Dept of Chemistry and Applied Chemistry, University of Salford, Salford 5, lancs. lntroduction Experimental techniques . . Pulse apparatus, 65 Acoustic resonator, 66 Brillouin scattering, 66 . . . . . . . . SOME CHEMICAL APPLICATIONS OF ULTRASONIC ABSORPTION MEASUREMENTS IN 'THE LIQUID STATE E. Wyn-Jones, B.Sc., Ph.D., D.Phil. . . . . 59 . . 64 * . . . . . * . . . 68 79 . . . . . . . . . . 81 . . 78 . . . . 83 . . 83 . . 81 . . . . . . . . .. . . . . .. Applications to fast chemical reactions . . .. Electrolytes in water, 70 Non-electrolytes in water, 72 Rotational isomerism, 74 . . . . . . . . Molecular association, 77 Applications to reactions of biological interest Vibrational relaxation Critical point phenomena .. Shear and volume viscosity phenomena Recommended reading for the specialized reader References . . . . . . . . .. . . INTRODUCTION This article discusses the type of chemical information that can be obtained on the molecular level from measurements of the absorption and velocity of sound in liquids. Although many chemists are not acquainted with the application of the ultrasonic absorption method or, as it is sometimes called, the molecular acoustic technique, it should be mentioned that Professor Manfred Eigen, who shared the Nobel Chemistry Prize in 1967, uses this technique in his relaxation studies on the kinetics of fast reactions. In this review article a brief introduction to the principles of sound absorption and relaxation is followed by an account of the experimental technique and various chemical applications.The review ends with a list of text-books and general, articles on the subjects, which are recommended to the specialized reader. Experimentally it is possible to produce sound waves which can have a frequency from as low as a few cycles per second to as high as 10 Gigahertz ( 1O1O Hz). This acoustic spectrum can be divided roughly into three regions- the audible range (< 14 kHz), the ultrasonic range (14 kHz-800 MHz) and the hypersonic range (> 109 Hz). When a sound wave passes through a liquid the excess pressure at any point will alternate periodically because in . . . . . . . . . . . . . . Wyn- Jones .. . . 59 each cycle of the sound wave a given volume of the liquid is successively compressed and then decompressed.In most liquids (except water) the specific heat at constant pressure, Cp, is greater than the specific heat at constant volume Cv, thus y (= C,/C,) > 1 and the passage of a sound wave is automatically adiabatic. This means that a periodic variation in the tempera- ture will accompany the compression-decompression cycle of the sound wave. The pressure amplitude, p , of a one-dimensional sound wave travelling in the positive x-direction is given by: 1 p = poexp(- ax) exp iW(t - x/c) . . I . where po is the pressure amplitude at x = 0 and t = 0, a is the absorption coefficient related to the intensity of the sound wave, c is the phase velocity and w = 2nf where f is the frequency of the sound wave.In equation 1 the quantity poexp(- ax) represents the maximum amplitude of the sound wave after it has travelled a distance x in the liquid and is, therefore, a measure of the sound intensity. Thus the loss of sound intensity in a liquid depends on both the distance travelled by the wave and also the absorption coefficient a. The term exp io(t - x/c) is simply a function which describes the periodic motion of the sound wave. The real part of equation 1 is analo- gous to the Beer-Lambert law governing light absorption. The velocity of sound in a liquid is given by the following relationships: and c2 = 1 / p p . . . . .. . . . . 2 c2 = (y - 1) Cp/@TpV . . . . . . 3 where p is the density, /3 the adiabatic compressibility, 8 the coefficient of thermal expansion and V the molar volume.From ‘classical’ theory Stokes showed that the absorption coefficient a of a sound wave passing through a liquid is a function of the viscosity and the contribution aclass. from shear viscosity, qs is aclass. = (2n2/pc3) (4~,/3)f . . . . 4 The action of a sound wave in successively compressing and decompressing a given volume of liquid will cause this volume to alternate periodically about a given value. When these volume changes occur, the molecules of the liquid have to flow from a more compact to a less compact structure in the direction of the motion imposed by the sound wave. Since viscosity is defined as resistance to flow it follows that a liquid can have a ‘volume’ or compressional viscosity in addition to the more familiar shear viscosity.In deriving equation 4 Stokes assumed that viscous losses in liquids due to a motion of uniform compression are zero. Although recognizing the possibility that a liquid could have a volume viscosity, yv, Stokes had to assume qv = 0 because of a lack of any suitable method to measure this quantity. It is now known from hydrodynamic theory that the full expression for a is . . 5 R.I.C. Reviews 60 In addition to viscous losses, Kirchoff showed that a is also a function of the thermal conductivity, K, of the liquid according to equation 6: For all liquids except liquid metals the contributions to 1~ from thermal conductivity given by equation 6 are negligible.The excess absorption in any liquid is generally termed as the difference between the measured value CII and the classical value aclass. given by equation 4. Thus, the volume viscosity qv is 4 (a - aclass.) %lass. \ Coupling mechanism I Conformational energy ~ V v = 37" When this volume viscosity arises from a perturbation of a molecular equili- brium due to the temperature or pressure changes imposed by a compres- sional wave, ultrasonic relaxation will occur. Several different kinds of mechanisms are responsible for volume viscosity in liquids and within the limit of present knowledge all are relaxational in behaviour. This statement can be illustrated if we consider the total energy of a liquid in terms of the 'energy box' introduced by Litovitz.2 The total energy is the sum of many components such as translational, vibrational and the less familiar energy Vibrational energy Energy transfer Fig.I. A modified version of the energy box introduced by Litovitz. Wyn- Jones 6 7 . . . . . . 61 due to the degree of order which is brought about as a result of molecules aggregating in a quasi-crystalline state. In addition, a liquid can have energy due to the occurrence of a chemical equilibrium such as that between two or more conformers of a molecule. In Litovitz’s energy box these dif- ferent energies are considered separately as illustrated in the diagram Fig. 1 where each separate energy is contained in a segment. When the total energy of the liquid changes, the different energies in each segment will also change.Each energy segment is in contact with the other so that energy can flow from one to the other. This flow of energy is governed by some coupling mechanism which is relaxational in behaviour. This means that energy will flow from one segment to another at an exponential rate and with a finite time constant, T, known as the relaxation time. This relaxation time depends on the nature of the coupling mechanism which governs the energy transfer between segments. Jf the temperature of the system is raised, this will automatically increase the translational energy of the molecules and, in turn, some of this energy will be transferred to all the other energy modes until a new equilibrium is