chemical shift data from the sulphur compounds previously reported are summarised in fig. 1. The limited success of previous workers encouraged us to undertake a systematic study of the feasibility of using sulphur-33 n.m.r. spectroscopy to solve problems in food chemistry, where sulphur-containing molecules occur in important flavour compounds and naturally occurring toxicants.In order to do this we have surveyed a wide variety of compounds using moderately high magnetic field strengths, short r.f. pulses and a special pulse sequence to minimise the effects of acoustic ringing. These conditions have enabled us to observe lines up to 5 kHz in width with relative ease and to extend the range of compounds that have been characterised by sulphur n.m.r. spectroscopy. A sulphur-hydrogen scalar coupling is reported for the first time, and was observed using a conventional solution-state Fourier-transform spec- trometer configuration. 63 64 EXPERIMENTAL 33sN.M.R. SPECTROSCOPY 400 200 0 -200 -LOO -600 1 I I I I I -Mo S, OtZ,(aq) -WS,O~I, (aq) Valenc y I SUBSTITUTED THIOPHENES -I ZnS (s)II IR2Sa I RSH I EtNCS I NqS, laq) 1 cs2 L I Et,S, Valen cy I Me2SOaIV f Val en c y VI 400 200 0 -200 -400 -600 8, (PPm) Fig.1. Sulphur-33 chemical shift data, giving a summary of the data reported in the literature and in this study. (a)The results obtained in this study (illustrated) lie ca. 75 ppm to higher frequency than the literature values (not ill~strated).~' EXPERIMENTAL The majority of samples were obtained from commercial manufacturers and were used without further purification. The thiotungstate and thiomolybdate species were gifts from M. J. O'Connor (La Trobe University) ;the ring sulphones (except sulpholane) were sent by J. Lambert and E. Block; di-t-butyl sulphone, t-butyl phenyl sulphone, di-n-butyl sulphone and p-chlorophenyl methyl sulphone were prepared at the University of East Anglia by J.Tarbin. The experiments were carried out using Bruker CXP 300 and CXP 200 spectrometers, operating at 23.009 MHz (7.05 T) and 15.34 MHz (4.70 T), respectively. The CXP 300 instrument was set up for broad-line studies (for linewidths> 50 Hz). A high-power solenoid coil was used to maximise the signal-to-noise ratio' and to obtain short 90" pulse lengths (typically 11 ps, which is equivalent to ca. 1 kW of r.f. power). Under these conditions the total dead-time due to receiver recovery and acoustic ringing is ca. 60 PS.~ Signals from lines wider than 1 kHz are seriously affected, and lines greater than 4 kHz in width are virtually lost unless steps are taken to improve matters (see below).The CXP 300 instrument accommodates a 10 mm sample tube containing a sample volume of 2 cm3. It was necessary to seal the sample tubes because many of the samples are toxic. Since the sample tube lies in a horizontal position in the solenoid coil this proved difficult, although feasible. In general, a large number of transients must be acquired to obtain an adequate signal-to-noise ratio. However, relaxation times are generally short for sulphur-33 ; since T, and T, are equal for most species and the linewidths are often >100 Hz both TI and T2are <3 ms. Therefore a rapid recycle time can be used, typically 50 ms, which is the P. S. BELTON, I.J. COX AND R. K. HARRIS TRANSMITTER RECEIVER 90: goyx TRANSMITTER Fig.2. Pulse sequences to minimise the effects of spectrometer dead-time. Typically 7=0.5 ps and T,= 50 ms. shortest recycle time consistent with the avoidance of instrument duty-cycle problems. Spectra were obtained (without field-frequency locking) at ambient probe temperature (294 K), and the samples were run as neat liquids (where possible) or concentrated solutions. The CXP 200 instrument was optimised for high-resolution solution-state studies. The standard high-resolution probe, requiring 10 mm sample tubes, was used. The 90" pulse length was 80 ps. The effects of acoustic ringing were reduced by increasing the delay before acquisition, without significant loss in the signal intensity from the relatively sharp signals studied.A deuterated solvent was used as an internal lock, and spectra were obtained at ambient probe temperature (294 K). Chemical shifts were measured relative to external 2 mol dm-3 aqueous caesium sulphate solution by the replacement technique, which was shown to give closely comparable results to a concentric arrangement. No bulk susceptibility corrections were made. The chemical shifts are reported using the high-frequency-positive convention.' The chemical shift of the sulphate ion has previously been reported to be independent of counter-ion and concentra- ti~n,~~although we have shown there is a small effect of these variables.8 The linewidth is also dependent on counter-ion and PH.~ It is narrow (6.5 Hz) for a 2 mol dmP3 aqueous solution of caesium sulphate at pH 7.5.However, the signal lies to the high-frequency end of the sulphur chemical-shift range, and the majority of species therefore have a negative chemical shift. The use of the sulphate ion as a reference is consistent with the practice of most other workers, although carbon disulphide has been occasionally used as the refer- en~e,~,~~particularly for the continuous-wave experiment^.^ EXPERIMENTAL 33sN.M.R. SPECTROSCOPY 1 I 1 -20kHz Fig. 3. Spectra of neat thiophene acquired (a)with and (b) without the RIDE sequence.'" Same vertical scale, no data manipulation. Spectrometer frequency 23.009 MHz;90" pulse length 11-25 ps; number of transients 20 000; recycle time 0.1 s; spectral width 125 kHz; dwell time 4 ps; 512 points acquired for each transient.Spectrum (b) was acquired using a simple repetitive series of 90" pulses. Quadrature phase cycling was not used. Various pulse sequences can be employed which minimise the ring-down.' The basic pulse sequence is shown in fig. 2(a)and relies on the fact that the ring-down following an r.f. pulse is essentially independent of any previous pulses, whereas the nuclear induction signal following an r.f. pulse depends critically on these. Under the assumption that the ring-down from a 180" pulse after a time 7 can be ignored to a first approximation, the sequence in fig. 2(a) yields positive ring-down and positive signal from the first half of the pulse but positive ring-down and negative signal from the second half.Hence the effects of ring-down can be reduced or eliminated by an addition-subtraction routine. Such pulse sequences do not eliminate the ringing completely because the ring-down following a pulse is not exactly reproducible. Our best results have been achieved with what we refer to as the RIDE (RIng-Down Elimination) sequence'" [fig. 2(b)],which additionally cancels out the ring-down following a 180"pulse. Quadrature detection combined with quadrature phase routing and cycling was normally used. This means that each complete cycle requires 24 pulses. P. S. BELTON, I. J. COX AND R. K. HARRIS 1kHz Fig. 4. Sulphur spectra acquired using a single 90" pulse. (a)Neat sulpholane; dwell time 250 ps; 128 points acquired.(b) Neat carbon disulphide; dwell time 17 ps; 128 points acquired. Spectrometer frequency 23.009 MHz; 90" pulse length 11.25 ps. Note the difference in frequency scales between the two spectra. A study of the relationship between pulse length and ring-down showed that the magnitude of the ring-down increases with pulse length. We thought that some advantage might be gained from using pulses shorter than 90" in order to obtain less ring-down. However, it is the sum of the ring-down following 8, and 8-, pulses that is crucial, and this appears to be approximately independent of pulse length. Indeed a 90" pulse gives optimum signal intensity, since relaxation times are short." In the RIDE sequence deviations of up to 5% from the optimum inversion pulse length are not significant. The relative phases of the X, -X, Y and -Y quadrature pulses are crucial, however, and there is significant loss when the pulses are not orthogonal. When the RIDE sequence is used the effective dead-time is reduced from 60 to 20 ps, and we have been able to observe signals of 5 kHz with ease.The importance of minimising the ring-down is illustrated in fig. 3 by a spectrum of thiophene acquired with and without RIDE. RESULTS The number of transients required to obtain an adequate signal-to-noise ratio varies considerably from compound to compound. In favourable cases a spectrum can be obtained by Fourier-transforming a single transient; fig. 4 shows such spectra for neat carbon disulphide and neat tetrahydrothiophene- 1,1-dioxide (sulpholane), I. On the other hand, a spectrum of dimethyl sulphoxide with a signal-to-noise ratio EXPERIMENTAL 33sN.M.R.SPECTROSCOPY Fig. 5. Typical wide-line spectra. All spectra were acquired with RIDE. Spectrometer frequency 23.009 MHz. No further data manipulation. (a)Carbon disulphide (neat); number of transients 2500; recycle time 0.1 s; spectral width 30 kHz; dwell time 8.7 ps; 512 points acquired. (b)Thiophene (neat) ;.number of transients 20 000; recycle time 0.1 s; spectral width 125 kHz; dwell time 4.1 ps; 512 points acquired. (c) Dimethyl sulphoxide (neat); number of transients 780 000; recycle time 0.1 s; spectral width 100 kHz; dwell time 5 ps; 80 points acquired.of 20 cannot be obtained in <SO00 transients. Fig. 5 illustrates typical broad-line spectra obtained using the RIDE sequence, including the signal from dimethyl sulphoxide after 780 000 transients. It is interesting to note that the spectrum from a single transient of carbon disulphide obtained in 0.1 s in a field of 7.05 T compares P. S. BELTON, I. J. COX AND R. K. HARRIS Table 1. Sulphur-33 chemical-shift and linewidth data" concentration chemical shift linewidth compound /mol dmP3 solvent (ppm) / Hzb caesium sulphate ammonium sulphate 2 2 water water 0 -0.8 6.5 (4~1) 3.5 sodium thiosulphate' 2 water 33.5 (34.5) 4f 37 (36) sulpholane, Id,' neat acetone 36.7 (37) 4a 32f (50) butadiene sulphone, IId9" dimethyl sulphoned+ diphenyl sulphoned*' di-t-butyl sulphoned*' t-butyl phenyl sulphone 2 1 1 0.6 0.8 acetone acetone acetone acetone acetone 26.5 (28) 4n -13.6 (-13) 4a -24.5 (-21) 4a 27.1 (33) 4a 5.9 18f (50) 42 (120) 46 (160) 8Sf (50) 144 di-n-butyl sulphoned7' 1.5 acetone 0.4 (3.0) 4a 52 (180) p-chlorophenyl methyl sulphone 1 acetone -19.4 87 sodium methyl sulphate dimethyl sulphate sulphuryl chloride p-toluene sulphonic acid' benzene sulphonic acid L-cysteic acid' g neat neat 2 2 1 water --water water water -10 -12.6 -47 -1 1.9 -10 (-9) 4e -12 (-10) 4e 700 1400 600 65 (90) 24 29 (80) dimethyl sulphoxided carbon disulphide ethyl isothiocyanate butyl mercaptan tetrahydrothiophened neat neat neat neat neat -20 (-100) 3c -333 (-33 1) 4d -340 -415 -354 (-422) 3c 4900 (2600) 350 (360) 4300 2100 4800 (2600) thiophened neat -119(-113)3c 1450 (620) ammonium polysulphide water -584 2200 cyclic sulphones, (CHdn-, so2 n=4 0.9 chloroform -2.3 15 n = 54' n = 6"' 0.9 0.9 chloroform chloroform 34.7 (37)4" -11.4(-11)4" 15 (50) 44 (50) n=ll 0.2 chloroform 6.0 60 thiotungstate anions,h WSfl04-HI'- water n=4 159 13 n=3 57 140 n=2 -56 220 n=l -186 300 thiomolybdate anions,h [MoS 04-J-n=4 n=3 water 345 (343) 4g 240 38 (40) 170 n=2 123 250 n=l -25 200 Literature value in brackets.Defined as the full width at half height. 'No adjustments to literature chemical shifts have been made even though 4 mol dmP3 ammonium sulphate (rather than caesium sulphate) was used as reference, but we note a variation in shift with counterion, pH and concentration.The chemical shift was initially reported using a different reference. 'The literature values were obtained using different solvent conditions. When broad-band proton decoupling is used linewidths become 3.3. Hz for 2 mol dm-3 tetramethyl- ene sulphone in acetone, 7 Hz for 2 mol dm-3 butadiene sul hone in acetone and 6 Hz for 1 mol dm-3 dimethyl sulphone in acetone. Not measured. 