Pressure dependence

Pressure Dependence.— There is growing interest in interpreting the effects of pressure on thermodynamic quantities, especially in relation to phase loci.  

Schneider has reviewed this for non-polymer systems, Noguchi and Nose for polymer solutions using corresponding states theories (p. 304), and Bogdanovic et al. have critically reviewed the representation of thermodynamic data for polyethylene-ethylene at high pressure. For the same system, Harmony et have extended corresponding states models to calculate phase equilibria data over a wide pressure range. 

Wolf and Jendm find a corresponding states model useful in representing pressure dependence of critical loci, but highly sensitive to the careful adjustment of system specific parameters. Ishizawa et al. have discussed the pressure dependence of UCST in polystyrene-cyclohexane in terms of equation-of-state models. 

Wolf and Blaum n i have obtained an impressive range of data on the phase behaviour of oligomer mixtures, including dependence on pressure. They conclude that a quantitative description is impossible on the basis of any available theory but find a lattice-gas model useful in evaluating their results. 

The pressure dependence of the bulk modulus, and Raman shift as well, under compression are usually described using the quadratic functions [13], 

The pressure dependence of a has been examined in the pressure range between ambient pressure and 3 kbar with a polyacetylene sample doped to maximum with iodine and stretched by a factor of 6.5. Helium gas was used as pressure medium. Within the accuracy of the measurement, which was only 3 % in this experiment, no change of 

Based on the aforementioned discussion, DiGuilio et al. (1990), DiGuilio and Teja (1992), Bleazard et al. (1994), and Bleazard and Teja (1995), proposed a simple scheme for the prediction of the thermal conductivity of aqueous solutions at high pressures. According to this scheme, the thermal conductivity of the aqueous solution at a high pressure is obtained by the equivalent one at atmospheric pressure by multiplying it with the ratio of the thermal conductivity of water at that high pressure over its value at atmospheric pressure. This idea produced values that deviated by up to 2% from the experimental data. 

In this case, one can also integrate (8.34c) to obtain the form of the Van t Hoff equation most familiar to beginning chemistry students, namely, 

One can also see that the Van t Hoff equation is consistent with the prior conclusion [cf. (8.29), Sidebar 8.1] that AH° controls the variations of Keq with temperature. 

From (8.34c) or (8.35), it is easy to see that if the chosen reaction is endothermic (AH° 0), then a T increase tends to promote product formation (the reaction shifts right ). Conversely, if the reaction is exothermic (AH° 0), a temperature increase promotes formation of reactants (the equilibrium shifts left ). Such conclusions appear intuitive from the perspective of Le Chatelier s principle, and indeed we shall show in Section 8.6 that such Le Chatelier-like conclusions arise from deep theoretical roots that permeate the Van t Hoff equation and many other thermodynamic relationships. 

For the pressure dependence of Keq (at constant T), we again differentiate (8.32) to obtain 

Graphically, (8.39b) tells us that a log-log plot of In Keq versus In P should lead to straight-line behavior with slope given by (the negative of) the reaction volume change AV°  

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The Langmuir-Hinshelwood picture is essentially that of Fig. XVIII-14. If the process is unimolecular, the species meanders around on the surface until it receives the activation energy to go over to product(s), which then desorb. If the process is bimolecular, two species diffuse around until a reactive encounter occurs. The reaction will be diffusion controlled if it occurs on every encounter (see Ref. 211) the theory of surface diffusional encounters has been treated (see Ref. 212) the subject may also be approached by means of Monte Carlo/molecular dynamics techniques [213]. In the case of activated bimolecular reactions, however, there will in general be many encounters before the reactive one, and the rate law for the surface reaction is generally written by analogy to the mass action law for solutions. That is, for a bimolecular process, the rate is taken to be proportional to the product of the two surface concentrations. It is interesting, however, that essentially the same rate law is obtained if the adsorption is strictly localized and species react only if they happen to adsorb on adjacent sites (note Ref. 214). (The apparent rate law, that is, the rate law in terms of gas pressures, depends on the form of the adsorption isotherm, as discussed in the next section.)... 

The viscosity, themial conductivity and diffusion coefficient of a monatomic gas at low pressure depend only on the pair potential but through a more involved sequence of integrations than the second virial coefficient. The transport properties can be expressed in temis of collision integrals defined [111] by... 

The correct treatment of the mechanism (equation (A3.4.25), equation (A3.4.26) and equation (A3.4.27), which goes back to Lindemann [18] and Hinshelwood [19], also describes the pressure dependence of the effective rate constant in the low-pressure limit ([M] < [CHoNC], see section A3.4.8.2). 