reached.It is well known that the mechanism of energy transfer between translational and vibrational modes occurs via molecular collisions. The local temperature rise accompanying the compression period of a sound wave will automatically increase the translational energy of the molecules and, in turn, all the other energies. After the sound wave has passed through the crest of the compression half cycle this energy begins to return to the sound wave. If all this energy returns while the sound wave is still in the same compression cycle there will be no net loss of energy during that cycle. On the other hand if some of this energy returns to the sound wave out of phase during the decompression period there will be a net loss in sound energy during that cycle and absorption in addition to the viscous contributions of equation 5 will occur.A factor which determines whether there is a loss in the sound energy per cycle is the rate of the energy transfer in relation to the frequency of the sound wave. Consider the passage of a sound wave through a hypothetical system which is composed of only a chemical equilibrium of the kind 8 A + B ( + AH) .. . . . . Because there is an enthalpy difference between two forms, the temperature fluctuations accompanying a compressional sound wave will impose a periodic change in the chemical equilibrium about a mean value. At very low acoustic frequencies the periodic changes in both the sound wave and the equilibrium will be in phase.This corresponds to the situation where enthalpy is taken and returned to the sound wave during the same cycle. If the acoustic frequency is progressively increased the equilibrium will not be able to respond instantaneously to the temperature fluctuations and will start becoming out of phase with the sound wave. This lag enables the equilibrium to accept enthalpy during the temperature crest of the sound wave and to give up enthalpy during the temperature trough with an inevitable attenu- ation of the sound wave intensity as the wave travels through the liquid. When the acoustic frequency is very high the equilibrium is unable to follow R.I. C. Reviews 62 \ \ /--q Velocity \ \ / / Frequency - _ _ _ _ - - l:.ig. 2.The variation of the ultrasonic parameters a/f2, p and c with frequency during a single relaxation process. the temperature fluctuations imposed by the sound wave, again resulting in a zero loss of sound energy. The maximum loss in sound energy will occur at an acoustic frequency fc given byfc = 1/27~T where T is the relaxation time of the chemical equilibrium. This time lag in the transfer of energy in rates for the two-state process equation 8, T = (k12 + kzl)-l where the k’s a chemical equilibrium is obviously related to the forward and reverse are the forward and reverse rate constants respectively. The simple processes described above are the basic principles involved in the relaxation techniques3 to study the kinetics of fast reactions.Two factors determine the magnitude of the loss of sound energy per cycle. These are the amount of energy shared and returned out of phase (in the chemical equilibrium 8 this is proportional to AH) and secondly the time constant T. The behaviour described above corresponds to acoustic relaxation in a liquid. In liquids there are several different molecular processes* that may give rise to ultrasonic relaxation such as a chemical equilibrium, vibrational relaxation and structural processes such as compressional and shear relaxa- tion. Provided the respective relaxation times are well separated then each process can be studied separately. The behaviour of the acoustic parameters during a single relaxation process is shown in Fig. 2.At any given temperature the quantity a / f 2 will vary with frequency according to the equation: . . 9 where A is a relaxation parameter, fc is the relaxation frequency (= 1/2m-) and B represents contributions to a/f2 which are not related to the relaxation. These contributions are shear viscosity (equation 4) and any excess absorption 63 Wyn- Jones 5 due to a relaxation process with a relaxation frequency much higher than fc. The loss per cycle or absorption per unit wavelength relating to the relaxation p is 10 p = afA . . . . . . . . where a' is the excess absorption for the relaxation process. Thus * - p = (a - Bf') h = ACf/(l + (flfc)') Whenf=f, (COT = I), p reaches a maximum value p m given by: pm = 4ACfc .. . . 11 . .. . . . 12 The dispersion in the sound velocity c is given by the expression 13 where the subscripts 0 and a: refer to the sound velocity at low and high frequencies. When ultrasonic relaxation occurs the sound velocity becomes frequency dependent in the relaxation region as shown in equation 13 (cf. Fig. 2). From equation 2 it also follows that P, the adiabatic compressibility of the liquid, must also be frequency dependent. This quantity varies with frequency according to the equation P = Pa3 + APl(1 + iwd where PO is the adiabatic compressibility at infinite frequencies and AP is the frequency independent part of the compressibility that arises from the relaxation process; this quantity is commonly referred to as the relaxing part of the adiabatic compressibility. From a consideration of the thermo- dynamic properties of a liquid where a single ultrasonic relaxation occurs it can be shown that 14 In the quantitative interpretation of ultrasonic relaxation processes the main problem is to relate AP/Po commonly referred to as the relaxation strength, and also T , to the properties of the actual process that is causing In the study of acoustic relaxation in liquids it is desirable to measure the velocity and absorption of sound over a frequency ranging from a few cycles per second to the hypersonic region.Experimentally this range can be more or less successfully covered and there are techniques available with which absorption and velocity can be measured in the range 102-1010 Hz.335 the relaxation.EXPERIMENTAL TECHNIQUES R.I.C. Reviews 64 The accuracy of these techniques varies from qualitative estimates to values considered accurate to within a few per cent. In this section a brief description of the principles of three versatile techniques covering the frequency range 105-1010 Hz is given. These techniques are also considered to give very accurate acoustic data. Pulse apparatus (106-109 Hz) The pulse technique is the most widely used for absorption and velocity measurements in the region 5-200 MHz. Figure 3 shows a block diagram of a typical apparatus. A train of pulses is produced by the pulse generator and fed into the transmitter which is set at the desired frequency. These pulses are used to excite this oscillator which in turn produces bursts of oscillations at the desired frequency, f.The resulting radiofrequency pulses are fed to a transducer which is acoustically coupled to a fused quartz delay line. The acoustic pulses produced by the transducer pass through the quartz delay line, through the liquid under test and into a second quartz delay line the upper end of which is attached to the detecting transducer. This second transducer retransforms the acoustic pulse into an electric pulse which is amplified and demodulated at the receiver and finally displayed on the screen of a cathode-ray oscilloscope. A second train of pulses is produced by the pulse generator and used to excite a comparison oscillator which is set at 0 Pulse generator Transmitter Detecting transducer n osci I lator Cathode-ray oscilloscope Fused quartz Fused quartz delay line u delay line Wyn-Jones Comparison Attenuator m 0 0 0 I Receiver Test liquid Fig.3. A block diagram of the pulse apparatus. 