'Results from ref. (1 I). EXPERIMENTAL 33sN.M .R. SPECTROSCOPY Table 2. Variation of relative chemical shift and linewidth with concentration for sulpholane in acetonen concentration chemical shift linewidth /mol dm-3 (PPm)b7c /Hzb 1 -1.7 17 2 -1.6 14 3 -1.2 16 4 -1.1 19 5 -0.9 16 6 -0.8 17 7 -0.6 19 8 -0.3 20 9 -0.2 24 10 0.0 32 a Data derived from measurements using the CXP200 spectrometer.Estimated error in chemical shifts, 0.1 ppm; estimated error in linewidths, 2 Hz. Relative to neat sul- pholane, which is ca. 10 mol dmP3. favourably with the spectrum obtained using a continuous-wave instrument3c operat- ing at 1.4T after a number of hours (88 scans). A general survey of sulphur-33 chemical shifts is given in table 1 and summarised in fig. 1. Careful choice of solvent and dilution is particularly important for some samples. In general, the spectra of the sulphones show a marked solvent dependence. The linewidth is especially sensitive to the nature of the solvent.For example, the linewidths for 5 mol dmP3 solutions of sulpholane in acetone and water are 16 and 60 Hz, respectively. A shift difference of 6.5 ppm is observed between 5 mol dm-3 solutions of sulpholane in water and dioxane. Table 2 shows how the chemical shift (quoted relative to that for neat sulpholane) and linewidth vary with concentra- tion of sulpholane in acetone. The chemical shift in the sulphone series, RR'S02, varies by at least 60ppm according to substituent, and fig. 6 shows that the signals for a mixture of sulphones may be well separated. This confirms a suggestion that sulphur-33 n.m.r. spectroscopy of sulphones ma be used for quantitative analysis, particularly of sulphones originating from oils+ Reference to table 1 shows that the chemical shifts of saturated-ring sulphones are sensitive to ring size.The present chemical-shift range is 930ppm. The extremes are given by ammonium polysulphide solution, which has a shift of -584ppm and aqueous ammonium thiomolybdate solution, which has a shift of 345 ppm. A number of trends are immediately obvious, although any analysis is restricted by the small volume of data currently available. Without the transition-metal complexes the shift range would be reduced by 300 ppm. The large shift of the tetrathiomolybdate ion may be indicative of the mixing of the sulphur and metal d orbitals. The resonance moves 123 ppm on average to lower frequency as a sulphur is replaced by oxygen, and when the molybdenum is replaced by tungsten the resonance moves to lower frequency by ca.180 ppm throughout the series." The position of the polysulphide ion at the lowest end of the frequency range can be rationalised, since it has a negative charge and is subject to no electron-withdrawing effects. Four distinct regions exist, at present, between these extremes (in order of increasing frequency): (i) when the sulphur is singly bonded, (ii) when the sulphur is multiply bonded, P. S. BELTON, I. J. COX AND R. K. HARRIS -50 40 30 20 10 0 -10 -20 -30 -40 8 (PPm) Fig. 6. Spectrum from an equimolar (0.4 mol dm-3 each component) mixture of butadiene sulphone, dimethyl sulphone and diphenyl sulphone (with chemical shifts decreasing in this order) in acetone.RIDE; spectrometer frequency 23.009 MHz; number of transients 18 500; recycle time 0.2 s; acquisition time 94 ms; spectral width 5400 Hz; 1024 points acquired. The integrated trace is also shown. (iii) when sulphur is participating in a delocalised system and (iv) when sulphur is bonded to one or more oxygens. Anomalous high-frequency shifts have been observed for carbon- 13 and other nuclei that are CY to the sulphonyl or carbonyl groups in four-membered rings.12 The anomaly is characterised by deviations in chemical-shift plots of the function- alised series against the parent hydrocarbons. The sulphur in the cyclic sulphone series is certainly sensitive to the ring size (table I), and the variation of the chemical shift with ring size may be related to the above-mentioned chemical-shift anomaly for four-membered rings.However, the corresponding ring sulphides have not yet been characterised by sulphur-33 n.m.r. spectroscopy, and there is no general trend when the chemical shift of the ring nucleus in the functional group in the ketone or furan series is compared with that of the sulphur in the sulphones, so it is difficult to make general deductions from the data. EXPERIMENTAL 33sN.M.R. SPECTROSCOPY Table 3. Relaxation parametersa compounds Thb T2ls ammonium sulphate (2 mol drnA3, aqueous, pH 5.7) 9.5 x (9.8 x 4rnol dm-3)4b 9.0 x caesium sulphate (2 rnol dm-3, aqueous, pH 7.5) 7.8 x (7.6 x lo-'; 2 mol dm-3)4c 6.5 x lop2 carbon disulphide 1.0 x 10-3 1.0 x 10-~ (neat liquid) p-toluene sulphonic acid 5.0 x 10-~ 4.9 x (2 mol dm-3, aqueous) a Estimated accuracy lo%, literature values in brackets.Measured using the inversion-recovery sequence. Measured using the Hahn spin-echo sequence. Measured from the linewidth. Linewidth data are also presented in table 1. No nuclear Overhauser effect is observed for sulpholane, which suggests that sulphur-hydrogen dipolar interactions are not significant as a relaxation mechanism even for narrow lines. The width of most lines, therefore, is likely to be dominated by nuclear quadrupole relaxation. In principle, the electric-field gradient can be calculated. Indeed it has been reported for carbon di~ulphide.'~ In general, however, problems in measuring the correlation time and asymmetry parameters restrict a more comprehensive analysis.In some cases the nuclear quadrupole coupling constant is small (in particular for the sulphate and sulphone groups). It is clear that the symmetry at the central sulphur in the sulphate ion is high. However, the sulphate linewidth increases as the pH is lowered; e.g. at pH 0.9 the width of the line is 200 Hz,~ suggesting that the electric-field gradient is sensitive to protonation at the oxygen. Indeed the electrical symmetry is easily reduced; e.g. aqueous sodium methyl sulphate has a linewidth of 700 Hz, and dimethyl sulphate has a linewidth of 1400Hz. By comparison, however, the electrical symmetry at the sulphur in sulphones and sulphonic acids must be relatively high (see table 1).In many other cases lines are presumably too broad to be observed. Indeed we have been unable to observe signals from a range of com- pounds, including aqueous sodium sulphite, aqueous sodium sulphide, aqueous sodium thiocyanate, butyl thiocyanate, glucosinolates, diphenylsulphoxide and L-cysteine. Consistent with other reported re~ults,~' only one signal has been observed from the thiosulphate ion, equivalent in intensity to precisely one of the two sulphur environments [the signals from an equimolar mixture (0.8 mol dm-3 of each com- ponent) of sodium sulphate and sodium thiosulphate have equal intensities]. It is still a matter of speculation, however, whether this arises from the central sulphur or the sulphur in the thio-position.Relaxation times have been measured directly for a small number of systems, and the results are presented in table 3. Values of TI were obtained using the inversion-recovery sequence ( 180-7-90). T2was measured using the Hahn spin- echo sequence and/or from the linewidth (the Hahn spin-echo sequence is sufficiently accurate for T2measurement since the error due to the diffusion term is negligible, because of the low magnetogyric ratio of sulphur). The two relaxation times, TI and T2,were found to be equal within experimental error in all cases, which is consistent with a quadrupolar relaxation mechanism in the extreme-narrowing condition. P. S. BELTON, I. J. COX AND R.K. HARRIS b + 50 Hz Fig. 7. Spectrum of 2 mol dmP3 butadiene sulphone in acetone with (c) and without (a,b) proton decoupling. Pulse sequence: 90-acquire. Spectrometer frequency 15.34 MHz ; 90" pulse length 80 pus; recycle time 1 us; spectral width 1000 Hz; dwell time 476 pus; 512 points acquired. (a,b) Coupled; number of transients 53 500; total experiment time 14.9 h. (c)Decoupled; number of transients 2750; total experiment time 0.8 h. Data manipulation: (a) line broadening -6 Hz; gaussian broadening 0.5; (b)and (c) none. We have observed sulphur-hydrogen scalar coupling for a solution of 2 mol dm-3 2,Sdihydrothiophene-1,l-dioxide (butadiene sulphone), 11, in acetone. A partially resolved triplet, of total width 18 Hz, collapses to a singlet, width 7 Hz, when proton decoupling is applied (fig.7; note that the data can be manipulated using a Gaussian broadening function to enhance the triplet structure as shown). The major coupling is attributed to a vicinal interaction between the sulphur and olefinic protons, by comparison of the carbon spectrum of diethyl ketone (the carbonyl carbon resonance exhibits a septet) and the sulphur spectrum of sulpholane (the sulphur peak shows no structure). This was confirmed by selectively decoupling at the two relevant proton frequencies in turn. The coupling constant is 6 Hz, and the equivalent EXPERIMENTAL 33sN.M.R. SPECTROSCOPY 0S 4%00 reduced coupling constant is 0.6 x lo2' N A-2 m-3 in magnitude. The coupling is not sufficiently resolved to enable polarisation to be transferred from the protons to the sulphur, as in the INEPT14 experiment for example.Evidence of coupling is also observed for a solution of 2mol dm-3 sulpholane in acetone, since the linewidth is decreased by 7 Hz on decoupling. No structure can be seen on the coupled peak at probe temperature, however. DISCUSSION This study shows that a reasonably wide range of sulphur groups can be characterised by sulphur-33 Fourier-transform n.m.r. spectroscopy, including inor- ganic and organic sulphates and sulphides, sulphones, sulphonic acids, thiophenes, thiols, sulphoxides, isothiocyanates, sulphuryl halides and transition-metal com-plexes. In a number of cases, particularly when the symmetry at the sulphur nucleus is low, only the smaller members of the group can be characterised (and only then in very high concentration). The main problem that continues to restrict the characterisation of compounds using sulphur n.m.r.spectroscopy is the interference of the acoustic ringing with the short free induction decays. Pulse sequences can be used to minimise the ring-down (for example the RIDE sequence) but can not eliminate it completely owing to the irreproducibility of the acoustic ringing. Clearly the problem can be further reduced when higher magnetic fields are employed, since generally the effects of dead-time are less critical at higher frequencies.6 Nevertheless, signals as broad as 5 kHz can now be reproducibly observed using the RIDE sequence.Indeed, use of the RIDE sequence in general, as illustrated in fig. 3, considerably improves the quality of the spectra. However, hardware changes15 are necessary to reduce the effects of ring-down further for a given frequency. The moderate chemical-shift range will allow useful chemical work to be under- taken in specific cases. Indeed the variation of the chemical shift with substituent, solvent and concentration can be readily observed for the sulphone group, and effects due to changes in the substituent in thiophenes have been ~onsidered.~' While it is encouraging to observe signals from the sulphur nucleus in a wide range of environments, many intrinsic difficulties remain which restrict the usefulness of sulphur n.m.r.spectroscopy. A significant amount of time spent with a considerably higher-field instrument is needed before problems in general sulphur chemistry and those specific to food science can be tackled using sulphur-33 n.m.r. spectroscopy. We thank J. Lambert and E. Block for samples of the cyclic sulphones, M. J. O'Connor for the thiotungstates and thiomolybdates, J. Tarbin for several of the sulphones and K. M. Wright for help in implementing the RIDE and INEPT sequences. I.J.C. thanks the S.E.R.C. and the A.F.R.C. for finance under a CASE studentship. 75P. S. BELTON, I. J. COX AND R. K. HARRIS Note added in proof: Several further studies using 33Sn.m.r. have been p~blishedl~-'~ since this paper was accepted. The single resonance observed from the thiosulphate ion has now been assigned" to the central sulphur atom, by consideration of the spectrum of a sample enriched with 32Sat this position.C. Rodger, N. Sheppard, C. McFarlane and W. McFarlane, in N.M.R. and the Periodic Table, ed. R. K. Harris and B. E. Mann (Academic Press, London, 1978), chap. 12; R. K. Hams, Nuclear Magnetic Resonance Spectroscopy (Pitman Books, London, 1983), appendix 2. 0.Lutz, in The Multinuclear Approach to N. M. R. Spectroscopy, ed. J. S. Lambert and F. G. Riddell (Riedel, Boston, 1983), chap. 19 and references therein. (a) S. S. Dharmatti and H. G. Weaver, Phys. Rev., 1951, 83, 845; (b) K. Lee, Phys. Rev., 1968, 172, 284; (c) H. L. Retcofsky and R. A. Friedel, J. Am. Chem. SOC.,,1972, 94, 6579. (a) D.L. Harris and S. A. Evans, J. Org. Chem., 1982,47,3355; (b)J. F. Hinton and D. Shungu, J. Magn. Reson., 1983, 54, 309; (c) E. Haid, D. Kohnler, G. Kossler, 0. Lutz and W. Schuch, J. Magn. Reson., 1983, 55, 145; (d) 0. Lutz, A. Nolle and A. Scwenk, 2. Naturforsch., Teil A, 1973, 28, 1370; (e) R. Faure, E. J. Vincent, J. M. Ruiz and L. Lena, Org. Magn. Reson., 1981, 15, 401; (f)0. Lutz, W. Nepple and A. Nolle, 2. Naturforsch., Teil A, 1976, 31, 978; (g) P. Kroneck, 0. Lutz and A. Nolle, 2. Naturforsch., Teil A, 1980, 35, 226. D. I. Hoult and R. E. Richards, J. Magn. Reson., 1976, 24, 71.' E. Fukushima and S. B. W. Roeder, Experimental Pulse N.M.R. (Addison-Wesley, Massachusetts, 1980). Pure Appl. Chem., 1976, 45, 219. P. S. Belton, I.J. Cox and R. K. Harris, unpublished work. (a)P. D. Ellis, paper presented in part at NATO AS1 C 103 conference, The Mulrinuclear Approach to N. M. R., Stirling, August, 1981 ; (b) S. L. Pratt, J. Magn. Reson., 1982, 49, 161 ; (c) D. Canet, J. Brondeau, J. P. Marchal and B. Robin-Lherbier, Org. Magn. Reson., 1982, 20, 51. lo R. R. Ernst and W. A. Anderson, Rev. Sci. Instrum., 1966, 37, 93. 'I P. S. Belton, I. J. Cox, R. K. Hams and M. J. O'Connor, to be published. J. L. Lambert, S. M. Wharry, E. Block and A. A. Bazzi, J. Org. Chem., 1983, 48, 3982. l3 R. R. Vold, S. W. Sparks and R. L. Vold, J. Magn. Reson., 1978, 30, 497. I4 G. A. Morris and R. Freeman, J. Am. Chem. SOC., 1979, 101, 762. l5 D. I. Hoult, J. Magn. Reson., 1984, 57, 394. l6 J. F. Hinton and D. Buster, J. Magn. Reson., 1984, 57, 494. J. F. Hinton and D. Buster, J. Magn. Reson., 1984, 58, 324. in R. Annunziata and G. Barbarella, Org. Magn. Reson., 1984, 22, 251. l9 R. E. Wasylishen, C. Connor and J. 0. Freidrich, Can. J. Chem., 1984, 62, 981. (PAPER 4/ 6 14)

