Effect of Pressure on Activity

It can be shown from equation (6.85) that pressure has only a small effect on the activity of liquids and solids. 

A change of 1 MPa in pressure changes cii by 2.66%. Thus, the change of a few kPa in pressure around ambient pressure has only a small effect on a,. 

---

IV. Effect of Pressure on Activity Coefficients Partial Molar Volumes. 160... 

In Section I, we indicated that significant progress in understanding high-pressure thermodynamics of mixtures requires a quantitative description of the variation of fugacity with pressure as given by Eq. (3). To obtain the effect of pressure on activity coefficient we substitute as follows ... 

The magnitude of the effect of pressure on activity coefficients is much less than is the case for solubility products (see previous discussion). The errors in ignoring the compressibility term for activity coefficients are, at most, 2-5% in a pressure range up to 1000 bars (Millero 1983 Krumgalz et al. 1999). For the broad-scale FREZCHEM model, these errors are acceptable. 

In considering the effect of pressure on activity, we must recall that the standard state pressure (P°) is not always the same as the system pressure (P), so that the differentiation with respect to pressure is not always completely analogous to differentiation with respect to temperature. First of all, for variable pressure standard states, those that do have P° = P, we have... 

Figure 4. Effect of pressure on activity of TGase endogenous to surimi. From (Shoji et al, 1990).

Some investigations of the effect of pressure on activity for a given activation temperature indicate that the maintenance of a very low pressure over the catalyst bed during activation is necessary for attainment of maximum catalytic activity. 

The standard-state fugacity of any component must be evaluated at the same temperature as that of the solution, regardless of whether the symmetric or unsymmetric convention is used for activity-coefficient normalization. But what about the pressure At low pressures, the effect of pressure on the thermodynamic properties of condensed phases is negligible and under such con-... 

A calculation of the effect of pressure on the activity that does not involve the assumption of constant Vm usually starts with the compressibility k. Integration of equation (1.39) that relates k to V, while assuming that k is independent of pressure, gives the equation... 

This form assumes that the effect of pressure on the molar volume of the solvent, which accelerates reactions of order > 1 by increasing the concentrations when they are expressed on the molar scale, has been allowed for. This effect is usually small, ignored but in the most precise work. Equation (7-41) shows that In k will vary linearly with pressure. We shall refer to this graph as the pressure profile. The value of A V is easily calculated from its slope. The values of A V may be nearly zero, positive, or negative. In the first case, the reaction rate shows little if any pressure dependence in the second and third, the applied hydrostatic pressure will cause k to decrease or increase, respectively. A positive value of the volume of activation means that the molar volume of the transition state is larger than the combined molar volume of the reactant(s), and vice versa. 

The solvent may be an important parameter for reactions carried out in solution, since the value of activation volume is often dependent on the solvent. A limitation may be due to the effect of pressure on the freezing temperature of... 

At pressures above a few atmospheres, the deviations from ideal behaviour in the gas phase will be significant and must be taken into account in process design. The effect of pressure on the liquid-phase activity coefficientmustalso be considered. A discussion of the methods used to correlate and estimate vapour-liquid equilibrium data at high pressures is beyond the scope of this book. The reader should refer to the texts by Null (1970) or Prausnitz and Chueh (1968). 

The fundamental fact on which the analysis of heterogeneous reactions is based is that when a component is present as a pure liquid or as a pure solid, its activity may be taken as unity, provided the pressure on the system does not differ very much from the chosen standard state pressure. At very high pressures, the effect of pressure on solid or liquid activity may be determined using the Poynting correction factor. 

From this equation, we see that if the rate constant is determined at a series of pressures, a plot of In k versus P should be linear with the slope being —AV /RT. Although this approach is valid, the graph obtained may not be exactly linear, but the interpretation of these cases does not need to be presented here. It is sufficient to note that the volume of activation can be determined by studying the effect of pressure on reaction rates. 