65 the same frequency, f, as the transmitter. These radiofrequency pulses, after suitable delay, are fed into an attenuator, then to the receiver and finally are displayed at the side of the acoustic pulse on the oscilloscope. The intensity and hence amplitude of the acoustic pulse is changed by varying the distance between the launching and detecting transducers. This change of amplitude can be measured by visually comparing the height of the acoustic pulse with that of the ‘comparison’ pulse which is attenuated by a known amount.By plotting the attenuation against path length, the absorption coefficient, 01, follows from equation 1 since The quantity ln(po/p) is directly proportional to the attenuator readings and x is the acoustic path in the liquid. The lower frequency limit of this technique is about 4.5 MHz. At frequencies below this value the sound beam travelling to excitation acoustic be corrected. By use of high frequency experimental thus in the liquid frequency starts of diverging - 4 x 1010 and Hz can the be produced. Using of measurements a quartz quartz resona- rod have an tors of this kind instead of transducers in the mechanical unit, absorption measurements at frequencies of 1 GHz have been measured with this tech- nique.The velocity measurements are done by using the apparatus as an acoustic interferometer. This is done by counting the number of beats between two pulses which have travelled respectively one and three paths in the liquid sample over a known change in path length. Acoustic resonator This is a relatively new technique6 which has been used successfully in the region 100 kHz-50 MHz giving very accurate velocity and absorption measurements. In principle, the liquid sample is kept in a cylindrical cell whose ends are sealed with transducers (quartz crystals) which are both matched in frequency and are accurately parallel. Acoustic waves can be generated in this system by exciting one of the transducers by means of an oscillator.When conditions are such that the length of the cell is an integral number of half wavelengths of the sound waves in the liquid, the system will resonate. The resonance frequency of the system is proportional to the velocity of sound in the liquid and the shape of the resonance peak is related to the absorption coefficient of the sound wave. In practice the resonance peak of the ‘resonator’ isnobtained by plotting the output voltage against frequency in the resonance region. This can be done graphically or on the screen of a cathode-ray oscilloscope. Brillouin scattering The discovery of laser action has been entirely responsible for the experi- mental advancement of this technique to study the acoustic properties of liquids at very high frequencies (109-1010Hz).In a liquid medium, an as- sembly of thermally excited sound waves (sometimes called Debye elastic waves) are present with a range of wavelengths varying from the dimensions of the container to interatomic distances. These thermally excited acoustic R.I.C. Reviews 66 Lens Test sample Fa b ry- Pe rot interferometer Lens Detector - Fig. 4. A block diagram of a Brillouin spectrometer. waves, sometimes called phonons, perturb the electrons and hence the polarizability of the liquid, i.e. they can be considered elastic and will com- press and rarify. This in turn produces a periodic fluctuation in the refractive index. Brillouin7 first showed that these phonons affect light scattering. If monochromatic light is incident on the medium, the Rayleigh scattered light contains, in addition to the incident frequency, the Brillouin components which arise from light scattered as a result of ‘reflection’ (or more correctly Doppler shifting) of these sound waves.The resulting spectrum consists of a central component which has the same frequency as the incident light and the Stokes and anti-Stokes Brillouin components which are separated from the central Rayleigh line by an amount A v given by Av = -J-- 2vo(v/c) sin 8/2 where v o is the frequency of the incident light, c is the velocity of light, v the velocity of the thermally excited sound waves of frequency Av (- 109- 101OHz) and 8 is the scattering angle. The width of the Brillouin line is related to the lifetime and thus absorption coefficient a of the sound wave involved in the scattering process.A block diagram of a simple arrangement is shown in Fig. 4. Monochromatic light from the laser source is passed through the liquid sample. The scattered light is then collected at the desired angle and analysed with a Fabry-Perot interferometer. The resulting triplet spectrum can be detected either photographically or recorded using a photo- multiplier. When recording, a pressure scanned Fabry-Perot interferometer must be used. This technique is still at a fairly early state of development and should prove very useful in the hypersonic range. There are several other techniques which have been used for measuring the absorption and velocity of sound in liquids.395J The operation of some of these techniques is very tedious, others require very large amounts of liquids and in many cases the accuracy of the absorption measurements can only be regarded as semi-quantitative. When acoustic relaxation occurs the dispersion in the quantity a/f2 is Wyn- Jones 67 -I 2200 0.6 c - 1800 h - N N 1 k U - 1400 ‘5 - 1000 - 0 - 600 200 - 8.0 7.0 log f .. .. 15 Fig. 5. Experimental data for the rotational isomeric relaxation in I ,2-dibromo-2-methyl- propane. often appreciable and in most instances the dispersion curve can be measured accurately. On the other hand the dispersion in the velocity is usually very small. For example, when COT = 1, equation 13 reduces to for most relaxation processes.This means that the velocity dispersion is less than .the experimental error in most techniques (acoustic resonator and Brillouin scattering excepted). Thus when investigating relaxation processes it is desirable to measure the absorption coefficient a over the frequency range which spans the relaxation region. When a single ultrasonic relaxation occurs the results (i.e. a/f2 at different frequencies) are fitted into equation 9 and the relaxation parameters p m found via equation 12. When a multiple relaxation occurs the results are fitted into an equation of the type f” i = n a = C (Ai/(I + (f/fd2> + B) A typical example of the single relaxation curves found for the rotational isomeric relaxation in 1,2-dibromo-2-methylpropane is shown in Fig.5. The velocity dispersion was negligible. These measurements were carried out with a pulse apparatus. APPLICATIONS TO FAST CHEMICAL REACTIONS The ultrasonic absorption method has been used widely as a relaxation technique to study the kinetics and thus mechanisms of very fast chemical R. I.C. Reviews 68 equilibria.334 In the simpler equilibria which involve a single acoustic relaxa- tion the relaxation times (T = 1/2.rrfc) are related to the forward and reverse rate constants, klz and kzl respectively, and also to the concentrations of the reactants C, as shown in the following model examples. A + B A +2B T = (kiz + kzi)-l . . . . . . . . 16 T = (kiz + 4 k z i C ~ ) ~ l .. .. . . 17 = (kiz(CA 4- CR) 4- kzi)-l . . . . 