ISSN:0300-9238    

DOI:10.1039/F29858100063

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. C'hem. Soc., Faraday Trans. 2, 1985, 81, 77-85 Some Criticism Concerning the Brusselator Model of an Oscillating Chemical Reaction BY BRIAN F. GRAY" AND TRACEY MORLEY-BUCHANAN School of Chemistry, Macquarie University, North Ryde, New South Wales 21 13, Australia Received 13th April, 1984 The Brusselator is well known to exhibit an instability of the Hopf-bifurcation type leading to the appearance of limit-cycle oscillations. Here we show that this behaviour disappears entirely when detailed balancing and conservation of mass are strictly applied. Violation of both of these principles simultaneously is necessary for the instability. We conclude that the Brusselator, in spite of its termolecular autocatalytic step, is not an acceptable oscillatory model because it violates both detailed balance and conservation of mass. Computer simulations confirm the conclusions reached analytically.Many papers have appeared over recent years on the hypothetical reaction scheme AsX B+X G Y+D Y+2X e 3x (3) XsE (4) with rate constants kl . . . k, in the forward direction and k-, . . . k-, in the reverse direction. In general the reverse rate constants are assumed to be negligibly small. The majority of discussions of this system (see King' for the most recent) treat only X and Y as variables (we denote their concentrations by x and y, respectively), regarding A, B, D and E as 'pool chemicals', treated as constant (in time) variable parameters. This approach has been criticised by Gray and Aarons2 and will be further criticised in a later section of this paper.However, in the next section of this paper we take the pool-chemical approach, insisting that detailed balancing is not violated. King' criticised the Brusselator on the grounds that reaction (3) (i) is termolecular and (ii) contains the same substance X on each side of the reaction. He shows that the model is structurally unstable with respect to the removal of these assumptions in certain ways, i. e. the limit-cycle oscillation disappears. Whilst we agree with King's conclusions, we cannot agree that termolecular reactions are not acceptable. It is clear though that if reaction (3) takes place it is very difficult to argue on physico-chemical grounds that the reaction Y+X e 2x should not occur also, not to mention Y s x.Similarly, any collision-related interpretation of reaction (3) would note that it is simply a double collision-induced transition of Y to X, i.e. two Xs are providing 77 CRITICISM OF THE BRUSSELATOR MODEL energy as second and third bodies. We must then open the door to Y+2M X+2M where M may be virtually anything else present. If the rate constants are independent of the nature of M, this reaction becomes kinetically first order in both directions. Of course Y+M X+M would also be a strong candidate for inclusion. This point is pursued el~ewhere;~ it is sufficient to say here that kinetically high-order reactions must generally be imbedded in a manifold of related reactions, and then much of the non-linear character (and oscillating regime) is lost.In this paper the above point is not brought to bear on the Brusselator since we show in the next section that the oscillatory instability (Hopf bifurcation) is a direct consequence of the violation of detailed balancing explicit in earlier discussions of the Brusselator. We then show that the instability is also a direct consequence of the ‘pool- chemical’ approach, and when mass conservation is strictly enforced (even with the violation of detailed balancing) no semblance of oscillation appears at the parameter values where limit cycles occurred previously. One might have expected that strict enforcement of mass conservation would simply convert a limit cycle to a damped ‘far-from-equilibrium’ oscillation followed by the necessary monotonic decay near equilibrium; however, as conjectured by Gray and Aarons,2 the oscillations depend on non-conservation of mass.The limit cycle only occurs when both detailed balancing and conservation of mass are ignored. THE TWO-VARIABLE SYSTEM Here we follow the established practice and consider only X and Y as variables, with A, B, D and E being treated as constants. For pedagogical reasons only, at this stage we take B = D. The two differential equations for the system are then dx -= kla-k,bx +k-,by +k3x2y-k-,x3 -k4xdt dy= k2bx-k-,by -k3x2y+k-,x3 (ii)dt where a, b etc. are the concentrations of A, B etc. We have neglected k-l and k-4, since their inclusion in this case makes no qualitative difference (they are included in the next section where mass conservation is applied). Previous treatments of this two-variable system (e.g.Nicolis and Prigogine4) invariably write k-3 =0, k-, =0. The net change in reaction (2) is the reverse of the net change in reaction (3), i.e.one reaction converts an X molecule into a Y molecule and the other achieves the reverse. In a closed system at equilibrium (iii) Of course the second form of eqn (iii) holds in general (providing we are not working in the shock-tube regime where K,, # kf/ kJ. It may be regarded as a consequence of thermodynamics or detailed balance which states that in a closed B. F. GRAY AND T. MORLEY-BUCHANAN 6 4 Y 2 0 2 xi 60 2 xi 6 Fig.1. (a) Limit cycle generated by the Brusselator reactions (1)-(4), k, = k3= k4= 1.0, k2= 4.0. (b)Trajectories in the x-y plane generated from reactions ( 1)-(4) with the detailed balance conditions, k2k3 = k-2k-3, enforced, k, = k3= k4= 1.0, k2=4.0, k-, = k-3 = 2.0. system at equilibrium, individual reactions balance. However, the resulting relations between the rate constants must hold in general not only at equilibrium. Detailed balance implies a relation between rate constants when the kinetic scheme either has closed cycles or two reactions between the same components with different kinetic properties. The Brusselator falls into the latter category (as do some other chemical oscillators), and it is physically quite unacceptable to have infinity on one side of eqn (iii) and zero on the other.It has long been recognized that the triangular reaction scheme may show complex characteristic roots if the conditions of detailed balance are not imposed, but this system has never been regarded as a damped oscillator since detailed balancing is mandatory in a physico-chemical system. The steady-state values of x and y from eqn (i) and (ii) are kl k2xs+k-3~z x,=-; y,=k4 k-2+k3x; but use of the second part of eqn (iii) gives k-3 ys=-xs. k3 The characteristic equation for the system is The roots of this equation, Al and hZ,are always real and negative. There is no instability of the type reported when detailed balance is violated. In this case therefore a Hopf bifurcation cannot occur.We must conclude that the reported limit cycle and instability are direct results of the violation of detailed balance. CRITICISM OF THE BRUSSELATOR MODEL Computations with various initial conditions [fig. 1(b)] show no limit-cycle behaviour at parameter values where limit cycles do occur when detailed balancing is ignored [fig. l(a)]. The PoincariLBendixon criterion can also be applied to this system and it may be shown’ that limit cycles cannot occur when the total concentra- tion of chemicals in the system is less than a critical amount. This is an argument in the large, not a local one. Here we simply confirm the local stability results by computation. THE FULL SYSTEM We now turn to the full system including the so-called ‘pool chemicals’ as variables.In all we have to deal with six variables, the concentrations of X, Y, A, B, D and E. E may be eliminated by conservation of mass and either B or D eliminated in addition since db/dt = -dd/dt. We also include the reaction BGD (5) in addition to reaction (2), since we are now going to allow the initial value of dbldt to vary. Detailed balance requires a relation between the equilibrium con- stants of reactions (2), (3) and (5), ie. which leads to a similar relationship between the rate constants: k2 k3 k-, = k-, k-3k5. (vii) The values of k5 and kP5 may of course be kept arbitrarily small provided their ratio is such that eqn (vii) is true. The kinetic equations describing the full system of four independent variables are dx -= k-4 M2+(kI -kP4)a -(k-+k, +k-J x -(k-, -M 3 k-2 )YJdt *=-k-2M3~+L,by +k,bx -k3x2y+k-,x3dt (viii) da -= k-,x-k,adt db-k-5 M3 -(k5+k-5) b +k-2 M3y -k-2 by -k2 bx dt where a+b+d+e+x+y=M, a +e +x +y = M2 b+d=M, and MI,M2 and M3 are constants. Of course we know in general for a closed chemical system that there is a unique stable equilibrium represented by a stable nodal point, which is always approached monotonically.This can be shown in particular for the differential eqn (viii) along with the mass constraints and detailed balancing. The characteristic B. F. GRAY AND T. MORLEY-BUCHANAN equation is a quartic with positive coefficients satisfying the usual stability criteria. The steady-state values of x, y, b and a are greatly simplified through the application of eqn (vii), reducing to kl k3k-4M2 x, = kl k3( k, +k-4) +k-4( kl k-3 +k-1 k3) 1 k-3Ys =-xsk3 M3( k-, k-3~s +k3 k-5) b,= (k2k3 +k-2k-3)~~+k3( k5 +k-5) k-1 a, =-xs.kl Our primary intention in this section is to investigate whether any legitimate approximations (as opposed to the usual physically unacceptable ones) can show oscillatory behaviour. Of course, at most this could only be transient oscillation, when sufficiently far from equilibrium, eventually decaying to monotonic behaviour sufficiently close to equilibrium. On returning to eqn (iii) we see that in effect, for B and D varying (B # D), the equilibrium expression is One obvious question to investigate, therefore, is whether or not oscillatory behaviour can occur when b(O)/d(O)(ie.the ratio of initial concentrations b and d) is made sufficiently large (>> 6/d).Whilst this is not equivalent to putting k-, = 0 and violating detailed balancing, it will have the effect of pulling reaction (2) to the right for a time period without affecting reaction (3). The projections for the full four-variable scheme in the x-y phase plane are shown in fig. 2(a) and (b). The initial concentrations of X, Y and A are held constant, and the values of the rate constants were chosen so as to favour the production of oscillations. Each trajectory in fig. 2(a) exhibits a single looped curve, the extent of which clearly increases as b(O)/ d (0) increases, before monotonically approaching equili- brium.Letting b(O)/d(O)approach 6/d diminishes the size of the loop until the system eventually proceeds immediately to equilibrium without excursion. A similar computation for different values of x(0)and y(0)at constant a(0) and b(0)is shown in fig. 2( b),which serves to illustrate that although initially the curve shows some resemblance to the original Brusselator [fig. 1 (a)], no oscillatory behaviour is observed. Numerical analysis shows in this case that however large the ratio b(O)/d(O)is made, oscillations do not occur. What does occur is that the ratio b/d drops very quickly to near its equilibrium value. By ‘very quickly’ we mean in a time period shorter than the period of oscillation at the same parameter values in the pool- chemical Brusselator without detailed balance.Clearly the pool-chemical approxi- mation entertained in the original Brusselator is quite invalid for this system for these parameter values. We suppose this is true in general. The second question arising when considering the full eqn (viii), is: ‘If detailed balancing is violated but conservation of mass enforced, will the system show any 82 CRITICISM OF THE BRUSSELATOR MODEL Y xs *Ys X X Fig. 2. (a) Projection in the x-y phase plane generated by the four-variable scheme [eqn (viii)] for constant initial concentrations x(O), y(0) and a(0) and varying ratios b(O)/d(O): k, = k3= k4 = 1.0, k-, = k-, = k, = 0.1, k2= 10.0, k_, = 0.4, k-, = 0.25, k-, = 0.001, M2= 20.0, M3= 15.0: (1) b(O)/d(O)=0.5; (2) b(O)/d(O)= 2; (3) b(O)/d(O)=4; (4) b(O)/d(O)= 149; (5) b(O)/d(O)+ c.o [i.e.b(0)= M3,d(0)= 01.(b)Projection in the x-y phase plane generated by eqn (viii) for different initial concentrations x(O), y(0); rate constants, M2, M3 and a(0)are the same as in (a) and b(O)/d(O)= 149. 12 8 Y 4 0 Fig. 3. Projection in the x-y phase plane generated by eqn (viii) when detailed balance is ignored(i.e.k,k,k_,#k_,k_,k,): k,=k3=k4=l.0,k~l=k~4=k5=0.1,k2=10.0,k-5=0.001, k-2 = kP3= 0, M2=20.0, M3= 15.0, a(0) = 10.0, b(0)= 14.9, x(0) = 8.0, y(0)= 2.0, X, = 1.80, ys= 4.59 x 1o-~. transient oscillation?’ Accordingly we have computed the solution to eqn (viii) with k-, = k-, = 0 under various conditions and no oscillatory behaviour has been found.Results are shown in fig. 3. The final question arising is: ‘Can we choose parameter values and initial conditions such that the pool-chemical approximation is valid for a finite period of time and the resulting two-dimensional system is unstable?’ 83B. F. GRAY AND T. MORLEY-BUCHANAN "'v1 0 t7 14 Fig. 4. (a) Graph of In (y/a)against t for the four-variable model when approximated to the two-variable model: x(0) = 1.0, y(0)= 1.0, a(0)= 10.0, b(0)= 15.0, k, = 0.8, kZ=1.2, k, = k, = 10.0. (6) Graph of y/ b against t for the four-variable model when approximated to the two-variable model: x(0) = 1.0,y(0)= 1.0, a(0)= 10.0, b(0)= 15.0, k, = 0.8, k2= 1.2, k3= k, = 10.0.This question has been asked before by Gray and Aarons,2 where it was concluded that the probable answer is 'no'. Mathematically we are asking whether it is possible to approximate the behaviour of the four differential eqn (viii), by two differential equations for x and y containing a and b replaced by a(0)and b(0). This clearly cannot hold for all times t but may hold for a finite period 0s t d T for suitable initial conditions. In ref. (2) it was shown that the necessary conditions for this to hold at all were probably incompatible with the condition for the two equations to be unstable, i.e. the pool-chemical idea cannot work when the reduced system is unstable and oscillatory. Generally one would expect the necessity of x(O)/a(O), y(O)/a(O),x(O)/b(O), y(O)/b(0)<< 1 for the pool-chemical approximation to be valid.Other conditions on the rate constants are also necessary,* but none of these conditions turns out to be sufficient for the approximation to be valid. CRITICISM OF THE BRUSSELATOR MODEL t Fig. 5. Graphs of y against t for: (a)the original Brusselator, reactions (1)-(4),in the region of instability (oscillations produced), k,a(O)= 8.0, k,b(O)= 18.0, k, = k4 = 10.0, x(0) = y(0)= 1.O (a and 6 are constant and x and y are variables) ; (b)the four-variable model, reactions (1)-(4) with stable equilibrium, a, b, x and y are variables, k, = 0.8, k, = 1.2, k, = k4 = 10.0, a(0)= 10.0, b(0)= 15.0, x(0) = 1.0, y(0)= 1.0; (c) the two-variable Brusselator with stable equilibrium, k,a(O)= 10.0, k,b(O) = 10.0, k, = k4 = 100.0, x(0) =y(O)= 0.1 ; (d) the four- variable model with stable equilibrium, k, = k, = 1.O, k3= k4= 100.0, a(0)= b(0) = 10.0, x(0) = y(0)=0.1.We have performed computations (neglecting all reverse rate constants and with k5= 0) on the full system eqn (viii) with initial conditions and rate constants chosen so that initially one might expect the second-order system to approximate eqn (viii). Fig. 4(a) and (b) show that this is not the case, and the necessary conditions are violated very quickly after the run commences. The concentrations of A and B fall rapidly and do not remain in large excess for very long. The ratio y/a exceeds unity in a time very much shorter than the oscillation period.The concentration of X, however, falls concurrently and so the ratios x/a and x/b remain small for a long time. The full system again does not develop oscillations, they only occur when a and 6 are kept strictly constant at the expense of the conservation of mass. The pool-chemical approximation can give good qualitative and quantitative description of this system in the stable region of the second-order system. The long-accepted idea that chemicals present in large excess in a reaction mixture can be assumed to have constant concentrations seems to be limited to cases where the resulting reduced-order system has the same stability characteristics as the full system. Fig. 5 shows computations in the Brusselator scheme for these conditions.Clearly if the pool-chemical approximation changes the stability characteristics of the system, it is likely not to be valid for any significant period of time. CONCLUSIONS The temporal oscillations of the Brusselator rely for their existence on the simultaneous violation of the two inviolable physico-chemical principles: detailed balancing and conservation of mass. The kinetic behaviour of the system is qualita- tively altered if either or both of these principles is enforced, and then shows no interesting behaviour at all. These conclusions are independent of the unrealisitc nature of one of the reactions employed, which is termolecular and autocatalytic. B. F. GRAY AND T. MORLEY-BUCHANAN As the Brusselator was shown some years ago6 to be the simplest possible isothermal oscillator, thus holding a unique position in this field, the present work throws considerable doubt on the possibility of the occurrence of homogeneous isothermal chemical oscillations. T.M-B.acknowledges support from the Australian Research Grants Scheme. ' G. A. M. King, J. Chem. SOC.,Faraday Trans. 1, 1983, 79, 75. B. F. Gray and L. J. Aarons, Faraday Symp. Chem. SOC.,1974, 9, 129.' B. F. Gray, P. Gray and S. K. Scott, J. Chem. SOC., Faraday Trans. 1, 1984, 80, 3409.' G. Nicholis and I. Prigogine, Faraday Symp. Chem. SOC., 1974, 9, 7. J. D. Murray, Lectures on Nonlinear Differential Equation Models in Biology (Clarendon Press, Oxford, 1977). J. J. Tyson and J. C. Light, J. Chem. Phys., 1973 59, 4164. (PAPER 4/6 15)