The effective rate law correctly describes the pressure dependence of unimolecular reaction rates at least qualitatively. This is illustrated in figure A3,4,9. In the lunit of high pressures, i.e. large [M], becomes independent of [M] yielding the high-pressure rate constant of an effective first-order rate law. At very low pressures, product fonnation becomes much faster than deactivation. A j now depends linearly on [M]. This corresponds to an effective second-order rate law with the pseudo first-order rate constant Aq ... 

Figure A3.4.9. Pressure dependence of the effective unimolecular rate constant. Schematic fall-off curve for the Lindemaim-FIinshelwood mechanism. A is the (constant) high-pressure limit of the effective rate constant...

Figure A3.6.1. Pressure dependence of imimolecular rate constant...

In the thennodynamic fomiiilation of TST the pressure dependence of the reaction rate coefficient defines a volume of activation [24, 25 and 26]... 

There is one important caveat to consider before one starts to interpret activation volumes in temis of changes of structure and solvation during the reaction the pressure dependence of the rate coefficient may also be caused by transport or dynamic effects, as solvent viscosity, diffiision coefficients and relaxation times may also change with pressure [2]. Examples will be given in subsequent sections. 

In a microscopic equilibrium description the pressure-dependent local solvent shell structure enters tlirough... 

For very fast reactions, as they are accessible to investigation by pico- and femtosecond laser spectroscopy, the separation of time scales into slow motion along the reaction path and fast relaxation of other degrees of freedom in most cases is no longer possible and it is necessary to consider dynamical models, which are not the topic of this section. But often the temperature, solvent or pressure dependence of reaction rate... 

Because of the general difficulty encountered in generating reliable potentials energy surfaces and estimating reasonable friction kernels, it still remains an open question whether by analysis of experimental rate constants one can decide whether non-Markovian bath effects or other influences cause a particular solvent or pressure dependence of reaction rate coefficients in condensed phase. From that point of view, a purely... 

Measuring tire pressure dependence of k at different temperatures shows that the apparent activation energy at constant viscosity decreases with increasing viscosity [46, ( figure A3,6,8). From a detailed analysis one... 

This interpretation of the experimentally detennined pressure dependence of the isomerization rate rests on... 

Basilevsky M V, Weinberg N N and Zhulin V M 1985 Pressure dependence of activation and reaction volumes J. Ohem. Soc. Faraday Trans. 1 81 875-84... 

Schroeder J, Schwarzer D, Troe J and Voss F 1990 Cluster and barrier effects in the temperature and pressure dependence of the photoisomerization of trans.stilbene J. Chem. Phys. 93 2393-404... 

Meyer A, Schroeder J and Troe J 1999 Photoisomerization of f/ a/rs-stilbene in moderately compressed gases pressure-dependent effective barriers J. Phys. Chem. A 103 10 528-39... 

Borkovec M, Straub J E and Berne B J The influence of intramolecular vibrational relaxation on the pressure dependence of unimolecular rate constants J. Chem. Phys. 85 146... 

Miller W H 1988 Effect of fluctuations in state-specific unimolecular rate constants on the pressure dependence of the average unimolecular reaction rated. Phys. Chem. 92 4261-3... 

Song K and Hase W L 1998 Role of state specificity in the temperature- and pressure-dependent unimolecular rate constants for H02->H+02 dissociation J. Phys. Chem. A 102 1292-6... 

A recent example of laser flash-lamp photolysis is given by Hippier etal [ ], who investigated the temperature and pressure dependence of the thennal recombmation rate constant for the reaction... 

Apart from the natural lifetime due to spontaneous emission, both uni- and bimolecular processes can contribute to the observed value of T. One important contribution comes from coiiisionai broadening, which can be distmguished by its pressure dependence (or dependence upon concentration [M] of tlie collision partner) ... 

The development of active ceramic-polymer composites was undertaken for underwater hydrophones having hydrostatic piezoelectric coefficients larger than those of the commonly used lead zirconate titanate (PZT) ceramics (60—70). It has been demonstrated that certain composite hydrophone materials are two to three orders of magnitude more sensitive than PZT ceramics while satisfying such other requirements as pressure dependency of sensitivity. The idea of composite ferroelectrics has been extended to other appHcations such as ultrasonic transducers for acoustic imaging, thermistors having both negative and positive temperature coefficients of resistance, and active sound absorbers. 

At high neutralization levels with alkaH metal ions, many ionomers spontaneously form coUoidal suspensions in water when stirred vigorously at 100—150°C under pressure. Depending on soHds content and acid level, the dispersions range in viscosity from water-like to paste-like. These provide convenient methods for applying thin coatings of ionomers to paper and other substrates. 

The experimentally measured dependence of the rates of chemical reactions on thermodynamic conditions is accounted for by assigning temperature and pressure dependence to rate constants. The temperature variation is well described by the Arrhenius equation. 

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