18 19 by: A + B e A B A + B + c + D 7 = (kiz(C~ + CB) + kzi(Cc 4- CD)>-' In the unimolecular reaction 16 which occurs in ideal solution the relaxa- tion strength Ap/po is related to the thermodynamic equilibrium parameters [TOZp Vcz/JCp2] 20 where AH and AY are the enthalpy and volume difference between B and A, J is the joule, V the molar volume of solution and K the equilibrium con- Ap/po = R[(AH/RT) - (AY. Cp/V. ORT)]' [K/(1 + K)'] stant. In an ionic equilibrium of the kind AB + A + + B- and the equilibrium constant K is .. . . . . 21 which usually occurs in dilute aqueous solution, Cp/Cv = 1, and thus the passage of the sound wave is isothermal. In these types of reactions AV is finite and thus the equilibrium will be able to impose a volume contraction during the compression (or pressure) cycle of the sound wave and a volume expansion during the decompression cycle.This will also lead to an extra attenuation of the sound wave at frequencies around 1/277T. For those reactions where VOAH = CpAY occurring in a solvent where C,/C, > 1 there will be no acoustic relaxation because the temperature rise and compression crest of the sound wave have equal and opposite effects on the equilibrium. For the equilibrium 21 occurring in dilute aqueous solution, the full relationship between the relaxation strength AP/Po and the equilibrium parameters is : d a . . 22 where a is the degree of dissociation, the y's are the activity coefficients and PO the adiabatic compressibility at low frequencies.In most examples of equilibrium 21 the concentration of the ions are sufficiently dilute to assume a value of unity for the activity coefficients and also dln y :/do = 0. Some examples of different types of fast reactions studied by acoustic methods are described below. 69 Wyn- Jones Electrolytes in water In the dissociation of the weak electrolyte ammonium hydroxide in dilute aqueous solutions : NH,OH + NHZ + OH- . . . . . . 23 the equilibrium constant K is given by:9 K = 02C/(l - a) (= 1.75 x 10-5 at 20 "C) where C is the concentration of the ammonium hydroxide and (T the degree of dissociation. The above equilibrium is characterized by a single relaxation time T given by 18: T = [klz + kzl(CNH$ -k COH-)]-' where the C's are the equilibrium concentrations of the ions.For reasonably dilute solutions : and thus CNH$ (= COH-) = C a % 1/KC T = [kiz + k2121/KC]-l . . .. . . 24 Tamm et ~ ~ 1 . ~ 0 observed an ultrasonic relaxation in this system which was attributedll to a perturbation of the above equilibrium. The relaxation frequency was found to increase from 22 to 55 MHz as the concentration of ammonium hydroxide increased from 0.5 to 2.5 molar. A plot of 7-l against 2 d K C (equation 24) was a straight line whose slope gave k21 = 3.4 x 1010 s-1 mol-1 From Debye's theoretical calculationslz on the rates of collisions of ions in solution a rate constant of the order 1010-1011 s-1 mol-l is expected for a diffusion controlled reaction showing that the recombination rate in equation 23 is diffusion controlled.From the relationship K = k12/k21 it is found that klz = 6 x 105 s-1. The value of this dissociation rate constant shows that the reaction proceeds as shown in equation 23 and not via the alternative route NH, + H2O $ NH, + Hf + OH- + NH; + OH- which involves the dissociation of water. The dissociation of such a reaction would be governed by the dissociation rate constant for pure water3 which has a value of 3.4 x 10-5 s-1. This is much lower than the observed value of kl2. By use of equation 22 the volume change AV was found to be - 28 cm3 mol-1. This is in excellent agreement with the value derived from partial molar volumes :I1 AV = VNH; - VOH- - V N H ~ - V N ~ O = - 28 cm3 mol-1 Further ultrasonic measurements carried out at different pressures13 showed that k21 was independent of pressure.The recombination of NH4f and OH- is diffusion controlled and has no activation energy. The effects of pressure on diffusion coefficients are relatively small and so the experimental observa- tion that k21 was independent of pressure is expected. An ultrasonic relaxation observed in aqueous solutions of potassium R.I.C. Reviews 70 cyanide was attributed14 to the hydrolysis reaction : CN- + H2O (excess) + HCN + OH- . . . . 25 and analysis of the experimental data gave: klz = 5.2 x lo4 s-l, kzl = 3.7 x lo9 mol-l s-l and A V = - 12.4 cm3 mol-I. From the temperature dependence of the ultrasonic relaxation times an activation energy barrier, AH$, of 25 & 8 J mol-1 was found.The rate constant for the reverse hydrolysis k21 is much less than that found for the corresponding diffusion controlled recombination rate constant in equation 23. This can be understood if we consider the simple mechanisms for the proton transfer processes which are respectively : and m 5 N T 0 X h 4.0 NH; + OH- -+ H,N***H+*..OH- -+ NH,OH NCH + OH- + NC-***H+***OH- -+ CN- + H20 . . 27 for equations 23 and 25. In 26 the proton leaves behind an uncharged species (NH,OH) as it bonds chemically to the hydroxyl ion. On the other hand, in 27 the proton leaves behind a negatively charged ion as it transfers to the hydroxyl ion to form water.On purely electrostatic considerations more energy is needed for the proton transfer in equation 27 and the rate is, therefore, expected to be slower. In multi-step chemical reactions of the kind: A1 s A2 + A3 + A4 etc. a spectrum of relaxation times and frequencies is found with the number of relaxation times corresponding to the number of independent ~ t e p s . 1 ~ Because of the coupling between each state the relaxation times will not be the same I I 7.0 e ref. 10,16 0 ref. 17 A ref. 18 A ref. 19 I 5.0 5.0 log f Fig. 6. Ultrasonic relaxation data for magnesium sulphate. Absorption cross-section (Qh) per wavelength (QX is proportional t o p, the absorption per wavelength). [from: Physical Acoustics Vol.II, part A (Academic Press, New York), 1965. Fig. 22, p. 430. J. Stuehr and E. Yeager by courtesy of the publishers.] 71 Wyn-Jones 26 . . I 8 .O as those found by treating each step as an isolated two-state equilibrium. A typical example of a multi-step chemical reaction occurs in magnesium sulphate solutions where a double ultrasonic relaxation10J6-19 was found experimentally as shown in Fig. 6. This double relaxation has been inter- preted by Eigen and Tamml5 as a perturbation of the multi-state dissociation which takes place in three steps, thus State 0 is the diffusion controlled approach of the two hydrated, dissociated ions and contributes to the electrical conductivity of the solution. These ions can associate into three different ion pairs states 0, 0 and @, which are dissolved in solution and are electrically neutral.In steps 11 and 111 these ions come closer together owing to the stepwise removal of the waters of hydration trapped between the ions in step I. In state @ the ions are in direct contact with each other. This four-step model has been successful in quanti- tatively describing the multiple relaxation found in several 2,2 electrolytes. In magnesium sulphate ultrasonic measurements at different pressures21 also support the above model. Many other examples of the application of the ultrasonic method to study ionic equilibria are given in several excellent review articles.3322-24 Non-electrolytes in water Ultrasonic absorption and velocity measurements have been made in several solutions of non-electrolytes