ISSN:0300-9238    

DOI:10.1039/F29858100077

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem.SOC.,Faradav Trans. 2, 1985, 81, 87-99 Time-resolved Emission Studies of the Kinetic Behaviour of Ca(4 3P') and Sr(5 "') following Pulsed Dye-laser Excitation Electronic Energy Exchange between Ca(4 3PJ)and Sr(5 3PJ),Energy Pooling and Decay Measurements on Ca(4 3PJ),Sr(5 'P,) and Sr(6 3S1) BY DAVID HUSAIN* AND GARETHROBERTS Department of Physical Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EP Received 9th May, 1984 The kinetics of Ca(4 'P,), Sr(5 'P,) and Sr(6 3S1)have been studied by time-resolved emission following pulsed dye-laser excitation. Ground-state calcium atoms, Ca[4s2( ' So)], in the presence of Sr[5s2(' So)],were optically excited to the first electronically excited state, Ca[4s4p('P1)], by means of a dye-laser operating at A = 657.3 nm [Ca(4 3P,) t Ca(4 'So)].Following rapid Boltzmann equilibration within the Ca(4 'P') manifold, fast, near-resonant energy transfer took place with Sr(5 'So)to yield Sr(5 'PJ),together with energy pooling to yield Sr(63S,). The following transitions were monitored in the time-resolved mode using pre-trigger photomultiplier gating and boxcar integration with repetitive pulsing on a slow-flow system, kinetically equivalent to a static system: Ca(4 3PI)+Ca(4 'So)+ hv (A = 657.3 nm), Sr(53P1)+Sr(5 'So)+ hv (A = 689.3 nm) and Sr(6'S1)-+ Sr(5 'Po)+ hv (A = 679.1 nm). Detailed measurements of the time dependence of the emission profiles at A = 657.3 and 689.3 nm demonstrated the rapid establishment of the equilibrium Ca(4 'P') + Sr(5 'So)$ Ca(4 'So)+Sr(5 'PJ) and its sustenance throughout kinetic decays.A further comparison of the first-order decay coefficients for the decays of Ca(4 'P') and Sr(5 'P,) with those for Sr(6 'S,) demonstrated energy pooling between Ca(4 '4)+ Sr(5 'P') and Sr(5 'P,) + Sr(5 'P'). Finally, the measured first-order decay coefficients, k', for Ca(4 'PJ) and Sr(5 3PJ)in chemical equilibrium with one another are seperated quantitatively into their component parts of spontaneous emission, diffusion, the equilibrium constant connecting the 3PJ states and collisional quenching rate constants. Rate constants for quenching of Ca(4 'P,) and Sr(5 'PJ)by Kr, Xe, H2 and D, derived from earlier measurements on the individual atoms are found to be quantitatively consistent with the decay profiles under equilibrium conditions.Direct kinetic studies of the low-lying electronically excited atoms Ca[4~4p(~ PJ)] and Sr[5s5p(3PJ)], 1.888 and 1.807 eV above their ns2('So)ground states, respec- tively,' are now well established.2 The most recent papers of Husain and S~hifino~-~ on dye-laser excitation of ground-state Ca[4s2(' So)]and Sr[5s2( 'So)]to their respec- tive nsnp('P,) states, with subsequent time-resolved monitoring of the spin-forbidden emission processes n~np(~P,)ns2('So)+ hv, include summaries of a variety of-+ earlier investigations in which these species have been monitored in the time domain by a range of spectroscopic techniques following dye-laser excitation. Similar considerations also apply to Mg( 3 3PJ).7,8These recent papers3-' also indicate studies of the collisional behaviour of the 3PJstates by two further complimentary techniques, namely 3P1-+ 'So emission in a flow-discharge system, used primarily for the study of Mg(3 3PJ),9,10and molecular beams.The latter group of measure-87 KINETIC BEHAVIOUR OF ~a(4 'P')'P') AND ~r(5 ments for Mg, Ca and Sr ('P') is far too large for summary here. A particularly extensive investigation by Engelke' ' describes a range of studies of these electroni- cally excited Group IIA elements employing molecular beams. Detailed papers by Dagdigian and co~orkers'~-'~ give examples and by Menzinger and coworker~'~~'~ of the work of but two major research groups on these nsnp('PJ) atomic states.In the context of the present paper, both the flow-discharge technique and, more importantly, the molecular-beams techniques, which provide details of the individual collisional processes not available from other methods, are characterised, neverthe- less, by one significant limitation with respect to the electronically excited species 'PJ. Production of the 'P, states is not specific and it is usually necessary to make a variety of corrections for the r6les played by higher-lying states generated in the low-voltage discharges employed for generation of the 'P' states [see ref. (13)-( 16) and references contained therein]. Energy pooling between the metastable 'PJ states is therefore difficult to study. Taieb and Broida' and Bourguignon et a1." did succeed in demonstrating a squared dependence between the emission intensities of Mg(3 'P,-3 'So+hv, A = 285.2 nm) and Mg(3 3P1-+3 'So+hv, A = 457.1 nm), consistent with Mg(3 'PJ)+ Mg(3 'P,) annihilation in a flow-discharge system.Benard et all7 have further showed that electronic energy transfer takes place from Mg(3 3PJ)to Ca(4 'So) in a flow-discharge system. In this paper we present a quantitative kinetic investigation of both energy pooling, electronic energy transfer and the kinetics of both Ca(4 'PJ) and Sr(5 'P') when these two states are in electronic equilibrium with one another following specific dye-laser excitation of Ca(4 'So) to Ca(4 'PI). Near-resonant electronic energy exchange between Ca(4 'P') and Sr( 5 ' So), following rapid Boltzmann equilibration within the Ca(4 'PJ)mani-fold, is also shown to occur.This type of transfer process has been investigated by Malins et a1.'' following pulsed dye-laser excitation of Ca(4 'So)in a flow system. However, measurements were not carried out in the time domain, but rather employed phase-shift techniques which are model dependent and which, indeed, led to quenching rate constants for Ca(43 P,) and Sr(5 '2'') by the lighter noble gases that have been shown to be too Further, the system was not treated as an equilibrium-coupled population between Ca(4 'P') + Sr(5 'P,), which is found to be the case here (vide infra). Neither, for that matter, was equilibrium coupling between the 'PO-'P1-'P2 spin-orbit components in the nsnp('PJ) manifolds of Ca('P') and Sr(3PJ) included in the quantitative account of the 3P1+ 'So+hv emission, '' which is also necessary.'-' Emission from the higher-lying Sr( 5 ' P1, 6 'S1) and Ca(4 'Pl,5 3S1)states was noted and attributed to energy-pooling proces- ses.l' In this paper, time-resolved studies of Ca(4 3P1-+ 4 'So)+ hv (A = 657.3 nm) and Sr(5 3P1--+ 5 'So)+ hv (A = 689.3 nm) following pulsed dye-laser excitation at A = 657.3 nm in a system containing Ca(4 'So)+ Sr(5 'So) are described quantita- tively.The kinetics of the equilibrium system is established and the components of the decay of each atomic state, namely spontaneous emission, diffusion and col- lisional quenching, together with the equilibrium term connecting these states are quantitatively elucidated for various noble gases.Decays in hydrogen and deuterium were also studied as examples of added quenching gases. This type of coupled system as a general tool for a complete range of quenching studies for Ca(4'P') and Sr(5 'PJ)is presented here and applied in the following paper. Energy pooling is also quantitatively described by a comparison of the time dependence of the emissions Ca(4 'PI-+ 4 'So) (A = 657.3 nm), Sr(5 'P1-5 'So) (A = 689.3 nm) and Sr(6 3S1-5 'Po)(A = 679.1 nm), together with estimates of the fractional yields of the 'S1 state arising from energy pooling between Ca(4 'P,) + Sr(5 'P') and ~r(5 'P').'P') + ~r(5 D. HUSAIN AND G. ROBERTS EXPERIMENTAL Ca(4 3PJ),Sr(5 3P,) and Sr (6 3S1)were monitored as a function of time following pulsed dye-laser excitation of Ca(4 'So)to Ca(4 3P1)at A = 657.3 nm in a mixture of calcium and strontium vapour, which was itself in equilibrium with a mixture of the two solids at 950 K, and in the presence of an excess of buffer gas.The experimental arrangement was essentially a combination of that described hitherto for the study of Ca(43PJ)3'4 for the purpose of pulsed dye-laser excitation and that for Sr(5 3P,)5,6for the purpose for monitoring the time-resolved emissions at A = 657.3 nm [Ca(4 3P1)--* Ca(4 *So)+ hv], A = 689.3 nm [Sr(5 'P1)+Sr(5 'So)+ hv] and A = 679.1 nm [Sr(6 3S1)+Sr(53P0)+ hv]. The Einstein coefficients for the first two of these transitions have been investigated in detail by Husain and Schifino?.' yielding A,, = (2.94* 0.17) x lo3 and (4.98 f0.10) x lo4sA1,respectively.The Einstein coefficient for the transition at A = 679.1 nm is taken from the compilation of Corliss and Bozmann" (gA= 0.27 x lo8s-l). Its large magnitude is primarily employed here to justify placing [Sr(6 'S1)] in a stationary state, even though the atomic states from which it is derived, Ca(4 3PJ)and Sr(5 3PJ),are functions of time (vide infra). It is also used to estimate the ratios [Sr(6 3Sl)]/[Sr(5 3Pl)] and [Sr(6 3Sl)]/[Ca(4 'P,)] at what is effectively t = 0 on the kinetic timescales employed here. Thus Ca(4 'So),in a mixture of Ca(4 So)+ Sr(5 'So),is excited to the 4 3P1state by pulsed dye-laser excitation at A = 657.3 nm using the dye Cresyl Violet Perchlorate 670 (Exciton Chemical Co.Inc., U.S.A.; pulse length ca. 15-20 ns; pulse width 0.1 cm-' ; output ca. 10 mJ per pulse), pumped by means of the second harmonic (A = 532 nm; E = 150 mJ per pulse) of a Nd-YAG primary laser (J.K. Lasers Ltd, model DLPY4 200 YAG) at a repetition rate of 10 Hz. The three transitions indicated above were all monitored using the photomultiplier tube with the long-wavelength 'S20' response (p.m. tube E.M.I. 9797B) described previou~ly.~ This includes the use of the 'pre-trigger photomultiplier gating system' which permits recording of the photoelectric emission signals ca. 1-5 ps subsequent to the dye-laser pulse. The basis of the gating circuitry is as described in the earlier paper o,f Husain and Schifino.' Some variation in the applied photomultiplier voltage (ca.100V) to that normally used in monitoring the transition at A = 689.3 nm (750 V, Brandenberg power supply 415R, 0-2.2 kV)' was found to be convenient when monitoring the other two transitions, particularly that at A = 679.1 nm [Sr(6 3S1)--* Sr(53P0)+ hv] on account of the low concentration of Sr (6 3S1)resulting from energy pooling. Of course, this does not affect the time constant for an exponential decay process. The voltage applied to the photomultiplier tube in each case is taken into account when estimating ratios of what are effectively initial concentrations of Ca(4 3P1),Sr(5 3P1) and Sr(6 3S1),i.e. on the justified basis that Boltzmann equilibration within the 3PJmanifolds and equilibrium between the 'So and 3PJ states of Ca and Sr are effectively established instantaneously within the timescales of the present measurements (vide infra).We crudely employ the published curve of the gain-voltage relationship for the 9797B photomultiplier tube (E.M.I. catalogue), which is approximately logarithmic, in order to estimate [Sr(6 3Sl)]/[Sr(5 3Pl)]and [Sr(6 3S1)]/[Ca(4 3Pl)]. The detailed arrangement of the stainless-steel reactor with cooled Pyrex end windows, and into which was placed approximately equal quantities of solid calcium and strontium, constituting part of a calibrated slow-flow system kinetically equivalent to a static system, has been described in previous publication^.^,^ Whilst we may readily calculate an equilibrium atomic density above pure calcium or strontium at a given temperature,20-22 we may not conveniently or reliably do this for a mixture in this type of system.Even if the individual atomic densities above solid mixtures of defined composition had been characterised by previous measurement, this could not be controlled in a flow system (as opposed to a carefully controlled static system) where deposition is taking place within the reactor. The use of approximately equal quantities of calcium and strontium, added to the reactor when the material had been depleted to the point of losing the emission signal of interest, simply ensures approximate reproducibility in experimental conditions ( i.e. the intensity of the observed signals) from one series of kinetic runs to another.Like Malins et aZ.,18we take the ratio of [Sr(5 'So)]/[Ca(4 'So)]to be constant in a series of kinetic measurements made with a given mixture; unlike Malins et who employ the nominal vapour pressures of KINETIC BEHAVIOUR OF Ca(4 'P') AND ~r(590 'pJ) Ca and Sr for the pure elements when using mixtures in order to estimate quenching constants for Ca(4 'So)and Sr(5 'So)as the quenching gases, we do not assume the magnitude of this ratio. Instead it is measured from the rate data on the basis of the electronic equilibrium which is established (vide infru) and it is found to vary with the addition of further sample. Most measurements were carried out at 950 K, as this yields approximately equal emission intensities from Ca(4 3P,)and Sr(5 3P,),but a number of decays were monitored in a series of experiments in which the temperature was deliberately varied over the range 850- 1050 K.Hence, following dye-laser excitation at A = 657.3 nm, the emission signals at this wavelength and at A = 689.3 and 679.1 nm were optically isolated, in turn, using the small grating monochromator described previously ('Minichrom' 1-MC 1-02, Fastie-Ebert mount- ing; f/4; focal length 74mm; grating 20mm square; 1800 lines per mm)' mounted on the photomultiplier housing. The emission signals were recorded by means of boxcar integration (Brookdeal Electronics linear gate type 415; scan delay generator 425A; repetition rate 10 Hz). For the majority of decays a 'sample time' of 50 ys was employed with a 'reading time' of 200 s, a typical decay trace thus being constructed from 2000 individual shots.For some of the faster decays, sample times and reading times of 20 ps and 100 s, respectively, were employed, with decay traces being constructed from 1000shots. The output from the boxcar integrator was fed to an XY recorder (Farnell Instruments Ltd) for kinetic analysis and all rate data were subjected to a computerised linear least-squares treatment using the University of Cambridge I.B.M. 3081 model D mainframe computer. Finally, it must be emphasised that the comparisons of decay traces derived from signals at A = 657.3, 689.3 and 679.1 nm were always made on the same sample.All materials were prepared as described in previous publication^.^-' RESULTS AND DISCUSSION 'P') AND ~r(5EQUILIBRIUM BETWEEN ~a(4 'pJ) Fig. l(a) and (6) give examples of the XY output indicating the decay of the time-resolved atomic emission at A = 657.3 nm [Ca(4 'PI)+Ca(4 'So)+ hv] and A = 689.3 nm [Sr(5 'P1)-+ Sr(5 'So)+ hv], respectively, following pulsed dye-laser excitation of Ca (4 'So)to Ca(4 'P,)at 950 K in a mixture of Ca(4 'So)+Sr(5 'So) and in the presence of an excess of helium buffer gas. Two conclusions may be drawn immediately: first, energy transfer from Ca(4 'PJ)to Sr(5 'So)has taken place, and secondly, visual inspection shows that the decays of Ca(4 'PJ)and Sr(5 'PJ) are essentially equal, whereas for mixtures of Ca(4 'So)+ He and Sr(5 So)+ He these would be expected to be in the ratio of approximately one to ten for this pressure of helium, primarily on the basis of the Einstein coefficients for spontaeous The decay of Ca(4 'P') has become faster and that of Sr(5 'P,) slower.Fig. 2(a) and (6) show the first-order kinetic plots for Ca(4 'PJ)and Sr(5 'PJ), respectively, constructed from the data of fig. 1(a)and (b),and yield the first-order decay coefficients of k'[Ca(4 'P')] = [4.26 f0.01 (1a)]X lo3 and k'[Sr(5 'P')] = [4.12 * 0.01 ( 1a)]x lo3s-'. Husain and Schifino have demonstrated in previous both in terms of the energy to be transferred on collision with He, for e~arnple,~' thatand, in the case of Ca(4 'PJ),by reference to direct mea~urernent,~~ following excitation to the 'P1 level, spin-orbit relaxation within the 'PJ manifold is effectively instantaneous compared with the timescales of the present measure- ments [AElcm-': Ca, 3Po-1 == 52, = 106; Sr, 'Po-' = 186, 3P1-2395).' Those have further stressed that as the 'Po and 3P2states are highly optically metastable reservoir state^,^' and that whilst emission need only be considered in either case for the two transitions nsnp('P1)-+ ns'( 'So)+ hv, each characterised by its An, Einstein coefficient, the effective decay constant of the 'P, state for emission alone is given by k:rn=Anml(l+ 1/K,+ K2) (i) D.HUSAIN AND G. ROBERTS 91 2 00 150 h .-v)* 5 roo 4 v L .s 50 0 250 500 750 1000 0 250 500 750 1000 0-100 200 300 400 500 flPS Fig.1. Examples of the output of the XY recorder indicating the decay of the time-resolved atomic emission (IF)at (a) A = 657.3 nm [Ca(4 3P,)+Ca(4 'So)], (6) A = 689.3 nm [S(5 3P1)-+ Sr(5 'So)]and (c) A -679.1 nm [Sr(6 3SI)-+ Sr(5 3P0)]following the pulsed dye- laser excitation of calcium vapour [Ca(4 3P,)+Ca(4 's,)]in the presence of strontium vapour and helium (pHe= 30 Torr, T = 950 K). 