Processes with gaseous reactants are excluded here. Due to the large compressibility of gases an increase of pressure (up to 1 kbar) leads essentially only to an increase of gas concentration, and hence to an acceleration of bimolecular processes in which gases are involved as reactants. The effect of pressure on a chemical reaction in compressed solution is largely determined by the volume of reaction (AV) and the volume of activation (AV ). It is not the purpose of this chapter to provide a complete survey of reactions of dienes and polyenes which have been investigated at elevated pressures. There are many excellent monographs (e.g. References 1-4) and reviews (e.g. References 5-16) on this topic which cover the literature up to early 1990. After a short introduction into the basic concepts necessary to understand pressure effects on chemical processes in compressed solutions, our major objective is to review the literature of the past ten years. 

The effect of pressure on chemical equilibria and rates of reactions can be described by the well-known equations resulting from the pressure dependence of the Gibbs enthalpy of reaction and activation, respectively, shown in Scheme 1. The volume of reaction (AV) corresponds to the difference between the partial molar volumes of reactants and products. Within the scope of transition state theory the volume of activation can be, accordingly, considered to be a measure of the partial molar volume of the transition state (TS) with respect to the partial molar volumes of the reactants. Volumes of reaction can be determined in three ways (a) from the pressure dependence of the equilibrium constant (from the plot of In K vs p) (b) from the measurement of partial molar volumes of all reactants and products derived from the densities, d, of the solution of each individual component measured at various concentrations, c, and extrapolation of the apparent molar volume 4>... 

The observation that the transition state volumes in many Diels-Alder reactions are product-like, has been regarded as an indication of a concerted mechanism. In order to test this hypothesis and to gain further insight into the often more complex mechanism of Diels-Alder reactions, the effect of pressure on competing [4 + 2] and [2 + 2] or [4 + 4] cycloadditions has been investigated. In competitive reactions the difference between the activation volumes, and hence the transition state volumes, is derived directly from the pressure dependence of the product ratio, [4 + 2]/[2 + 2]p = [4 + 2]/[2 + 2]p=i exp —< AF (p — 1)/RT. All [2 + 2] or [4 + 4] cycloadditions listed in Tables 3 and 4 doubtlessly occur in two steps via diradical intermediates and can therefore be used as internal standards of activation volumes expected for stepwise processes. Thus, a relatively simple measurement of the pressure dependence of the product ratio can give important information about the mechanism of Diels-Alder reactions. 

In contrast to the effects of temperature, the effect of pressure on c/w is relatively small and can be neglected for reasonable pressure differences. Based on thermodynamics, a change in total pressure of a system affects the vapor pressure. The change in water activity with pressure, at constant moisture content, can be calculated using Eq (8) (Bell and Labuza, 2000) ... 

Figure 8-8 Effect of Pressure on AES Cell Performance at 1000°C [(22) 2.2 cm diameter, 150 cm active length]... 

The effect of pressure on chemical equilibria and reaction rates is described by the following standard equations which define the reaction volume A V and activation volume of a reaction ... 

It should be noted that the condition of a dilute solution was introduced into the considerations for two reasons primarily, in order that it would be possible to replace the activities by concentrations and thus determine the equilibrium concentrations on the basis of (2.3.3) and, secondarily, in order for it to be possible to neglect the effect of pressure on the chemical potentials of the components whose electrochemical potentials appear in (2.3.2). Because of the differing ionic concentrations in solutions 1 and 2, the osmotic pressures in these solutions are not identical and this difference must be compensated by external pressure. A derivation considering the effect of pressure can be found, for example in [9] or p. 191 of [18]. 

Chain-transfer reactions would be expected to increase in rate with increasing pressure since transfer is a bimolecular reaction with a negative volume of activation. The variation of chain-transfer constants with pressure, however, differ depending on the relative effects of pressure on the propagation and transfer rate constants. For the case where only transfer to chain-transfer agent S is important, Cs varies with pressure according to... 

R16H selectivity and activity kinetics were fit over a wide range of temperature and pressure. Reforming selectivity is shown in Figs. 16 and 17, where benzene and hexane are plotted against C5-, the extent of reaction parameter. The effect of pressure on reforming a 50/50 mixture of benzene and cyclohexane at 756 K is shown in Fig. 16. Selectivity to benzene improves significantly when pressure is decreased from 2620 to 1220 kPa. In fact, at 2620 kPa, hexane is favored over benzene when the C5 yield exceeds 10%. This selectivity behavior can be seen in the selectivity rate constants ... 

---