in water such as amines, alcohols, ethers and ketones. Andrae and his collaborators25 were amongst the first to attempt a quantitative explanation of the experimental data.In these systems the experi- mental observations were as follows. As the concentration of the solute is increased from zero there is a sharp rise in the sound velocity until a certain concentration is reached when the sound velocity reaches a maximum and then falls off with further increase in concentration. An ultrasonic relaxation was also observed in these systems and the maximum absorption per wave- length was found to reach a maximum at a concentration termed the peak sound absorption concentration (PSAC). The solute concentration relating to the maximum in sound velocity is not the same as the PSAC.The sharp rise in the sound velocity at the lower solute concentrations shows that there is a sharp drop in the compressibility of the solution (cf. equation 2) although the solute that is added is more compressible than water. This is explained in terms of a breakdown in the hydrogen-bonded structure of water, which results from a solute molecule preventing a number of the water molecules around it from maintaining hydrogen-bonded contact with the highly orientated water structure nearby. These ‘free’ water molecules pack closely around each solute molecule with a subsequent decrease in their own compressibility and that of the solute. On the other hand the ultrasonic relaxation is attributed to a perturbation of the equilibrium 72 R.I.C. Reviews between solute-water complexes and solute and water. Several model equili- bria were used in an attempt to explain quantitatively the absorption data. These were respectively : and A + Bm +ABm A + B + A B A + B + AB + B + AB2 + B + AB3 + B etc. Model 3 Model 4 A + B + A B and B + B * where A represents the solute and B and B* ‘free’ and ‘bound’ water respec- t ivel y . The general agreement found between experiment and theory in the application of some of these model equilibria to explain the experimental data is a good indication of their basic correctness. However, several details Mole fraction acetone - Fig. 7. Experimental points and theoretical curves for model IV used by Andrae et a/. (refer- ence 25) Acetone-water.[ from : J. H. Andrae, P. D. Edmonds and J. F. McKellar. Acustica. 1965, 15, p. 74-88, Fig. 6 by courtesy of the publishers, S. Hirzel Verlag, Stuttgart.] Wyn-Jones Model 1 Model 2 73 have to be worked out in order to get a more complete quantitative explana- tion. One of the drawbacks in testing these models was the lack of data on thermodynamic quantities such as Cp and 8. In the system acetone-water where most data were available the agreement between experiment and theory for model 4 is shown in Fig. 7, where the solid lines are theoretical and the points experimental. In some aqueous solutions of aliphatic amines a double ultrasonic relaxa- tion was observed.The lower relaxation frequency (< 10 MHz) was only found in those solutions whose temperatures were fairly close to the critical solution temperature, and it was attributed to a phase separation process. The higher frequency relaxation was found to follow the pattern described above. In the system dioxan-water Hammes and Knoche26 observed a double relaxation in the frequency region 5-200 MHz. This was attributed to the two reactions : DW+W+DW2 and DW2 + D + D2W2 where D represents dioxan and W is water. The ultrasonic relaxation observed at very low amine concentrations in amine/water mixtures3 has been attributed to the reaction : RzNH + H2O + R2NH; + OH- which corresponds to model I used by Andrae and his collaborator^.^^ Rotational isomerism Ultrasonic relaxation also occurs in molecules that exhibit the phenomenon of rotational isomerism.27928 Most of the studies of these molecules have been carried out in the pure liquids. The ultrasonic relaxation is attributed to a perturbation of the equilibrium that exists between conformers that differ in energy.The equilibrium is disturbed by the temperature fluctuations accompanying the passage of the sound wave. The rotational isomers of 1,Zdibromoethane are shown in Fig. 8 together with the potential energy diagram for rotation. In the liquid phase the molecules of this compound exist as an equilibrium mixture of the trans and two optically active and energetically equivalent gauche isomers. This equilibrium can be represented as a two-state process A + B where A represents the more stable trans isomer and B the gauche isomers.B is a degenerate state owing to the optical activity of the gauche isomers. The relaxation time, T, and hence the relaxation frequency,f,, of a two-state equilibrium is related to the forward and reverse rate constants kl2 and k21 respectively by T = (k12 + k21)-l = 1/2nfc Thus k21 = 2nfc/(l + K) where K(= k12/k21) is the equilibrium constant. By measuring the temperature dependence of the relaxation time and also 74 R. I. C. Reviews t Potential energy 60 240 I80 I20 3 60 0 300 Azimuthal angle - Fig. 8. Rotational isomers and potential energy diagram for 1,2-dibromoethane (x = Br). making use of the Eyring first-order rate equation: kzi = K [y] exp (AH&/RT) exp (ASZIR) the slope of a plot of log (kzl/T) against 1/T will yield AHZ the potential barrier hindering rotation for the reverse isomerization B -+ A.In many cases K, the equilibrium constant, is not known, but a plot of log (fc/T) against reciprocal temperature29 will still give a very accurate value for AHZ. Thus by use of the ultrasonic technique the barrier height hindering rotation in molecules can be obtained directly from experimental data. The equilibrium thermodynamic parameters, AH, AS and A V, the enthalpy, entropy and volume differences between the isomers, are related to the maximum absorption per wavelength p m through equations 14 and 20. The manipulation of these equations to yield the equilibrium thermodynamic parameters requires many approximations29 and in many cases the resulting enthalpy differences are found to be incompatible with those from spectro- scopic methods.Ultrasonic relaxation studies leading to values for the activation energy barriers relating to conformational changes in other systems include ring inversion in cyclohexane derivatives30 : Axial Equatorial Wyn-Jones 75 cyclic sulphites :31 Axial Equatorial dioxans : Axial Equatorial 0 \ as well as cis .+ trans isomerism in aliphatic esters:33 R2 / R Trans 0 Cis In Table 1 a selected list of the potential energy barriers for conformational changes in different types of molecules is given. The n.m.r. technique can also be used to determine the exchange rates (kJ mol-1) 18.8 between different conformers of a molecule.For 1-fluoro-1 lY2,2-tetra- Table 1. Some energy barriers hindering rotation in molecules found from molecular acoustic studies Type of isomerism AH2 potential barrier hindering rotation in reverse isomerization Mo/ecu/e I ,2-dichloro-2-methyl propane I ,2-dibromo-2-methyl propane acrolein cinnamaldehyde methyl formate ethyl formate met h y I cyclo hexane ch lorocyclohexane 4-methyl- I ,3-dioxan 4-phenyl- I ,3-dioxan trimethylene sulphite 4-methyl trimethylene sulphite 23 .O 23 .O 20.9 23.4 32.6 25.5-33.5 axial-equatorial 45.4 chair inversion axial-equatorial 50.2 chair inversion axial-equatorial 37.2 chair inversion axial-equatorial 18.I chair inversion axial-equatorial chair inversion 19.2 trans-gauche trans-gauche cis-trans cis-trans cis-trans cis-trans axial-eq uatorial chair inversion R . I.C. Reviews 76 4 c 3.a t- Y M - & 2.c 1.c C - I .