0 250 500 750 1000 0 250 500 750 1000 0 100 200 300 400 500 UPS Fig. 2. Examples of first-order kinetic plots for the decay of the time-resolved atomic emission [In(IF) against t] at (a) A = 657.3 nm [Ca(4 3P1)-+Ca(4 'So)], (6) A =689.3 nm [Sr(5 3P1)-+ Sr(5 'So)]and (c) A =679.1 nm [Sr(6 3S1)-+Sr(5 3P0)]following the pulsed dye- laser excitation of calcium vapour [Ca(4 3P,)+-Ca(4 's,)]in the presence of strontium vapour and helium (pHe= 30 Torr, T = 950 K).where K, and K2are the respective equilibrium constants connecting the spin-orbit components K, 3P0e3P, (1) KINETIC BEHAVIOUR OF ~a(4 'P')'P,) AND ~r(5 The function (1+1/K,+K,) takes the values of 2.78 and 2.36 for Ca (4'P') and Sr(5 'P'), respectively, at 950 K, and each approaches the value of 3 at infinite temperature, being determined by the statistical weights of the spin-orbit com-ponents. Thus, the recorded emission signals of the 'PI-'SOtransitions monitor the 'P' states of Ca and Sr both with respect to emission itself and with respect to the chemical kinetics of these 'Boltzmannised' species. Further, the equality in k' for Ca(4 'P') and Sr(5 'P') from the examples in fig.2( a) and (b)reflects an equilibrium system connecting Ca(4 'P,) and Sr(5 'P,), and this is now expressed in quantitative terms. The foregoing observations support the rapid establishment of the equilibrium K3'P') +~r(5'so)F= ~a(4~a(4 'so)+Sr(5 'P,). (3) We cannot, however, employ the rate constants reported by Malins et a2." for the forward and reverse processes involved in equilibrium (3)as their analysis, restricted to lifetime measurements on Sr(5 'P') from phase shifts, did not treat the system as coupled through equilbrium (3). Furthermore, the omission of equilibrium within the Sr(5 'PJ)spin-orbit manifold led to the use of A,, as the effective spontaneous emission decay constant," rather than eqn (i) above, and the conclusions of large effects of radiation trapping in the transition Sr(5 'PI)-+Sr(5 'So)+hv." In fact, emission lifetime measurements on Sr(5 'PI)as a function of [Sr('So)] reported by Husain and Schifino' have shown that for the density of atomic strontium given by Malins et a2." (1.5 x lOI5 atom cm-') the effect of trapping on the lifetime is only ca.The rate measurements on both Ca(43P') and Sr(5'P') do, however, indicate that equilibrium (3) is set up effectively instantaneously on the timescales employed here and maintained throughout the kinetic measurements. The value of AE for equilibrium (3) of -654 cm-' is taken from the average spin-orbit energies within each 'P, state, and K, = [w5 '~~)i[ca(4 'so)l/[ca(4 '~,)i[sr(5 3~0>i (ii) takes the value of 2.56 at 950 K.For the kinetic analysis, we may combine equilibrium (3) with the processes describing the overall first-order decays specific to Ca(4 'P,) and Sr(5 'P,), namely ka Ca(4'PJ) -products (4) kb Sr(5 'P,) -products (5) characterised by decay coefficients k, and kb. The products in the case of the noble gases will, of course, be the 'So ground states. As the 'P, states are equilibrium coupled, we must consider the total removal of both states, namely -d{[Ca(4 'P')]+[Sr(S 'P,)]}/dt = k,[Ca(4 3P,)]+kb[Sr(5 'PA. (iii) Combining eqn (ii) and (iii) we obtain -d(ln {[Ca(4 'P')]})/dt = -d(ln {[Sr(5 'P')]})/dt D. HUSAIN AND G. ROBERTS 1 3 5 7 9 11 k"Ca(4 3~J)3/lo3s-* Fig.3. Comparison of the pseudo-first-order rate coefficients (k')for the decay of Sr(5 'P,) and Ca(4 'P') obtained by monitoring Sr(5 'PI) and Ca(4 'P,) at A = 689.3 nm [Sr(5 'P1)---* Sr(5 'So)]and A = 657.3 nm [Ca(4 3P1)---* Ca(4 'So)],respectively, following the pulsed dye-laser excitation of calcium vapour [Ca(4 'P1)+Ca(4 'So)]in the presence of strontium vapour and helium at various temperatures (pHe= 30 Torr, T = 850- 1050 K). Fig. 3 shows the plot of the measured values of k'[Ca(43P,)] against those of k'[Sr(5 3P,)]obtained by varying the temperature in the range 850-1050 K and also by variation of the pressure of helium. The slope of this line is unity within experimental error [slope = 0.999 f0.009 (1a)], clearly establishing an equilibrium system as indicated by eqn (iv).ENERGY POOLING Fig. l(c) shows an example of the XY output for the decay of the emission Sr(6 3SI)-+ Sr(5 3P0)+ hv(A=679.1 nm) in the system described above. Visual inspection shows the decay to take place on a timescale approximately one-half those shown in fig. 1(a)and (b),suggesting near-resonant energy pooling: Ca(4 'P,)+Sr(5 'P,) kc __+ Sr(6 3S,)+Ca(4 'So), AE = -762 cm-' Sr(53P,)+Sr(53P',) kd +Sr(63S1)+Sr(5'So), AE=-108cm-' Writing [Ca(4 3P,)]l = [Ca(4 3PJ)]t=o exp (-k't) "P,)], = [~r(5[~r(5 3~,)]t=o exp (-k't) placing [Sr(6 'S,)], in stationary state (see Experimental section), namely = kC[ca(4 3~,~1t[~r(5 drsr(6 3sS,)itidt 3~J)it + kd[Sr(5 3P,)]:- A,,[Sr(6 3S,)]t= 0 KINETIC BEHAVIOUR OF ~a(4 3~,)'pJ)AND ~r(5 Fig.4. Comparison of the pseudo-first-order rate coefficients (k')for the decay of Sr(6 'S,), Ca(4 'PJ) and Sr(5 'PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(4 'PI)+-Ca(4 'So),A = 657.3 nm] in the presence of strontium vapour and helium (data for various Ca/Sr samples; pHe= 30 Torr, T = 950 K). (a)k'[Sr(6 'S,)]against k'[Ca(4 '191; (b)k'[Sr(6 'q)]against k'[Sr(5 'PJ)]. as spontaneous emission from Sr(6 'S,)is considerably more rapid than any other removal process (gA= 0.27 x lo8s-'),'' we may calculate IF(A = 679.1 nm) = A,,[Sr(6 'Sl)]t 3~J)],=0+kd[Sr(5 'P,)];=~}= {k,[~a(4 'PJ)],=~[S~(S exp (-2k't). (viii) This final result is, of course, independent of A,, for the 679.1 nm transition assuming the absence of any significant collisional quenching of Sr(6 'S,),a justifiable assump- tion in this system in view of the magnitude of the Einstein coefficient.Collisional quenching by a gas Q would only lead to a term (1 + kQ[Q]/A,,)in the denominator of eqn (viii) and would not affect the time dependence of the form IF(A = 679.1 nm) exp (-2k't). The above analysis, however, does not enable us to seperate the relative contributions from reactions (6) and (7). Fig. 2( c) shows the first-order kinetic plot constructed from the decay profile for Sr(6 3S,)given in fig. I (c),yielding a slope D. HUSAIN AND G. ROBERTS 95 of [8.66*0.02 (lo)] x lo3s-', double within experimental error of those in fig.2(u) and (6). Fig. 4(a) and (6) show plots of k'[Sr(63S,)] against k'[Ca(43P')] and k'[Sr(5 3P')], respectively, for all recorded decays in noble gases, yielding slopes of 1.908f0.019 (1 a)and 1.946 f0.025 (1 a),clearly establishing energy pooling from measurements in the time-domain but not, of course, separating the terms in k, and kd. The ratios [Sr(6 'Ss,)]/[Sr(5 3P,)] and [Sr(6 3SI)]/[Ca(4 'PI)] at t =0 may be estimated from intercepts of plots of the type given in fig. 2 using the Einstein coefficients for all the transitions indicated above and the gain (G) of the 9797B photomultiplier tube as a function of voltage. This latter property is a sensitive function of the applied voltage. Thus, for the conditions of fig. 1, the transitions at A = 657.3 and 689.3 nm were monitored with a p.m.voltage of 650 V where G =7 X lo4 (E.M.I. catalogue), whereas for the A = 679.1 nm transition [fig. 2(c)] a p.m. voltage of 900 V was employed (G= 1.2 x :06).As these three transitions are so close in wavelength and in view of the crudity of the estimate, no correction was made for the small variation in the sensitivity of the p.m. tube with wavelength. Thus, from the intercepts in fig. 2 we obtain [Sr(6 3S,)]/[Sr(5 'P,)]-4= lop4 and [Sr(6 'SI)]/[Ca(4 'PI)]=4 x Note that the A =679.1 nm transition can only be detected under limited experimental conditions since the signal is lost in the presence of significant pressures of quenching gases such as H2 or D2 (vide infru) and can only be monitored at the low end of the range of quenching gas concentrations employed, for reasons implicit in eqn (viii).SPONTANEOUS EMISSION, DIFFUSION AND COLLISIONAL QUENCHING WITHIN THE 3~J)-~r(5~a(4 'P') EQUILIBRIUM SYSTEM The overall first-order rate coefficients for the decays specific to Ca(43PJ) and Sr(53PJ), k, and kb in eqn (iv), can be seperated into their component parts due to spontaneous emission, diffusion and collisional quenching, i. e.3-s K, and K2[reactions (1) and (2)] are specific to calcium and strontium, respectively, in eqn (ix) and (x) as are the appropriate terms in p/p which include the diffusion coefficients for the atoms. In the case of light noble gases such as He, the collisional quenching terms, ka[Q], are negligible for both Ca(4 3PJ)3and Sr(5 3PJ).5 Hence, for He, k, and kb are constructed from terms due to spontaneous emission and diffusion, and each of these terms has been quantitatively characterised in the previous studies of Husain and Schifino for Ca(4 3PJ)3and Sr(5 3PJ).5Thus, for a given sample, one measurement of k' of both Ca(4 3P,) and Sr(5 3PJ)in He, coupled with the calculated values of k, and kbfrom the previously published data for A,, and p,395together with the calculated value of K3, yields [Sr(5 'So)]/[Ca(4 'So)], which for the conditions of fig.1 is of magnitude 0.03, though values as high as 0.22 have been obtained. This analysis may then be extended to include the collisionai-quenching term, ka[Q]. Even for the heavier noble gases Kr and Xe, collisional quenching is relatively small [Ca(4 'PJ): kKr<4 x and kxe= (2.4*0.3) x cm3 molecule-' s-';~ Sr(5 3PJ): k,<5.5 x and k,,<3.1 x cm3 molecule-' s-' 5] and the variation of k' with p(nob1e gas) primarily arises from the diffusional term, P/p.'*' Thus, having determined [Sr( 5 'So)]/[Ca(4 'So)] from one measurement on a given sample in He, k' [eqn (iv)] can be calculated for 3~J)AND ~r(5KINETIC BEHAVIOUR OF ~a(4 'pJ) 10 8 6 4 4 6 a 10 4 6 8 10 k;a,c/ 103 s-l Fig.5. Comparison of the pseudo-first-order rate coefficients (k')for the decay of Ca(4 3PJ) and Sr(5 3PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(4 3Pl)+Ca(4 'So),A =657.3 nm] in the presence of strontium vapour and krypton at various pressures (T=950 K).(a)Ca(43PJ):kLxpt,against kLalC;(b)Sr(5 'PJ):kLxptlagainst kLC. all other mixtures using the components in eqn (ix) and (x) to yield kLa,cand the result compared with the experimental result, kdxptl,for a given experiment. Fig. 5(a) and (6) show plots of k:,,,, against k& for Ca(4 'PJ)and Sr(5 3PJ), respectively, for the decay in Kr where collisional quenching is negligible and where the various points on the plots arise from the different diffusional contributions at different pressures. Fig. 5(a) and (b)are characterised by slopes of 1.12f0.062 (1a)and 0.970 *0.014 (1a),respectively, the limited accuracy resulting from the relatively small variation in k' due to diffusional loss when the pressure is varied in this type of system.Given that the dominant factor governing the slope of a plot of k:xptl against kLalcin the case of noble bases arises from the variation of the diffusional term with pressure, the effect, even of a small contribution due to collisional quenching, as in the case of xenon, yields a larger range of values of k' and an improved correlation between kLxptland kLalc. This is shown in fig. 6(a)and (b)for Ca(4 3PJ)and Sr( 5 3PJ),respectively, yielding slopes of 1.001f0.026 (1a) and 1.003 f0.030 (1 a). The method can, of course, be applied to gases where collisional quenching is significant and which are carried out at effectively constant pressure with an excess of He so that the terms in A,,, p and [Sr(5 'So)]/[Ca(4 'So)]are held constant with a given sample.This will be dealt with in detail in a subsequent paper for all the collisional-quenching partners that have been studied individually with both Ca(4 3PJ)and Sr(5 3PJ);3-6it will also deal with the general problem of extracting individual rate data from coupled systems. In this paper, we restrict the examples to the two isotopes H2 and D2, whose collisional behaviour with Ca(43PJ) and Sr(5 'PJ)are of wide fundamental interest [ref. (3) and (5) and references contained therein]. Hence, following the procedure described above using a constant total pressure with He and the quenching rate data for these gases reported by Husain and Schifino [Ca(4 3PJ): kH2(1000 K) =(6.0f0.6) x and kD2(1000 K) = (2.7 f0.3) x lopt4 cm3 molecule-' s-I ;Sr(5 3PJ):kH2(950K) =(3.7 f0.4) x and D.HUSAIN AND G. ROBERTS 2 10 18 26 2 10 18 26 K,JI o3s-1 Fig. 6. Comparison of the pseudo-first-order rate coefficients (k') for the decay of Ca(4 'PJ) and Sr(5 3PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(4 'P1)+-Ca(4 'So),A =657.3 nm] in the presence of strontium vapour and xenon at various pressures (T=950 K). (a)Ca(4 3PJ):kk,,,, against kLalc;(b) Sr(5 'PJ):kkxptlagainst kLalc-12 20 28 36 6 12 18 24 k&/ 1o3s-Fig. 7. Comparison of the pseudo-first-order rate coefficients (k') for the decay of Ca(4 'PJ) and Sr(5 3PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(4 'P,)+-Ca(4 'So),A =657.3 nm] in the presence of strontium vapour, helium buffer gas and hydrogen as a quenching gas (pHe= 30 Torr, T= 950 K).(a)Ca(4 3PJ):kLxptlagainst (b)Sr(5 'P~):kkxptlagainst kLalc. kD2(950K) = (3.0*0.2) x cm3 molecule-' s-~],~*~we compare k~,,,,and as indicated. The results are shown in fig. 7 and 8. The smallest value of k' with a particular symbol represented on a diagram describes the result with no quenching gas for a new sample and therefore determines the value of [Sr(S 'So)]/[Ca(4 'So)] for that sample. The slopes of fig. 7(a) and (b)and 8(a)and (b)are 0.997*0.006 (I u),0.999f0.007 ( 1a),1.004f0.007 (I a)and 0.991 *0.006 (1a),respectively. Thus the correlations given in fig. 5-8 justify the data used to construct values of 'P~) 'pJ)KINETIC BEHAVIOUR OF ~a(4 AND ~r(5 -22 -'v) 19-m2 h -~16 -0X Y 7 10 13 16 19 22 25 103s-I Fig.8. Comparison of the psdueo-first-order rate coefficients (k') for the decay of Ca(4 'P') and Sr(5 3PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(4 'P,) 4-Ca(4 'So),A = 657.3 nm] in the presence of strontium vapour, helium buffer gas and deuterium as a quenching gas (pHe= 30 Torr, T = 950 K). (a) Ca(4 'PJ):kLxptlagainst kLalc;(b) ~r(5'P'): k&l against k&. including the collisional-quenching data of Husain and Schifino for H2 and D2.'.' The sensitivity of kLalcto the errors in the collisional-quenching data will be dealt with in the following paper for all the quenching gases that have been investigated in pure systems of Ca and Sr.In general terms, whilst an equilibrium system is effectively used for collisional rate-data determination by difference methods through eqn (iv), and the sensitivity depends on the magnitudes of the components in that equation, including the ratio [Sr(5 'So)]/[Ca(4 'So)] for a given sample, it is found that the fractional error in collisional rate constants generates comparable fractional errors in kLalc. That the correlation between kLxptland kLa1c is so good, supporting the earlier individual rate measurements,'~' justifies the extension of the application of the equilibrium system to a wide range of collision partners. We thank the S.E.R.C. for an equipment grant for the purchase of the dye-laser system and for a Research Studentship held by one of us (G.R.), during the tenure of which this work was carried out.Atomic Energy Levels, Natl Bur. Stand. Re$ Data Ser. 35, ed. C. E. Moore (U.S. Government Printing Office, Washington D.C., 1971), vol. 1-111. W. H. Breckenridge and H. Umemoto, in Advances in Chemical Physics: Dynamics ofthe Excited State, ed. K. P. Lawley (Wiley, New York, 1982), vol. L, p. 325. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans. 2, 1983, 79, 1265. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans. 2, 1983, 79, 1677. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans. 2, 1984, 80, 321. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans. 2, 1984, 80, 647. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans.2, 1982, 78, 2083. D. Husain and J. Schifino, J. Chem. SOC., Faraday Trans. 2, 1983, 79 919. G. Taieb and H. P. Broida, J. Chem. Phys., 1976, 65, 2914. lo B. Bourguignon, J. Rostas and G. Taieb, J. Chem. Phys., 1982, 77, 2979. I' F. Engelke, Chem. Phys., 1979, 44,213. D. HUSAIN AND G. ROBERTS 99 " J. W. Cox and P. J. Dagdigian, J. Phys. Chem., 1982, 86, 3738. l3 L. Pasternack and P. J. Dagdigian, Chem. Phys., 1978, 33, 1. l4 J. A. Irvin and P. J. Dagdigian, J. Chem. Phys.,.1981,74, 6178. l5 A. Kowalski and M. Menzinger, Chem. Phys. Lett., 1981, 78, 461. 16 M. Menzinger, in Advances in Chemical Physics: Potential Energy Surfaces, ed. K. P. Lawley (Wiley, New York, 1980), vol. XLII, p. 1. 17 D. J. Benard, P. J. Love and W.D. Slafer, Chem. Phys. Lett., 1977, 48, 321. R. J. Malins, D. Logan and D. J. Benard, Chem. Phys. Lett., 1981, 83, 605. l9 C. H. Corliss and W. R. Bozmann, Experimental Transition Probabilities for Spectral Lines of Seventy Elements, Natl Bur. Stand. Monogr. 53 (U.S. Department of Commerce, Washington D.C., 1962). 2o A. N. Nesmiyanov, Vapour Pressure ofthe Elements (Academic Press, New York, 1963). 'I C. Smithels, Metals Reference Handbook (Butterworths, London, 5th edn, 1976). 22 R. Hultgren, P. D. Desai, D. T. Hawkins, M. Gleiser, K. K. Kelly and D. D. Wagman, Selected Values of the Thermodynamic Properties of the Elements (American Society for Metals, Metals Park, Ohio, 1973). 23 A. B. Callear and J. D. Lambert, in Comprehensive Chemical Kinetics: The Formation and Decay of Excited Species, ed. C. H. Bamford and C. F. H. Tipper (Elsevier, Amsterdam, 1969), vol. 3. 24 T. J. McIlrath and J. L. Carlsten, J. Phys. B, 1973, 6, 697. 25 W. L. Wiese, M. W. Smith and B. M. Miles, Atomic Transition Probabilities (U.S. Government Printing Office, Washington D.C., 1962), vol. I and 11. (PAPER 4/747)

