Activity coefficient pressure variation

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 ... 

For a dilute solution at high pressure, the variation of activity coefficient with pressure cannot be neglected. But when x2 is small, it is often a good approximation to assume, as above, that the activity coefficient is not significantly affected by composition. If we also assume that v2 the partial molar volume of the solute, is independent of both pressure and composition... 

When the Krichevsky-Kasarnowsky equation fails it may be because of either changing activity coefficient of the solute gas with composition, changing partial molal volume of the gas with pressure, or both. The Krichevsky-Ilinskaya equation takes into account the variation in the activity coefficient of the solute gas with mole fraction by means of a two-suffix Margules equation. 

FIGURE 9.4 The variation of the molar free energy of a real gas with its partial pressure (orange line) superimposed on the variation for an ideal gas. The deviation from ideality is expressed by allowing the activity coefficient to vary from 1. 

Eqn.(3.6) provides a good insight into the variation of retention with temperature in GLC. Both the activity coefficient and the vapour pressure of the solute vary with temperature in an exponential way. For the activity coefficient we can write... 

VARIATIONS OF ACTIVITY, ACTIVITY COEFFICIENTS AND EQUILIBRIUM CONSTANTS WITH TEMPERATURE AND PRESSURE... 

Equation (24.4) shows that the heat of mixing can be calculated from the temperature variation of the activity coefficients. This may be done by measuring the partial vapour pressures of the solution at different temperatures. It is generally preferable, however, to employ a direct calorimetric method, by which h is found without having to employ the activity coefficients. 

In general, the deviation between different methods and groups is expected to be quite high, due to the dependencies on various parameters of the apparatus (e.g. carrier gas flow), the use of literature data (e.g. vapour pressure) for the calculations, and variations in the equations used for the inter- and extrapolation of the activity coefficients. In addition, the above-mentioned deviation of the activity from linearity at concentrations approaching zero may play a role. 

It should be noted that equation (31.6) also gives the variation of the activity coeffideni Yn viith temperature. This follows from the definition of 7n as amole fraction of the given constituent, is constant, the variation of the activity coefficient with temperature will be exactly the same as that of the activity at constant compodtion and pressure. 

The variation of the activity coefficient with pressure is negligible. 

While the theoretical treatment we have presented for non-ideal solutions are all formally exact, they are of no practical use until we have some kind of model for the variation of the activity coefficients with the system composition, temperature, and pressure. In the next section, we present various models that can be employed to describe the activity coefficients. 

This is one case in which the separation of azeotropes may be carried out without the use of a separating agent. The method applies to minimum- or maximumboiling homogeneous azeotropes that are characterized by an appreciable variation in the azeotropic composition with pressure. The effect of pressure on the azeotropic composition may be estimated for a binary, provided the activity coefficients can be adequately represented by one of the activity coefficient equations (Section 1.3.4). 

At the start of the calculations the liquid flow rates in the column, Q and E in Equation 12.45, may be assumed equal to the inlet feed and solvent rates. The initial values for the activity coefficients may also be based on the inlet compositions and thermal conditions of the streams. The temperature and pressure variations in the extractor column are usually small, but the compositions will vary, and this may require recalculating the activity coefficients. The column calculations may be repeated with updated values of Q and E, taken as respective averages of each phase inlet and outlet stream flow rates calculated in the first trial. The activity coefficients can also be refined by recalculating them at the column top and bottom compositions for each phase. Averages of the top and bottom coefficients for each phase can be used in Equation 12.47 to calculate the new E-values. The extraction factors are then recalculated with the new values of Q, L, and E by Equation 12.45. The product component flow rates E v and Q i are Anally calculated by Equations 12.43 and 12.44. If large variations appear between the first and second trials, more trials may be considered. 

When calculating gas partial pressures it quickly becomes apparent that the principal temperature effect lies in the change of Kh rather than in the activity coefficients. In Equation 9 a constant 5C°p for the solubility equilibrium is assumed. While this is unlikely to result in appreciable errors over the temperature range 0-40°C, further from 25°C the variation of 6C°p has a significant effect on Kh, requiring a more complex form of Equation 9 (55). However, the necessary data are not always available, particularly for C°p of the aqueous solute which generally accounts for most of the change. 

In fact, the species activity coefficients also depend on temperature see Eq. 9.3-22. However, since this temperature dependence is usually small compared with the temperature variation of the vapor pressure, it is neglected here. 

It is clear from Eq. 11.1-9 that the Henry s law constant will vary with pressure, since f - and y are functions of pressure. The common method of accounting for this pressure variation is to define the Henry s law constant to be specific to a fixed pressure- Pq (frequently taken to be atmospheric pressure) and then include a Poynting correction for other pressures. Independent of whether we apply the correction to the fugacity of. the solute species in solution f T, P,x —> 0) or separately to the pure component fugacity and the infinite-dilution activity coefficient (see Eq. 9.3-20), we obtain... 

The change in activities as a response to changes in temperature and pressure of the system naturally depends to a large extent on the standard states involved. Note too, that any variation of activity with change in T or P is actually due to variation of the activity coefficient, because these effects are normally calculated for constant composition conditions. 

Non-uniformity of the column packing does affect the retention volume because of the pressure variation along the column. Measurements must therefore be made on both forward and reversed columns and the average taken, if this effect is to be eliminated. If the difference between the activity coefficients from the two sets of measurements is large the column must be repacked. 

Approximately straight lines are obtained in a plot of the logarithm of the activation coefficient vs. 1/7 or, better, vs. 1 / r, cf Fig. 2.1-25. Figure 2.1-25 shows the activity coefficient vs. the vapor pressure and the respective boihng temperature for the system water/acetic acid. Provided a small variation of the partial molar mixing enthalpy with temperature, the following simple relation is applicable to recalculate the activity coefficient at different temperatures ... 

Even the corresponding peak temperatures of the blown bitumens show very small variances in the tests in 10 bar methane and also permit the calculation of statistical means. The resulting coefficients of variation are 3.0 % maximum. This is also true for the colloid components, except for the dispersion medium of the two bitumens 85/40 (sample III) and 85/25 (sample IV). Here again a weight loss caused by distillation even occurs under pressure with the consequence of low values for the activation energy and frequency factor. Only the data of the other three samples was included in the statistics. The average values of the Arrhenius coefficients calculated in this manner and the means of the conversion aie shown in Table 4-93. 

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