( Fig. 9. An Eyring rate plot for the gauche + trans isomerization in I-fluoro-I, ,2,2-tetra- chloroet hane. chloroethane both ultrasonic and n.rn.r. techniques have been used to study the temperature dependence of the exchange rates between the isomers. The different results are compared in the form of an Eyring rate plot in Fig. 9 and show that the agreement between these two different approaches is very g00d.34 .O Molecular association The first complete treatment of an ultrasonic relaxation process leading to the energetics of an equilibrium involving molecular association was carried out by Lamb and Pinkerton35 in a study of pure acetic acid.The relaxation process was attributed to a perturbation of the equilibrium between monomer and dimer molecules. The interpretation of the results for the pure acids, however, is not unique36 and there is also a possibility that internal rotation 3.0 Wyn- Jones 0 Ultrasonic data I 1 I 1 I I 6 .O 4 .O 0 n.m.r. data I f I O - ~ K- 77 about the C-0 bond (as found in esters) may contribute to the relaxation process.37 The monomer + dimer equilibrium in benzoic acid:3 has been measured in both carbon tetrachloride and toluene solutions.The rate constants for the dimerization reactions (cf. model equation 17) involving the formation of the hydrogen bonds are of the order 109 s-l mol-l i.e. almost diffusion controlled. Ultrasonic relaxation has also been observed in solutions of 2-pyridone,3* caprolactam and N-methyl acetamide39 in organic solvents. The mechanism of these relaxation processes is a perturba- tion of the following type of monomer-dimer equilibrium : APPLICATIONS TO REACTIONS OF BIOLOGICAL INTEREST While the examples discussed in the above sections are of interest because of their own intrinsic value they also form the basis of some recent work involving chemical reactions in the biological field. It has been shown that ultrasonic absorption and velocity measurements can be used as a probe to study microscopic solvent structure; in particular the behaviour of water in certain aqueous solutions (cf.non-electrolytes in water). Successive additions of urea, guanidine chloride, ammonium chloride and sodium chloride to water appear to have similar effects in causing a breakdown in the hydrogen-bonded structure of ~ a t e r . ~ O , ~ l On the other hand, an ultrasonic relaxation was observed in an aqueous solution of the synthetic polymer polyethylene glycol which was attributed42 to a hydrogen- bonded equilibrium between the polymer and water. When the behaviour of this relaxation was investigated in the presence of urea, guanidine chloride, ammonium chloride and sodium chloride it was found that the effects of urea and guanidine chloride were different from those of ammonium chloride and sodium chloride.Since urea and guanidine chloride are well known protein denaturants it was concluded that their distinctive effects on the relaxation process in affecting the local solvent structure around the hydro- phobic groups of the polymer could be related to their denaturing ability.41 these At molecules serve as simple models for solutions The the behaviour isoelectric of pH amino - 5-6, acids an in various ultrasonic the relaxation polypeptide is also was of chain interest observed43 in proteins. because in aqueous solutions of glycine, diglycine and triglycine. The relaxation time was independent of concentration and was also unaffected by the addition of protein denaturing agents and neutral salts.The relaxation process was R. I. C. Reviews 78 found to be a unique property of the zwitterion structure and was attributed to a perturbation of an equilibrium of the kind: where G represents glycine, n = 1, 2 or 3 and the number x was unspecified. aqueous d i g l y ~ i n e ~ ~ In another independent solution investigation, at pH - 11 was an ultrasonic attributed relaxation to the proton observed transfer in equilibrium : -0OCCHzNHCOCH2NH; + OH- + -0OCCH2NHCOCH2NHz + H2O which is equivalent to model equilibrium 18 when occurring in excess water. The rate constant klz was of the order 1O1O s-l mol-l which means that the recombination rate is diffusion controlled.In many biological systems, especially those involving macromolecules containing the -NHCO- group, the role of equilibria where hydrogen bonding occurs is important. In systems containing proteins there are several equilibria, both intra- and intermolecular in nature, competing for hydrogen bonds. In order to make an assessment of the importance of these equilibria it is desirable to study several simple equilibria which serve as model steps in these complicated systems. Progress in this direction is discussed in the section on molecular association. In addition, simple competitive reactions between molecules with purine and pyrimidine structures, such as that between caprolactam and 2-aminopyrimidine in various solvents, will serve as models for base-pairing reactions in nucleic a c i d ~ .~ 5 The kinetics of the conformational changes involving the helix-coil transi- tion has also been observed in the polypeptide poly-~-lysine.~~ This confor- mational change was one of the mechanisms discussed in the interpretation of the relaxation observed in poly-L-glutamic acid solutions.46 VIBRATIONAL RELAXATION An ultrasonic relaxation in liquids occurs due to the time delay in the transfer of energy from the translational to the vibrational degrees of freedom. This phenomenon is usually called vibrational relaxation and is associated with the vibrational specific heat of the molecule. The redistribution of energy between the vibrational and translational modes is induced by temperature changes and at constant temperature and pressure does not involve volume changes.The time delay in the transfer of energy occurs because it takes many collisions before a molecule loses one quantum of vibrational energy. This energy is coupled into the molecule through the lowest frequency vibrational mode and then spreads rapidly to all the other modes. Since this process is a thermal effect (i.e. it is initiated by the temperature changes accompanying the sound wave) the total specific heat becomes frequency dependent in the relaxation region. The relaxation contribution to the specific heat, AC, usually termed the vibrational specific heat, is related to the relaxation strength AS/&, by APlP = (Y - 1) ACPIC, .. . . 28 79 and thus p m = ~ ( r - 1) ACp/2Cp The quantity (y - 1) can be calculated from equation 3. The vibrational specific heat ACp can be calculated from the usual Planck-Einstein formula: ACp = C niRxi exp(- xi)/(l - exp(- xi)) i .. 29 where X i = hvi/kT and vi is the frequency of the ith vibrational mode of degeneracy ni. The relaxation strength AP/Po can be determined from experi- ment via equation 14 and can also be calculated through equations 3, 29 and 28. In some molecules the total vibrational specific heat is associated with a single relaxation time whereas in others a multiple relaxation is observed resulting from the different relaxation times of different vibrational modes. In liquid carbon disulphide a single ultrasonic relaxation was attributed47 to the relaxation of the total vibrational specific heat involving the two non- degenerate vibrational modes at 1523 and 657 cm-1 and also the doubly degenerate deformation mode at 397 cm-l.Analysis of the relaxation led to the following data: 0.165 0.085 25 "C -63°C 25 "C 25 "C 0 "C 0.167 0.085 In methylene chloride, on the other hand, a single relaxation in the ultra- sonic range48-50 was attributed to the relaxation of all the normal modes except the lowest at 283 cm-l. The following data were obtained to justify this conclusion. 