ISSN:0300-9238    

DOI:10.1039/F29858100087

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1985, 81, 101-113 Time-resolved Emission Studies of the Collisional Quenching of Ca(4 "') and Sr(5 'P') by Various Gases in an Equilibrium-coupled System following Pulsed Dye-laser Excitation at A = 657.3 nm [Ca(4 ",) +-Ca(4 'So)] BY DAVIDHUSAIN*AND GARETHROBERTS Department of Physical Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EP Received 9th May, 1984 The collisional quenching of Ca(4 'P,) and Sr(5 'P,) has been studied for an equilibrium system coupled by the process 'P,) + ~r(5'so)e ~a(4 'P~)~a(4 'so)+ ~r(5 and established following pulsed dye-laser excitation of Ca(4 'So)to Ca(4 'PI)at A = 657.3 nm in mixtures of calcium and strontium vapour in the presence of an excess of helium buffer gas and added quenching gases.Ca(4'P1), Sr(5'P,) and Sr(6'S,) were monitored by time-resolved emission at A =657.3, 689.3 rSr(5 'P,)-+ Sr(5 'S,)+hv) and 679.1 nm [Sr(6 'S,)+ Sr(5 3P0)+ hv],respectively, using boxcar integration. The measured first-order decay coefficients for Ca(4 'P,) and Sr(5 'P,) under equilibrium-coupled conditions were found to be in quantitative agreement with previously measured values of absolute second- order rate constants for the quenching gases Kr, Xe, H2, D2, N2, CO, NO, N20, C02,NH3, CH4, CF4, C2H2, C2H4 and C6H6, obtained using the present system, employing signal averaging, on decays of Ca(4 'P,) and Sr(5 'P,) individually. Finally, energy pooling accord- ing to the processes Ca(43P,)+Sr(5 'P,) Sr(6 3S,)+Ca(4 'So)-+ Sr(5 'PJ)Sr(5 'P') -+ Sr(63S,)+Sr(5'So) was investigated quantitatively from measurements at A = 679.1 nm in the time domain and in the presence of the quenching gases.Kinetic studies of fundamental collisional processes undergone by atoms in specific electronic states in an equilibrium-coupled system are governed by two basic experimental requirements. First, the coupling between the electronic states must be established sufficiently rapidly and the quasi-localised Boltzmann equilib- rium sustained throughout kinetic measurements and not disturbed by the collisional processes. This normally requires that the energy spacing between electronic states of a given species be sufficiently small so as to maintain Boltzmann equilibrium by collisions with a buffer gas.Alternatively, the electronic states of different chemical species, Ca(4 3PJ)and Sr(5 3PJ)in the present case, must be at energies so as to be coupled by a near-resonant exchange process. Secondly, a much more restrictive condition is the accompanying design of the experimental system which permits separation of the individual kinetic processes pertinent to the specific electronic states in the coupled system. A recent example is seen in the time-resolved vacuum 101 COLLISIONAL QUENCHING OF ~a(4 'P')'P') AND ~r(5 ultraviolet resonance absorption measurements by Clark and Husain on the spin- orbit components of atomic chlorine.'-' At short times following the pulsed irradi- ation of C,-chlorofluorocarbons ('freons'), CF,CI,-,, the low-lying spin-orbit state, C1[3p'(Pl/2)], 882 cm-' above the 3p5(2P3/2) ground ~tate,~ can be monitored as a separate kinetic entity before the onset of Boltzmann equilibrium.With the establish- ment of the equilibrium in the longer time domain, both spin-orbit components will then demonstrate the identical kinetics of a coupled system, incorporating in the decay coefficients for either state rate constants specific to both Cl(3 'P3/2)and Cl(3 2P1/2).1-3 A particularly important practical example of the kinetic study of a coupled system involving electronic states of different chemical species is the pioneering investigation of Derwent and Thrush' of the electronic energy exchange process: 'Zg) * 1(52P3/2)+02(~1(5 2P1/2)+02(X 'Ag), AE =315 cm-' 677 (1) which laid the basis of later work on the continuous-wave electronic-transition laser operating on the iodine The exchange reaction (1) sustained a Boltzmann equilibrium in a a flow-discharge system and hence the equilibrium condition, coupled with measurements of the relative intensities of the emission process 1(52P1/2)-+ 1(5'P3/2)+hu (A = 1.315 pm) (2) 02(a'A,)(v'=O) -+ Oz(X'Z,)(v"=O)+hv (A = 1.27 pm) (3) enabled Derwent and Thrush' to measure the ratio of the Einstein coefficients for reactions (2) and (3).This type of emission-ratio measurement is used in the reverse manner in the present investigation to measure the ratio of populations in the electronic states Sr(6 3S1),Ca(4 'PI)and Sr(5 'PI)using known Einstein coefficients. Of course, a whole range of investigations have been reported in recent years on separate studies of the collisional quenching of 1(5 2P1/2)10711and 02(a and reference can be made to a variety of reviews.In the present study we describe the collisional quenching of Ca(4'P') and Sr(5 'PJ),1.888 and 1.807 eV, respectively, above the ns2('So)ground states,6 with a range of gases. The electronically excited atoms are generated in an equilibrium system via Ca(43PJ)+Sr(51So) Ca(4'SO)+Sr(5'PJ), AE =-654cm-I6 (4) following pulsed dye-laser excitation of calcium vapour at A =657.3 nm [Ca(4 'PI)eCa(4 'So)]in the presence of strontium vapour and an excess of helium buffer gas. This employs the experimental arrangement described in the preceding paperI4 using repetitive pulsing on a slow-flow system, kinetically equivalent to a static system, pre-trigger photomultiplier gating, boxcar integration and XY record-ing.It was established in this earlier paper from the time dependence of the decay measurements on Ca(4 'PI)--+Ca(4 'So)+hv (A =657.3 nm) and Sr(5 'PI)--, Sr(5 'So)+hu (A =689.3 nm) that the equality in the measured first-order decay coefficients for Ca(4 'P') and Sr(5 'PJ)were so highly correlated, namely unity to an accuracy of 1-2%, that the decay coefficients for each 'P' state could be broken down quantitatively into the kinetic components of which they are comprised, i.e. spontaneous emission, diffusion, the equilibrium condition for reaction (4) and collisional quenching.The system can then be used as a kinetic tool for collisional quenching studies. In this paper, the decay coefficients for Ca(4 'P') and Sr(5 'P,) are measured as a function of the concentration of some fifteen added gases, including Xe, Kr, H2 and D2,as described previo~sly,'~ and are compared with the D. HUSAIN AND G. ROBERTS results using absolute rate data derived from signal averaging measurements on systems involving Ca(4 'PJ)and Sr(5 'P,) individually, as described by Husain and Schifin~.'~-'~The sensitivity of the system is demonstrated by comparison of the calculated first-order decay coefficients, k', for both Ca(4 'P') and Sr(5 'PJ)with the maximum and minimum values of k' obtained from the error limits for the individual quenching rate constants in the input data.Finally, the influence of collisional quenching of Ca(4 'P,) and Sr(5 'PJ)by various gases on the energy- pooling processes Ca(4 'PJ)+ Sr(5 'PJ)-+ Sr(6 'S1) +Ca(4 'So) (5) Sr(5 'P,)+Sr(5 'PJ)-+ Sr(6'S1)+Sr(5 'So) (6) demonstrated in the previous paperI4 is investigated quantitatively by comparison of the measured first-order decay coefficients for Ca(4 'P') (A =657.3 nm), Sr(5 'P') (A = 689.3 nm) and Sr(6 'S1) [A =679.1 nm, Sr(6 'S1) -+ Sr(5 'So)+ hv] on a given sample of Ca/Sr solid. The present overall system is compared with that described by Malins et aZ.,I9 who report phase-shift measurements on Sr(5'PJ) using the A =689.3 nm transition, but not on Ca(4 'P,) itself, following pulsed dye-laser excitation of Ca(4 'So)to Ca(4 'P,) in the presence of strontium vapour and noble gases only, and who employ the kinetic conditions neither of an equilibrium-coupled system through reaction (4) nor of an equilibrium in the spin-orbit components of Sr(5 3Po-1-2)to describe the kinetic decay by emission from Sr(5 'PI).EXPERIMENTAL The experimental arrangement has been described in the preceding paper14 and is briefly summarised here. Ca(4 3P1)is generated by pulsed dye-laser excitation (A = 657.3 nm, repeti- tion rate 10 Hz) of Ca(4 'So)in the presence of strontium vapour, an excess of helium buffer gas and an added quenching gas in a slow-flow system, kinetically equivalent to a static system, at an elevated temperature (950 K).Following rapid Boltzmann equilibration within Ca(4 P')20 and the energy-transfer reaction (4), time-resolved emission, optically separated by means of a small monochromator, was monitored at A = 657.3, 689.3 and 679.1 nm, using boxcar integration to measure the decays of Ca(4 3PJ),Sr(5 'P,) and Sr(6 'S1), respectively. The Einstein coefficients for these transitions have been described in the previous paper.I4 The output was fed to an XY recorder for kinetic analysis. All materials [Ca(solid), Sr(solid), He, b,Xe, H2, D2, N2, NO, CO, COz, N20, NH3,CH,, CF,, C2H2, C2H4, C6H6 and Cresyl Violet Perchlorate 6701 were prepared essentially as described in previous publications. 14-18,21,22 RESULTS AND DISCUSSION Individual kinetic studies on the collisional quenching of Ca(4 'P,) and Sr(5 3PJ) by molecular nitrogen, 16,'* as an example, using time-resolved atomic emission following pulsed dye-laser excitation, have shown the removal rates of these two electronically excited atoms to differ by a factor of approximately two orders of magnitude (k[Ca(4'P')+N2] (1000 K)= (8.9k0.9) x lo-'' l6 and k[Sr(5 'PJ)+N2] (950 K) = (3.2 f0.2) x lo-" cm' molecule-' s-l Is}.However, in an equilibrium sys- tem coupled by reaction (4), the decays of these atomic states in the presence of molecular nitrogen should show identical pseudo-first-order rates (vide infra). This is shown qualitatively in fig. l(a) and (b),which give examples of the XY output for the time-resolved atomic emission at h = 657.3 nm [Ca(4 'PI)-+ Ca(4 'So)+ hv] and A = 689.3 nm [Sr(5 'PI)--+ Sr(5 'So)+hv] in the presence of a relatively low pressure of N2 where the decays of both atomic states are measured over the same COLLISIONAL QUENCHING OF ~a(4 'P~)3~J)AND ~r(5 250 500 750 1000 0 250 500 750 1000 Fig.1. Examples of the output of the XY recorder indicating the decay of time-resolved atomic emission (IF) at (a) A =657.3 nm [Ca(4 3P,)-+ Ca(4 'So)], (b) A = 689.3 nm [Sr(5 3P1)-+Sr(5 'So)]and (c) and (d) A =679.1 nm [Sr(6 3S1)-+ Sr(5 3P0)]following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P,)+Ca(4 'So)]in the presence of strontium vapour, helium buffer gas and nitrogen as a quenching gas (ptota,with He =30 Torr, T=950K).[N2]/1014m~leculecm-3; (a) 1.5, (b) 1.5, (c) 1.5 and (d) 3.1. timescale. Fig. 2(a) and (b) show the first-order kinetic plots constructed from such decays from which the pseudo-first-order rate coefficients, k', for each atomic state are derived. In view of the energy-pooling processes to yield Sr(6 3S,) according to reactions (5) and (6), and the time dependence of this state of the form [Sr(6 'S1)lf= [Sr(6 3Sl)]f=oexp (-2k't) demonstrated in the preceding paper,14 the study of the 'S1 state on timescales one-half of those employed for monitoring Ca(4 'PI)and Sr(5 3PI)under identical conditions is to be expected and is demon- strated in fig. 1(c) with the appropriate first-order kinetic plot presented in fig. 2( c). D. HUSAIN AND G.ROBERTS 6 5 4 3 2 Fig. 2. Examples of first-order kinetic plots for the decay of the time-resolved atomic emission [In(IF) against 13 at (a) A = 657.3 nm [Ca(4 3P1)-+Ca(4 'So)], (b) A = 689.3 nm {Sr(5 3P1)--* Sr(5 'So)] and (c) and (d) A = 679.1 nm [Sr(6 3S1)-+ Sr(5 3P0)]following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P1)+Ca(4 'So)]in the presence of strontium vapour, helium buffer gas and nitrogen as a quenching gas (ptota,with He = 30 Tom, T = 950 K). [NJ/ 1014molecule ~m-~: (a) 1.5, (b) 1.5, (c) 1.5 and (d) 3.1. COLLIS~ONALQUENCHING OF ~a(4 'P')'P') AND ~r(5 5 4 h 4-v 53 2 1 I 1 I ~~ 0 250 500 750 1000 tlw Fig. 3. Examples of first-order kinetic plots for the decay of the time-resolved atomic emission [ln(l,) against t] at A = 657.3 nm [Ca(4 3P,)-+Ca(4 'So)]indicating the decay of Ca(4 'P') following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P1)+Ca(4 'So)]in the presence of strontium vapour, helium buffer gas and relatively high concentrations of N2 (ptotalwith H~ = 30 Tom, T = 950 K).[NJ 1 Oi4 molecule cmS3: 0,0.0; 8,6.1;0,15.2. The dependence of [Sr(6 'SJ, on [Ca(4 'PJ)],[Sr(5 'PJ)ltand [Sr(5 'P,)]: clearly prevents the study of Sr(6'SI) over a large concentration range of quenching gas, such as N2, necessary for investigating the decay of the two 'P' states. Hence, energy-pooling measurements are confined to low concentrations of quenching gases [e.g. fig. l(c) and (d)],whereas kinetic studies of Ca(4 'PJ)and Sr(5 'PJ)by such quenching gases in the coupled system have to be made in concentration ranges over most of which Sr(6'S1) cannot be monitored on the basis of sensitivity. A coupled system particularly requires the true coefficients for decay specific to Ca(4 'P') and Sr(5 'P,) to be relatively large, as these must be extracted effectively by difference from decays for the coupled system.We may note that the ordinate of diagrams of the type shown in fig. 1 is in arbitrary units and reflects electronic amplification as well as emission intensity. Fig. 3 shows the pseudo-first-order kinetic plots for the decay of Ca(43PJ) in the coupled system at the higher concentrations of N2 better suited for quenching studies, with the analogous plots for the decay of Sr(5 3PJ)under identical conditions shown in fig.4. The decay coefficients, k'[Ca(4 'P')] and k'[Sr(5 3PJ)],for the coupled reaction (4) should be identical in a given experiment and expressed by:14 -d{ln [Ca(4 'P,)]}/dt = -d{ln [Sr(5 'P,)]}/dt = k' D. HUSAIN AND G. ROBERTS 0 250 500 750 1000 ~IW Fig. 4. Examples of first-order kinetic plots for the decay of the time-resolved atomic emission [In(l,) against t] at A = 689.3 nm [Sr(5 3P,)+Sr(5 'So)]indicating the decay of Sr(5 3P,) following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P1)cCa(4 'So)] in the presence of strontium vapour, helium buffer gas and relatively high concentrations of N2(ptotalwith He =30 Torr, T = 950 K). [NJ/ lOI4 molecule cmP3: 0,0.0; 8,6.1 ; 0,15.2.where ka and kb describe pseudo-first-order decay coefficients specific to Ca(4 'P,) and Sr(5 'P,), re~pective1y.I~These are given by the general form ka,b=Anm/(l+l/Kl +K~)+P/PH~+~Q[QI (ii) where the symbols have their usual meaning and where He would be the buffer gas in this case,I4 and the ratio [Sr(5 'So)]/[Ca(4 'So)] is determined from a given sample of Ca( solid) and Sr(solid), as described previo~sly.'