0.065 0.069 0.055 0.022 0.066 0.066 0.055 0.022 -60°C The results of a spot absorption,5l and velocity measurements52 in the Gigahertz region using Brillouin scattering, indicate that the relaxation frequency associated with the deactivation of the lowest mode, the bending CCl2, is at approximately 10 GHz.The behaviour of the relaxation processes associated with vibrational specific heat is very similar for gases and liquids. This was demonstrated experimentally for carbon dioxide when the relaxation time was measured as a function of density in going from gas to Using some of the existing kinetic theories of the liquid phase and assuming that vibrational relaxation occurs due to binary collisions of molecules, Litovitz and his collaborators53-56 have been able to account quantitatively for the pressure dependence of the vibrational relaxation time of both CSz and COZ.In brief, an increase in R. I . C. Reviews 80 the pressure increases the number of binary collisions and thus makes the energy transfer from translational to vibrational degrees of freedom occur in a shorter time. This, in turn, increases the ultrasonic relaxation frequency. In the interpretation of the experimental data for COz a modified version of the ‘fixed wall model’ was proposed53 to calculate the mean free path of a molecule in the liquid state. In most liquids the binary collisions between the molecules are so efficient that the relaxation frequencies associated with the vibrational specific heat are well into the Gigahertz region. This means that it has only been possible to study a handful of molecules in the liquid phase. CRITICAL POINT PHENOMENA Ultrasonic relaxation also occurs around the critical points in liquids ; the critical point being the normal gas/liquid critical point in a one-component system or the point of critical phase separation in binary liquid mixtures.Most of the work reported on this subject has involved velocity and/or absorption measurements in binary liquid mixtures near the critical point or, as it is sometimes referred to, the consulate temperature.57358 In most of the studies involving sound absorption, the observed relaxation cannot be represented by a model involving a single relaxation time. Two mechanisms have been put forward to account for the anomalous behaviour of sound absorption and velocity near the critical point. In the first mode159 the relaxation is thought to arise from a scattering of the sound energy due to density or composition variation around the critical point.This theory predicts that alf2 should increase with increasing frequency. An alternative model has been proposed by Fixman60-62 and is associated with critical fluctuations in the liquid parameters that are induced by the temperature changes accompanying the sound wave. These fluctuations imposed by the sound wave affect the liquid parameters specific heat, compo- sition, the molecular level friction constant and the coefficients of the long- range correlation function. This theory predicts that the dependence of a/f2 with frequency is: alf2 cc f 514 in the critical region. Recent experimental data appears to be consistent with Fixman’s theory both for the gas/liquid@ critical point as well as the consulate temperature in binary liquid mixtures.G1363364 However, there are several details that still remain to be worked out before a full quantitative description of sound velocity and absorption around the critical point is obtained. SHEAR AND VOLUME VISCOSITY PHENOMENA In liquids where relaxation processes such as those described above are either absent or occur well outside the present available frequency range ( 104-1010 Hz) acoustic absorption and velocity measurements have been used to determine the volume viscosity of liquids by use of equations 4, 5 and, where applicable, 6.The values of the quantity rlv/rls for several classes of liquids are in Table 2.65 In liquids ranging from hydrocarbons, alcohols and Wyn-Jones 81 Table 2.Comparison of volume and shear viscosities for several liquids.65 Molecule Carbon tetrachloride Toluene Chlorobenzene Cyclohexanone Propane Cyclopentane n - Hexan e n-Heptane n-Octan e Acetone Benzene Dichloromethane Carbon disulphide lsobutyl bromide Hydrocarbon oil (MW m 300) Liquid polymers Associated I iq u ids Liquid metals Fused salts Liquid argon WIT8 I .27 long chain polymers to inorganic melts, molten metals and liquid argon the orders of magnitude of volume and shear viscosity are approximately the same. This result suggests that there is a close relationship between the mechanisms of shear and volume viscosity.In order to appreciate this relationship it is necessary to consider the general background of the theories of both liquid structure and viscosities. A liquid exists structurally in a quasi-crystalline form made up of submicro- scopic crystals containing from about five to 50 molecules. These crystalline groups are held together by ‘intermolecular bonds’ which are continually breaking up and reforming. At any temperature or pressure a degree of order exists in the liquid which describes its geometrical state. The liquid lacks long-range order and the short-range order that is present is imperfect result- ing in the presence of holes. A change in energy of the liquid by variation of temperature or pressure will affect the short-range order, hence the imper- fection, and also the number of holes.For example, a temperature rise will cause the short-range order to decrease and thus increase the number of holes. These changes, in turn, affect the thermal expansion coefficient, 8. On the other hand a pressure rise in the liquid will increase the short-range order, decrease the number of holes and thus affect the compressibility, ,8. In the current theories of shear viscosity of liquids the molecules under shear flow are involved in translational jumps from one lattice site to another. The mobility of the molecules that are invoIved in these translational jumps is determined by both the probability that a molecule has enough energy to break away from its neighbours (submicroscopic crystalline groups) and also the probability that there is sufficient free volume (holes) for the molecule to jump into.In a similar way volume viscosity arises as a result of the flow of molecules from one lattice site to another lattice position of closer packing 0 . 9 0.44 I I . 3 3 1-3 0.4-4 82 1-2.5 I .7 I .6 0.4 1 . 