~ Similarly the Einstein coefficients for the spontaneous emission processes 'PI --* 'S,+hv may be taken from the data published by Husain and Schifin~"~'' together with the diffusion decay parameters, P, for the Hence, we may calculate an expected first-order decay coefficient, kLalc,for Ca(4 'PJ)and Sr(5 'PJ) in a given coupled system using the input data for kN2in this case for either atomic state following Husain and Schifino.'6*'8 This may be compared with the observed decay coefficient, kLxptl,for Ca(4 3PJ)and Sr(5 'P,), respectively.The result is shown in fig. 5 for NZ. The decay coefficients (k')for Ca(4 'P,) and Sr(5 'P,) are clearly identical by visual inspection of fig. ~(a)and (b),and the correlation between kdxptland kLalc is satisfactory to within ca. 1% in both cases. Again, as previ~usly,'~ the symbol corresponding to the lowest value of k' on a diagram of the type shown in fig. 5 (e.g.0or 0)describes k' in the presence of He only and yields [Sr(5 'So)]/[Ca(4 'So)] for a given sample. Surface poisoning requires the use of more than one sample for most quenching gases investigated with Ca(4 'P,)and Sr(5 3PJ).15-18Comparable correlations between kLxptland kLalcfor given input quenching data16318 are shown in fig.6 for the other diatomic molecules investigated here, namely CO and NO. The sensitivity of to k, will be considered later. COLLISIONAL QUENCHING OF Ca(4 'P,) AND ~r(5108 'P,) I 8 12 16 20 24 k&/ 10' s-' Fig. 5. Comparison of pseudo-first-order rate coefficients (k')for the decay of Ca(4 'P,) and Sr(53PJ)following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P,)-Ca(4 'So), A = 657.3 nm] in the presence of strontium vapour, helium buffer gas and the quenching gas N2 (kixptlagainst kLalc,ptotalwith He = 30 Torr, T = 950 K).(a)Ca(4 'P') [slope = 0.987 f0.007 (1a)];(b)Sr(5 'P') [slope = 0.990f0.007 ( 1 a)]. Discussion of specific pathways to defined electronic states of products resulting from the collision between Ca(4 'PJ)and Sr(5 'P,) with a range of gases have been given in previous paper~,I'-~* particularly for the triatomic molecules N20 and C02,16.18where some comparison can be made with chemiluminescence measure- ments that have employed molecular-beam technique^.^'-^^ Here we restrict our considerations to the rate data themselves, presented for N20 and C02 in fig. 7 (k:,,,, against kLalc) based on the absolute second-order rate constants for the quenching of Ca(4 'P,) and Sr(5 3PJ)reported by Husain and Schifino.l6>" Similar measurements to those described above have been made for quenching of Ca(4 'P,) and Sr(5 'P,) in the coupled system for a range of polyatomic molecules with the correlation in the slopes of k:, tl against kLalcgiven in brackets in each case: NH3 [Ca(4'PJ), 1.002f0.004; Sr(5 9P,), 1.004f0.007], CH, [Ca(4'PJ), 1.010~0.010; Sr(5 'P'), 0.995 f0.01 13, CF, [Ca(4 'P,), 0.990 f 0.020; Sr(5 'P,), 0.980f0.0261, C2H2 [Ca(4 'P'), 1.032k0.032; Sr(5 'PJ),1.009*0.039], C2H4 [Ca(4 'P,), 0.995 f0.016; 'P,), 0.944It0.0181 and C~H~ 0.977*0.016; ~r(5 ~r(5 (~a(4'~,), 'P,), 0.969f0.014].Quenching of Ca(4'P') and Sr(5'P') by the molecules NH3, CH, and CF, has primarily been considered previously in thermochemical terms from the viewpoint of atom abstraction,l6,l* and the rapid collisional removal by C2H2, C2H4 and C6H6 in terms of chemical interaction with the welectron s4:stems.The kinetic effects of electronic quenching of Ca(4 PJ)and Sr(5 'PJ)on energy pooling [reactions (5) and (6)] are shown in fig. 8, which may be compared with energy pooling in He alone when collisional removal of the 'P, states by the noble gas is negligible. l4 The energy-pooling mechanism14 as indicated earlier in this paper predicts a plot of kdxptlfor Sr(6'S1) against k:xptl or kLalcfor Ca(4'P,) or Sr(5 'P') in the coupled system of slope 2. The resulting values were found to be 1.947 *0.044 ( 1a)and 1.976 f0.045 ( 1a),respectively. The points in fig. 8 represent data for all the quenching gases studied and are usually restricted to one or two points for each quenching gas at the lowest pressures of the quenching species D.HUSAIN AND G. ROBERTS '4 4 8 12 16 20 8 12 16 20 \ a X 2du 20 20 16 16 12 12 8 8 4 4 8 12 16 20 8 12 16 20 Fig. 6. Comparison of pseudo-first-order rate coefficients (k')for the decay of Ca(4 'PJ)and Sr(5'PI)following the pulsed dye-laser excitation of calcium vapour [Ca(4 3Pl)+Ca(4 'So), A = 657.3 nm] in the presence of strontium vapour, helium buffer gas and the quenching gases CO and NO (k& against kLalo ptotal He = 30 Torr, T = 950 K). (a) Ca(4 'P,) and CO [slope= 1.029*0.009 (lo)]; (b)Sr(5 'PJ)and CO [slope= 1.038*0.016 (lo)];(c) Ca(4'P') and NO [slope = 1.OOO f0.007 (1o)];(d) Sr(5 3PJ)and NO [slope =0.990*0.099 ( I o)]. employed.For some gases demonstrating particularly high collisional quenching efficiencies with Ca(4 3PJ)and Sr(5 'P'), it was not possible to monitor [Sr(6 'S1)] as a function of time via the h =679.1 nm transition for reasons given earlier in this paper. The general conclusion that may be drawn from fig. 8 is the continued sustenance of the energy-pooling reactions (5) and (6) coupled with reaction (4) during significant collisional quenching of the 'P' states. A convenient test of the sensitivity of k, for Ca(4 'P') and Sr(5 'P') on the observed values of kdxptlin the coupled system [eqn (i)] can be made using the quoted errors in the absolute second-order rate data reported by Husain and S~hifino.'~-*~We may note, first, that the errors (1(T) in the plots of lidxptl against kLalc were of the order of 1-2% and that the errors (la)in k, of Husain and COLLISIONAL QUENCHING OF ~a(4 'pJ)3~J)AND ~r(5 18 15 12 9 6 3 23 6 9 12 15 18 3 6 9 12 IS 18 18 2X 15 12 9 6 3 3 6 9 12 15 18 3 6 9 12 15 18 kLaIc/1 O3 s-' Fig.7. Comparison of pseudo-first-order rate coefficients (k')for the decay of Ca(4 3PJ)and Sr(5 3PJ)following the pulsed dye-laser excitation of calcium vapour [Ca(4 3P1)+-Ca(4 'So), A = 657.3 nm] in the presence of strontium vapour, helium buffer gas and the quenching gases N20 and C02 ( kLxptlagainst kLalc,ptotalwith He = 30 Torr, T = 950 K). (a)Ca(4 3P~)and N20 [slope= 1.012*0.016 (la)]; (b)Sr(5 3PJ)and N20 [slope=0.996*0.016 (la)]; (c) Ca(43PJ) and C02 [slope = 0.969i0.013 (la)]; (d)Sr(5 3PJ)and C02 [slope = 0.983 *0.012 (lu)].Schifino's-'8 were of the order of 5-10% or greater for the various quenching gases. Here we will employ the input data of Husain and Schifinols-" and consider the resulting variation in kLalcusing (a)k, + (1a)and (b)kQ-(1a)for both Ca(4 'PJ) and Sr(5 'P') using the experimental conditions (ie.[Q]) for the values of k' closest to the mid-point of each plot of k~,,,,against The results are presented in tables 1 and 2 for Ca(4 'PJ)and Sr(5 'PJ),respectively, for all the quenching species investigated by the equilibrium method. The result for k[Sr(5 'PJ)+ NO] = (1.8f 0.1) X 1OW' I cm3 molecule-' s-' (950 K), not measured by Husain and Schifino" or by any other investigator, was obtained from the best fit of the data for kLxptlobtained here.The sensitivity will, of course, depend on the magnitude of [Sr(5 'So)]/[Ca(4 'So)]in the particular experimental system. Whilst there is a marginal reduction in many cases in the fractional errors in kLalc compared with the input data of Husain and Schifino for kq,1s-18the errors are clearly comparable. From the correlation between kLalc and kLxptl(unity to within 1-2'/0), neither is further comparison with kdxptlnecessary, as seen from the figures containing the D. HUSAIN AND G. ROBERTS 3 5 7 9 11 3 5 7 9 11 kLaIc/1O3 s-' Fig. 8. Comparison of the pseudo-first-order rate coefficients (k')for the decay of Sr(6 3S1), Ca(4 3PJ) and Sr(5 3PJ) following the pulsed dye-laser excitation of calcium vapour [Ca(43P1)-Ca(4 'So),A = 657.3 nm] in the presence of strontium vapour, helium buffer gas with He = 30 Torr, T = 950 K).(a)and low pressures of various quenching gases (ptotal k'[Sr(6 'SI)] against k'[Ca(4 3PJ)] [slope = 1.947f 0.044 (1 a)];(b) k'[Sr(6 'S1)] against k'[Sr( 5 3PJ)][slope = 1.976f0.045 ( 1a)]. Table 1. Comparison of the variation in kLalc for Ca(43PJ) with the la errors in k, for different quenching gases, Q kQ/cm3(atom-' Q [Q]/(atom or molecule) cmP3 or molecule-') s-' ( la)'5716(1000 K) k' (k + la) cay1 o'Qs-1 kL,,,(k -la)/ 10%' k;,,,(av.)/ lo3s-i 8.1 x 1017 4 x - - - 8.1 x 1017 7.6 x 1015 2.4 f0.3 X lopi4 6.0*0.6 X 24.7 16.1 20.3 14.4 22.5 f 2.2 15.2f 0.7 1.1 x loi6 2.7 f0.3 X 23.4 21.6 22.5 f 0.9 1.5x loi5 1.5 X 10" 3.7 f0.5 X I Owi3 8.9f 0.9 x lopi3 12.5 11.9 11.6 11.3 12.1 f0.5 11.6f0.3 3.0 x 1015 1.5x 10'5 3.2f 0.3 X 6.5 *0.5 X 12.9 10.0 12.3 9.0 12.6 f 0.3 9.5 f0.5 3.0 x 1015 3.1 f0.3 X 8.9 8.3 8.6f0.3 1.5x 1015 2.5 *0.3 X 12.7 12.0 12.3 f 0.3 1.1 x 10l6 5.4zt0.4 X 14.4 12.7 13.6f0.8 1.2 x 10l6 1.9f0.2 x 9.1 8.4 8.8f0.3 7.5 x 1014 i.4+0.2 x 10-l~ 6.1 5.9 6.0 f0.1 7.6 x 1013 2.3f0.3 X 6.3 6.0 6.1 fO.1 1.5 X loi6 6.0 f0.2 X 8.0 7.5 7.7 f0.3 data, nor is the onerous procedure of extracting k, point by point, as the results lie well within the (la) errors reported by Husain and S~hifino.'~-'~ The present results have shown that the rate data for the decay of Ca(4 'PJ)and Sr(5 'PJ)under the conditions of an equilibrium-coupled system are quantitatively consistent, within COLLISIONAL QUENCHING OF ~a(4 3~,)3~J)AND ~r(5 Table 2.Comparison of the variation in kblCfor Sr(5 3P,) with the la errors in k, for different quenching gases, Q k,/ cm'( atom-' Q [Q]/(atom or molecule) cmP3 or molecule-') s-' (la)17*18(950K) k&( k + 1a) / 103Qs-' kLalc(k, -1u) kLalc(av.)/ lo3 s-l / lo3s-l Kr 8.1 x 1017 5.5 x 10-l~ 9.4 Xe 8.1 x 1017 3.1 x 10-l~ - - 22 H2 7.6 x 1015 3.7 f0.4 x 16.5 14.7 15.6f0.9 D2 1.1 x 10I6 3.0f0.2 x 23.5 21.7 22.6 f0.9 N2 1.5 x 1015 3.2f0.2 x lo-" 12.3 11.3 11.8*0.5 co 1.5 x 1015 2.0*0.1 x lo-'' 11.6 10.9 11.3 f0.4 NO 3.0 x 1015 1.8*0.1 x lo-" 12.9 12.2 12.6f0.4 N20 1.5 XlO" 1.5*0.2~10-~' 9.8 8.8 9.3 f0.5 C02 3.0~10'~ 1.3fO.l x lo-" 10.1 9.3 9.7f0.4 NH, 1.5 x 1015 2.2f0.1 x lo-" 12.6 12.0 12.3f0.3 CH4 1.1 ~10'~ 1.0f0.2 x 10-l2 14.7 12.9 13.8f0.9 CF4 1.2 x 10l6 7.8 f0.7 X 9.1 8.4 8.7 f0.4 C2H2 7.6 X 1014 1.3f0.1 x10-" 6.3 6.0 6.2 f0.2 C2H4 C6H6 7.6 x 1013 1.5 x 4.2 f0.3 x lo-'' 2.4 f0.2 X lo-'' 6.7 8.0 6.4 7.5 6.5 f0.2 7.8 *0.3 the errors, with the quenching rate data reported from measurements made individually on Ca(4 'P') and Sr(5 3PJ).15-18However, it must be stressed that the present type of system does not give rise to k, directly, but yields values of k,.by difference.This procedure is clearly optimised with the improved accuracy resulting from the use of signal averaging as employed in the present arrangement and in the individual measurements on Ca(4 'P') and Sr(5 3PJ)15-18and in contrast with previous quenching measurements on the two optically metastable states generated by pulsed dye-laser excitation.We thank the S.E.R.C. for an equipment grant for the purchase of the dye-laser system and for a Research Studentship held by one of us (G.R.), during the tenure of which this work was carried out. ' R. H. Clark and D. Husain, J. Chem. SOC., Faraday Trans. 2, 1984, 80, 97. 'R. H. Clark and D. Husain, J. Photochem., 1983, 21, 93. R. H. Clark and D. Husain, J. Photochem., 1984, 24, 103. P. R. Davies and K. D. Russell, Chem. Phys. Lett., 1979, 67, 440.R. G. Derwent and B. A. Thrush, Chem. Phys. Lett., 1971, 9, 591. Atomic Energy Levels, Natl Bur. Stand. Ref: Data Ser. 35,ed C. E. Moore (U.S. Government Printing Office, Washington D.C., 1971), vol. 1-111. K. P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure, IV. Constants of Diatomic Molecules (Van Nostrand Reinhold, New York, 1979). W. E. McDermott, N. R. Pchelkin, D. J. Benard and R. R. Bousek, Appl. Phys. Lett., 1978,32,469. R. F. Heidner 111, C. E. Gardner, T. M. El-Sayed and G. I. Segal, Chern. Phys. Lett., 1981,81, 142. D. Husain and R. J. Donovan, Advances in Photochemistry (Wiley-Interscience, New York, 1971), vol. 8, p. 1. I' P. L. Houston, in Advances in Chemical Physics: Photoselective Chemistry, Part 2, ed.J. Jortner, R. D. Levine and S. A. Rice (Wiley-Interscience, New York), vol. XLVII, p. 381. l2 R. P. Wayne, Advances in Photochemistry (Wiley-Interscience, New York, 1969), vol. 7, p. 31 1. l3 E. A. Ogryzlo, in Organic Chemistry Series Monograph 40, ed. H. Wasserman and R. W. Murray (Academic Press, New York, 1979). D. HUSAIN AND G. ROBERTS 113 14 D. Husain and G. Roberts, J. Chem. SOC.,Faruday Trans. 2, 1985,81, 87. l5 D. Husain and J. Schifino, J. Chem. SOC.,Faraday Trans. 2, 1983, 79, 1265. l6 D, Husain and J. Schifino, J. Chem. SOC.,Faraday Trans. 2, 1983, 79, 1677. 17 D. Husain and J. Schifino, J. Chem. SOC.,Faruduy Trans. 2, 1984, 80, 321. 18 D. Husain and J. Schifino, J. Chem. SOC.,Faraday Trans. 2, 1984, 80, 647.19 R. J. Malins, D. Logan and D. J. Benard, Chem. Phys. Lett., 1981, 83, 605. 'O T. J. McIlrath and J. L. Carlsten, J. Phys. B, 1973, 6, 697. 21 D. Husain and J. Schifino, J. Chem. SOC.,Faruduy Trans. 2, 1982, 78, 2083. 22 D. Husain and J. Schifino, J. Chem. SOC.,Faruday Trans. 2, 1983, 79, 919. 23 J. A. Irvin and P. J. Dagdigian, J. Chem. Phys., 1981, 74, 6178. 24 L. Pasternack and P. J. Dagdigian, Chem. Phys., 1978, 33, 1.*' J. W. Cox and P. J. Dagdigian, J. Phys. Chem., 1982, 86, 3738. 26 F. Engelke, Chem. Phys., 1979, 44, 213. 27 B. E. Wilcomb and P. J. Dagdigian, J. Chem. Phys., 1978, 69, 1779. (PAPER 4/748)