1 0.7 6 7 6 2 0.5 I .4 0.2 R. I . C. Reviews as a result of acoustic pressure. This flow takes a finite time and thus the volume changes are out of phase resulting in acoustic attenuation. In both shear and volume viscosity processes, molecules change their lattice positions and thus the same intermolecular ‘bonds’ must be broken in the molecular clusters, making these processes closely related as the data in Table 2 suggest.When a molecule changes from one lattice site to another an energy change is involved and since the molecules take a finite time over this movement this type of structural rearrangement will give rise to acoustic relaxation. In practice acoustic relaxation arising from shear and volume (or compres- sional) viscous processes have been observed. These relaxation processes have been studied by combining both longitudinal (i.e. ordinary ultrasonic) wave and also ultrasonic shear wave measurements at different temperatures and frequencies. The data are usually presented in the form of reduced plots of the liquid moduli.66 It has been found that the behaviour of both shear and volume relaxation data are very similar.With the exception of molten zinc chloride all the liquids studied give rise to a distribution of relaxation times which has been attributed to the co-operative behaviour of these structural decay processes, and the results have been used to test various theories of the liquid phase.66 RECOMMENDED READING FOR THE SPECIALIZED READER Books K. F. Herzfield and T. A. Litovitz, Absorption and Dispersion of Ultrasonic Waves. New York: Academic Press, 1959. V. F. Nozdrev, The Use of Ultrasonics in Molecular Physics. London: Pergamon, 1965. D. Sette (ed.), Dispersione ed assorbimento del mono nei processi rnolecolari (in English). New York: Academic Press, 1963. Warren P. Mason (ed.), Physical Acoustics. New York: Academic Press, 1965.Vol. I Part A and B: Methods and Devices, Vol. I1 Part A: Properties of Gases, Liquids and Solutions, Vol. I1 Part B: Properties of Polymers and Nonlinear Acoustics. A. B. Bhatia, Ultrasonic Absorption. Oxford: Clarendon Press, 1967. (Monograph on the Physics and Chemistry of Materials). Review articles R. 0. Davies and J. Lamb, Q. Rev. chern. Soc., 1957. 11, 134. G. Verma, Rev. mod. Phys., 1959, 31, 1052. G. Atkinson, S. Kor and S. Petrucci, Proc. Instn elect. Engrs, 1965, 53, 1355. J. E. Piercy, Proc. Znstn elect. Engrs, 1965, 53, 1346. M. Eigen and L. de Maeyer in Technique oforganic Chemistry, ed. S. L. Friess, E. S. Lewis and A. Weissberger, vol. 8, part 2, p. 895. New York: Interscience, 1963. T. A. Litovitz, J . acoust.SOC. Am., 1959, 31, 681. D. Sette in Handbuch der Physik, ed. S. Flugge, Sec. XI, Vol. I . Berlin: Springer, 1961. Aids to literature searches The Journal of the Acoustical Society of America publishes, every other month, an index of published papers on ultrasonic absorption in liquids under subject index 10.5. REFERENCES 1 G. Stokes, Trans. Camb. SOC., 1845, 8, 287. 2 T. A. Litovitz, J . acoust. SOC. Am., 1959, 31, 681. 3 M. Eigen and L. de Maeyer in Technique of Organic Chemistry, ed. S. L. Friess, E. S. Lewis and A. Weissberger, vol. 8, part 2, p. 895. New York: Interscience, 1963. 4 See, for example, Chapters 4, 5 and 6 in Physical Acoustics, ed. Warren P. Mason, Vol. 11, Part A. New York: Academic Press, 1965. Wyn-Jones 83 5 H. J. McSkimin in Physical Acoustics, ed.Warren P. Mason, Vol. I, Part A, Chapter4. New York: Academic Press, 1965. 6 F. Eggers, private communication. 7 L. Brillouin, Annls Phys., 1922, 17, 88. 8 K. Tamm, Z. Elektrochem., 1960, 64, 73. 9 W. A. Poth, quoted in Landolt-Bornstein, Physical Chemical Tables EG 111,1936,2818. 10 K. Tamm, G. Kurtze and R. Kaiser, Acustica, 1954, 4, 380. 11 M. Eigen, Z. phys. Chem. Frankf. Ausg., 1954, 1, 176. 12 P. Debye, Trans. electrochem. SOC., 1942, 82, 265. 13 E. Carnevale and T. A. Litovitz, J. acoust. SOC. Am., 1958, 30, 610. 14 J. Stuehr, E. Yeager, T. Sachs and E. Hovorka, J. chem. Phys., 1963, 38, 587. 15 M. Eigen and K. Tamm, Z. Elektrochem, 1962,66, 93 and 107. 16 G. Kurtze and K. Tamm, Acustica, 1953, 3, 33.17 0. Wilson and R. Leonard, Tech. Report 4, Office of Naval Research Contract N6 onr 27507, University of California, Los Angeles, California. 18 C . Mulders, Appl. scient. Research, 1949, B1, 341. 19 M. Smith, R. Barrett and R. Beyer, J . acoust. SOC. Am., 1951, 23, 71. 20 J. Stuehr and E. Yeager in Physical Acoustics, ed. Warren P. Mason, Vol. 11, part A, p. 351. New York: Academic Press, 1965. 24 G. Verma, Rev. mod. Phys.. 1959, 31, 1052. 21 F. H. Fisher, J. acoust. SOC. Am., 1958, 30, 442; 1963, 35, 805. 22 K. Tamm in Dispersione ed assorhimento del suono nei processi molecolari, ed. D. Sette, p. 175. New York: Academic Press, 1963. 23 G. Atkinson, S. Kor and S. Petrucci, Proc. Instn elect. Engrs, 1965, 53, 1355. 25 J. H. Andrae, P.D. Edmonds and J. F. McKellar, Acustica, 1965, 15, 74. 27 R. 0. Davies and J. Lamb, Q. Rev. chem. SOC., 1957, 11, 134. 26 G. G. Hammes and W. Knoche, J. chem. Phys., 1966,45,4041. 28 J. Lamb in Physical Acoustics, ed. Warren P. Mason, Vol. 11, part A, p. 203. New York: Academic Press, 1965. 29 E. Wyn-Jones and W. J. Orville-Thomas, Chem. SOC. Special Publication No. 20 (Molecular Relaxation Processes), 1966, 109. 30 J. E. Piercy, J . acoust. SOC. Am., 1961, 33, 198. 31 R. A. Pethrick, E. Wyn-Jones, P. C. Hamblin and R. F. M. White, J. mol. Struct, 1968, 1, 333. 32 P. C. Hamblin, R. F. M. White and E. Wyn-Jones, Chem. Communs, 1968, 1058. 33 J. Bailey and A. M. North, Trans. Faraday Soc., 1968, 64, 1497. 34 R. A. Pethrick and E. Wyn-Jones, J. chem. Phys., 1968 (in the press). 35 J. Lamb and J. M. M. Pinkerton, Proc. R. Soc., 1949, A197, 114. 36 J. E. Piercy and J. Lamb, Trans. Faraday SOC., 1956, 52, 930. 37 J. E. Piercy and M. G. Seshagiri Rao, J . acoust. SOC. Am., 1967, 41, 1591. 38 G. G. Hammes and H. Olin Spiney, J. Am. chem. SOC., 1966, 88, 621. 39 R. A. Pethrick, D. Grimshaw and E. Wyn-Jones, unpublished data. 40 G. G. Hammes and P. R. Schimel, J. Am. chem. SOC., 1967, 89,442. 41 G. G. Hammes and J. C. Swann, Biochemistry, N . Y., 1967, 6, 1591. 42 G. G. Hammes and T. B. Lewis, J. phys. Chem., 1966, 70, 1610. 43 G. G. Hammes and C. Nick Page, J . phys. Chem., 1968,72, 2227. 44 R. C. Parker, L. J. Slutsky and K. R. Applegate, J . phys. Chem., 1968, 72, 3177. 45 L. de Maeyer, M. Eigen and J. Suarez, J . Am. chem. SOC., 1968, 90, 3 157. 46 J. J. Burke, G. G. Hammes and T. B. Lewis, J. chem. Phys., 1965, 42, 3520. 47 J. H. Andrae, E. L. Heasell and J. Lamb, Proc. phys. SOC., 1956, B69, 625. 48 J. H. Andrae, Proc. phys. Soc., 1957, B70, 71. 49 J. H. Andrae, P. L. Joyce and R. J. Oliver, Proc. phys. Soc., 1960, 75, 82. 50 J. L. Hunter and H. D. Dardy, J. chem. Phys., 1965, 42, 2961. 51 G. R. Hanes, R. Turner and J. E. Piercy, J. acoust. Soc. Am., 1967, 38, 1057. 52 N. A. Clark, C . E. Molter, J. A. BucaFo and E. F. Carome, J. chem. Phys., 1966,44,2528. 53 W. M. Madigosky and T. A. Litovitz, J. chem. Phys., 1961. 34, 489. 54 T. A. Litovitz, E. Carnevale and P. Kendall, J. chem. Phys., 1957, 26, 465. 55 T. A. Litovitz, J. chem. Phys., 1957, 26, 469. 56 T. A. Litovitz and E. A. Carnevale, J . acoust. SOC. Am., 1958, 30, 134. 57 D. Sette, Nat. Bur. Std. (US.) Misc. Pub/., 1965, 273, 183. 58 D. Sette in Handbuch deer Physik, ed. S. Flugge, Sec. XI, Vol. 1. Berlin: Springer, 1961. 59 A. E. Brown and E. G. Richardson, Phil. Mag., 1959, 4, 705. 60 M. Fixman, Adv. chem. Phys., 1964, 6, 175. 61 M. Fixman, J. chem. Phys., 1962, 36, 1957, 1961. 62 M. Fixman, J. chem. Phys., 1965, 42, 196, 199. R. I.C. Reviews 84 63 A. V. Anantarman, A. B. Walters, P. D. Edmonds and C . J. Pings, J. cheni. Phys:. 1966,44, 2651. 64 G. D. Arrigo and D. Sette, J. chem. Phys., 1968, 48, 691. 65 J. E. Piercy and M. G. Seshagiri Rao, J . acuust. SOC. Am., 1967, 41, 1063. 66 T. A. Litovitz and C . M. Davies in Physical Acoustics, ed. Warren P. Mason, Vol. TI, part A, p. 281. New York: Academic Press, 1965. Wyn-Jones 85