ISSN:0300-9238    

DOI:10.1039/F29858100101

出版商:RSC    

年代:1985

数据来源: RSC

摘要:

J. Chem. SOC.,Faraday Trans. 2, 1985, 81, 115-121 Irregular Behaviour of Kinetic Equations in Closed Chemical Systems Oscillatory Effects BY GYORGYPOTA Physical Chemistry Department, Institute of Chemistry, Kossuth Lajos University, H-4010 Debrecen, Hungary Received 14th May, 1984 The use of irreversible reaction steps may lead to unacceptable dynamical behaviour when modelling closed chemical systems. In some cases the kinetic equations may generate sustained concentration oscillations instead of convergent behaviour. Since the existence of irregular oscillators represents some dangers in the modelling of kinetic results, it is of interest to estimate how many such reaction schemes exist. This problem is studied for three-component bimolecular reaction schemes.It is shown that there is only one reaction scheme which exhibits irregular oscillatory behaviour. For the sake of comparison the possibility of regular oscillations is also studied. The reactions considered cannot provide sustained oscillations under the conditions of an isothermal continuous-flow stirred-tank reactor. It is generally accepted that the kinetic equations for closed chemical systems should be compatible with the equilibrium concepts. When all the reactions of a closed isothermal-isobaric model system are taken as being reversible and the principle of detailed balance holds, this requirement is fulfilled and the concentration against time functions tend to their equilibrium values.'-3 In practice, however, schemes containing irreversible reaction steps are also frequently used in the modelling of kinetic results.In most of such cases, though the kinetic equations do not satisfy the equilibrium requirements, the concentration against time curves are qualitatively the same as in the corresponding balanced model. The theory of oscillatory reactions has recently focused attention on the possibility of irregular kinetic behaviour in closed reaction models. We will consider the following reaction scheme, which involves three irreversible autocatalytic reac- tions: X+Y + 2Y Y+Z 22 (1)--b z+x --* 2x. The kinetic equations for this reaction scheme formally produce sustained oscilla- tions in a closed isothermal-isobaric Since the detailed balanced model KINETIC EQUATIONS IN CLOSED SYSTEMS tends to an equilibrium, it is reasonable to believe that the oscillations generated by scheme (1) do not correspond directly to any real phenomena.This is an accepted view which will be adopted here. Thus scheme (1) may be an excellent example of the dangers that arise from the application of irreversible reaction steps. It shows that under some circumstances reaction schemes may exhibit unacceptable dynamical behaviour in closed systems. An interesting question is how many irregular oscillatory schemes [like scheme (l)] exist among the regular models. What is the possibility of meeting such a case? The general solution of this problem appears difficult. In this paper we present a partial answer: we study the class of closed three-component reaction schemes to which schemes ( 1) and (2) belong.For convenience, the reaction schemes considered must satisfy only two requirements: they must obey the law of mass conservation and must not involve reactions with a molecularity greater than two. THE FORM OF IRREGULAR OSCILLATORS In case of mono- and bi-molecular reactions the three-component isothermal- isobaric systems can be described by the following equations: i= a, +a,x +a2y+a3z+a,xy +asxz +a6yz+ a7x2+a8y2+a9z2 j = b, +bly+b2x+b3z+b4xy+bsyz +b6XZ +b7y2+b8X2+ b9z2 (3) According to the chemical requirements, the signs of the coefficients in eqn (3) obey the following relations: a2, a39 a69 a8, a930 Coefficients al, a4, a5, b,, b4, b5, c,, c4 and cs, if they do not vanish, may be of either sign, depending on whether they represent a decay or an autocatalytic increase.Let us denote the right-hand sides of eqn (3) by P(x, y, z), Q(x, y, z) and R(x, y, z), respectively. We can assume that none of the functions P, Q and R is identically zero and that they are defined in the non-negative octant of the three-dimensional Euclidean space R3. Since the systems considered are closed to the transport of matter, eqn (3) have a linear first integral H(x, y, z) =Ax +By + Cz (5) so that aH dH a~--I' +-Q +-R = AP + BQ + CR =0 ax ay az is fulfilled over the entire domain of eqn (3). For our purposes it is enough to assume that A, B and C are not all zero. The relations a, = bo= co= 0 and a,, b,, c1 0 implied by the closed form of eqn (3) will not be used here.Consequently, the results will also cover some open systems. We shall apply a planar version of the GY. POTA 117 criterion of Demidowitsch’ which may be formulated as follows: Let i= P(x,y, z) j = Q(x,Y,z> (7) i= R(x,y, z) be a continuously differentiable system on a cube G of R3. Assume that eqn (7) has a linear first integral of the type H(x,y, z)= Ax +By + Cz. H(x,y, z)= h defines a set of parallel planes in R3 for variable values of h. If the expression aP aQ aRD=-+-+-ax ay az has a constant sign in cube G then G does not contain closed trajectories of eqn (7). The full Demidowitsch’s criterion’ covers more general manifolds. However, discussion of its validity and applicability is beyond our present aims.? We apply the planar Demidowitsch’s criterion to eqn (3) and (6).Suppose first that eqn (3) does not contain autocatalytic terms. Thus we find that a,, a49 a5 d 0 hb4, b’ =s 0 (9) c1, c4, c5s 0 and the other coefficients obey eqn (4). In a cube G of the positive octant of R3 we get the following formula for Demidowitsch’s function D: D(x,y, z)= a,+a4y+a5z+2a,x +bl +b4x+bgz +2b7y +c1+c4x+ c5y +2c7z. (10) Taking into account eqn (4) and (9) it follows that cube G can contain closed trajectories only if all coefficients in D(x,y, z) vanish. In this case, however, the time derivatives in eqn (3) are positive, which rules out the existence of positive periodic solutions.Thus, the presence of direct autocatalytic steps is a necessary condition for irregular oscillatory behaviour in three-component closed and bimolecular schemes. This statement, in another context, is originally due to Motova.6 In the following we prove a stronger theorem, which asserts that every closed and bimolecular reaction scheme which exhibits irregular oscillatory behaviour in three dimensions can be described by kinetic equations of the same form. To prove this assertion let us consider the following system: i= P(x,y, z)/xyz where P(x,y, z), Q(x,y, z) and R(x,y, z) denote the right-hand sides of eqn (3). Naturally, the domain of definition is restricted here to the positive octant of R3. It is evident that every linear first integral of eqn (3) in eqn (5) is also a first integral for eqn (1 1).Furthermore, if eqn (3) and (6) have a positive periodic solution, eqn t Note added in proof: After submitting this paper a reference was noted29 from which a corrected form of the full Demidowitsch’s criterion follows. This remark does not affect the planar criterion applied here. KINETIC EQUATIONS IN CLOSED SYSTEMS (1 1) and (6) also do. [For the general theorem see ref. (7).] Therefore, instead of eqn (3) and (6) we may apply Demidowitsch's criterion to eqn (1 1) and (6). If eqn (1 1) and (6) have a closed trajectory in a cube G of the positive octant of R3 then the expression bo b2 b b b bgX 692+-+,-+++LL +-+7xy2z y z xy y2 xz y2z xy +2+2xyz yz xz2 z2 xy yz xz")+2!2-+2L+l+C6-c? c8x (12) must vanish here.This.means that eqn (3) has the following form: x = a,x +a4xy+a5xz y = bl y +b4xy+b5yz (13) 2 = c1z +c4xz +csyz. If two coefficients, say A and B, in the first integral eqn (5) are not zero but C is, eqn (3) must be of the form x = a4xy y = b4xy (14) i = c1z +c4xz +csyz. Here, the two time derivatives being of constant sign, positive periodic solutions cannot occur. Using similar arguments we find that eqn (3) must have the form i= a4xy+a,xz y = b4xy+b5yz (15) i = c4xz +c,yz where all the coefficients are non-zero and the same is valid for the first integral eqn (5) too. In eqn (15) we can rule out sign combinations which cannot be realized by bimolecular reactions. So all such combinations are ruled out in which a4, b4 >0 or a,, c4>0 or bs, cs>0.Let us assume, for instance, that a4c0. For the existence of positive periodic solutions then a5>0 must be fulfilled, which, because of the forbidden sign combinations, implies that c4<0. From this we get the relation c5>0 and finally b, <0, 64 >0. Let a4,a,,p4,p,, y4 and ys be positive constants. With this notation eqn (15) has the form k = -a4xy +a5xz 9 =P4XY -PSYZ (16) i = -y4xz +ysyz which is just the form of the kinetic equations of scheme (1). If we assume a4>0, the form of the resulting system differs from eqn (16) in notation only. In this sense we may say that among the closed and bimolecular reaction schemes there is GY. POTA 119 only one scheme which exhibits irregular oscillatory behaviour in three dimensions.Thus, the danger of meeting an irregular oscillator in these reactions is limited. Note, however, that we considered only the case of undamped oscillations and did not perform a complete qualitative analysis of the kinetic equations. An additional problem may be, for example, the occurrence of damped oscillation^.^^^ Some of the models considered here belong to the set of weakly reversible three-complex systems.’ In case of short complexes this set does not involve oscillators.2 The class of closed and bimolecular reaction schemes is almost empty in two dimensions and it does not contain irregular oscillators. In fact, among the two- variable and bimolecular reaction systems there exists only one oscillator, the open Volterra-Lotka The same may be suggested for a class of population dynamic ~ystems.~’ The uniqueness of oscillatory behaviour which we have found is in accordance with a result of Jenks, who purely quadratic population models in the presence of a linear first integral.Jenks has shownI7 that under these circumstances eqn (16) is the only three-dimensional model which has a centre-type stationary state. Some related considerations may also be found in ref. (18) and (19). We have seen that sustained oscillations do occur in the the three-dimensional class of closed and bimolecular models, though they cannot possess an immediate physical meaning. An interesting related question is what happens if the reactions considered take place in open systems.In this case we have no a priori reasons to exclude the possibility of physically meaningful sustained oscillations. For the sake of comparison we perform such an investigation under the conditions of an isother- mal continuous-flow stirred-tank reactor (c.s.t.r.). Such systems were studied by Gray and Scott2’ in two dimensions. A thorough mathematical analysis of non- isothermal c.s.t.r. was given by Chicone and Retzloff.2’ It is interesting to remark that recently a number of reactions were found to be oscillatory in a c.~.t.r.*~-~’ THE POSSIBILITY OF OSCILLATIONS IN A C.S.T.R. Let us denote by P(x,y, z), Q(x, y, z) and R(x,y, z) the right-hand side of eqn (3) and assume that a linear first integral such as eqn (5) holds.According to the argument above we may consider the following transformed equations: where ko is the inverse of the mean residence time and xo, yo and zo are the appropriately defined input concentrations of the components. Let (x, y, z):[0,00) + R: be a positive solution of eqn (17) defined over the time interval [0,m). Then the function defined by HI(t)= H[x(t),y( t), z( t)] [see eqn (5)] satisfies the differential equation KINETIC EQUATIONS IN CLOSED SYSTEMS From eqn (18) we find that HI(t) may be expressed as where H, = ko(Axo+By, + Cz,) and (o( t) = [x( t)y(t)z(?)]-I. When the solution (x, y, z) is bounded, L’Hospital’s rule gives the following limit: HOlim HI(?)=-. f+o3 k0 Let us assume that (x, y, z) is a non-trivial positive periodic solution of eqn (17).Thus eqn (20) can be fulfilled only if HI(t) = AX(t) +By( t) + CZ(t) =-HO k0 holds for each non-negative value of t. This means that if eqn (17) has a non-trivial positive periodic solution, this must lie in the plane AX+By +CZ=-.Ho k0 Furthermore, in the positive region of this plane the following equation is valid: AP* +BQ* +CR* =O (23) where-P*, Q* and R* denote the right-hand sides of eqn (17), respectively. As the proof5 shows, the planar variant of Demidowitsch’s criterion may also be applied under these conditions. Since at least one of the flow terms in eqn (17) does not vanish, Demidowitsch’s function D takes negative values in the entire positive octant.This means that closed and bimolecular reaction schemes in three dimensions do not provide periodic concentration changes under conditions of an isothermal c.s.t.r. Thus, in a paradoxical formulation we may say that sustained oscillations do occur in the reaction class considered when they are forbidden and do not occur when they are allowed. The connection between batch reactors and c.s.t.r. may be approached by applying some transformation methods.26 In this paper we have studied the irregular kinetic equations for a limited class of reactions. However, these simple considerations emphasize that in general the kinetic equations are not necessarily compatible with the thermodynamic require- ments. The uniqueness of irregular oscillatory behaviour which we found appears to be a satisfactory result.It is further problem as to whether the differential equation systems, having a more or less limited chemical interpretation, can serve as approxi-mations to more detailed model^.^,^^,^* The author thanks Jiinos T6th and P6ter Erdi for their valuable comments. ’ D. Shear, J. Theor. Bid, 1967, 16, 212.’ F. Horn and R. Jackson, Arch. Rat. Mech. Anal., 1972’47, 81. A. I. Volpert and S. I. Khudaev, Analysis in Classes of Discontinuous Functions and the Equations of Mathematical Physics (Nauka, Moscow, 1975), (in Russian), pp. 351-387. A. M. Zhabotinskii and M. D. Korzukhin, in Oscillatory Processes in Biological and Chemical Systems, ed. G. M. Frank et al. (Nauka, Moscow, 1967) (in Russian), pp.223-231. W. B. Demidowitsch, Z. Angew. Math. Mech., 1966, 46, 145. GY. POTA 121 M. I. Motova, in Oscillatory Processes in Biological and Chemical Systems, Part 2 (Institut Biologicheskoi Fiziki, Puschinoon Oka, 1971) (in Russian), pp. 265-268. I. G. Petrovskii, Lectures on the Theory of Ordinary Diflerential Equations (Nauka, Moscow, 1970) (in Russian), p. 201. ' J. Z. Hearon, Ann. N.Y. Acad. Sci., 1963, 108, 36. B. F. Gray, Trans. Faraday SOC.,1970, 66, 363. 10 D. A. Frank-Kamenetskii and I. E. Salnikov, Zh. Fiz. Khim., 1943, 17, 79. 'I L. Ya. Fukshanskii, G. I. Yozefovich and V. A. Yangarber, in Oscillatory Processes in Biological and Chemical Systems, ed. G. M. Frank et al. (Nauka, Moscow, 1967) (in Russian), pp. 395-404.P. Hanusse, C.R. Acad. Sci., Ser. C, 1972, 274, 1245. I3 J. J. Tyson, and J. C. Light, J. Chem. Phys., 1973, 59, 4164. l4 Gy. Pota, J. Chem. Phys., 1983, 78, 1621. l5 J. Schnute and P. van den Driessche, Appl. Math. Notes, 1975, 1, 75. l6 R. D. Jenks, J. Dig Eqs, 1968, 4, 549. R. D. Jenks, J. Dig Eqs, 1969, 5, 497.'' L. A. Cherkas, Dig Uraun., 1977, 13, 779. l9 A. Fernandez and 0. Sinanoglu, J. Math. Phys., 1984, 25, 406. 2o P. Gray and S. K. Scott, Chem. Eng. Sci., 1983, 38, 29. 21 c. Chicone and D. G. Retzloff, Nonlin. Anal., 1982, 6, 983. 22 M. Orban, C. Dateo, P. DeKepper and I. R. Epstein, J. Am. Chem. SOC.,1982, 104, 591 1. 23 P. DeKepper and K. Bar-Eli, J. Phys. Chem., 1983, 87, 480. 24 M. Orban, P. DeKepper and I. R. Epstein, J. Am. Chem. Soc., 1982, 104, 2657. 25 W. Geiseler, J. Phys. Chem., 1982, 86, 4394. 26 A. G. Pogorelov and N. F. Kononov, Dokl. Akad. Nauk SSSR, 1978, 238, 154. 27 M. D. Korzukhin, in Oscillatory Processes in Biological and Chemical Systems, ed. G. M. Frank et al. (Nauka, Moscow, 1967) (in Russian), pp. 231-251. 2' G. A. M. King, J. Chem. SOC.,Faraday Trans. 1, 1983, 79, 75. 29 K. R. Schneider, 2. Angew. Math. Mech., 1969, 49, 441. (PAPER 4/791)

ISSN:0300-9238    

DOI:10.1039/F29858100115

出版商:RSC    

年代:1985

数据来源: RSC