Mixing at constant T and

It gives the enthalpy change when pure species are mixed at constant T and to form one mole (or a unit mass) of solution. Data are most commonly availa for binary systems, for which Eq. (13.4 ) solved for H becomes ... 

Thus the mixing of two components to form a perfect solution takes place at constant enthalpy. This means that if the components are mixed at constant T and p, no absorption or evolution of heat occurs. For since p is constant, the first law gives, cf. (2.15),... 

Generally mixing at constant T and P and mixing at constant T and V are quite different. However, for the ideal gas we have... 

Table 9.1-1 Properties of anTdea) Gas Mixture (Mixing at Constant T and P)...

This implies that the entropy does not change when ideal gases are mixed at constant T and v. But rather than T and v, usually we want to use T and P as the independent variables. Here P represents the pressure when the mixture has temperature T and molar volume v. Therefore, (4.1.29) can be written as... 

When pure components are mixed at constant T and P, an energy balance shows that measures the heat effect. In the Lewis-Randall standard state, ideal solutions have no heat effect on mixing, = h = 0 but for real mixtures, the heat effect may be exothermic q = h < 0) or endothermic q = h > 0). 

When pure ideal gases mix at constant T and p to form an ideal gas mixture, the molar entropy change A5 (mix) = -R yt In (Eq. 11.1.9) is positive. 

We can now understand why the entropy change is positive when ideal gases mix at constant T and p Each substance occupies a greater volume in the final state than initially. Exactly the same entropy increase would result if the volume of each of the pure ideal gases were increased isothermally without mixing. 

The behavior of binary hqnid solutions is clearly displayed by plots of M, AM, and In % vs. X at constant T and P. The vohime change of mixing (or excess vohime) is the most easily measured of these quantities and is normally small. However, as illustrated by Fig. 4-1, it is subject to individiiahstic behavior, being sensitive to the effects of molecular size and shape and to differences in the nature and magnitude of intermoleciilar forces. 

A solution is a single-phase mixture of more than one compound, and the driving force for its spontaneous formation from the pure compounds at constant T and p is the negative Gibbs free energy change of the mixing process, —AG, as... 

Mixing two gases at constant T and p increases the disorder, and hence, the entropy. 

The standard state for the chemical potential of each component in the solution is thus defined as the pure component at the same temperature and pressure and in the same state of aggregation as the solution. We say that the chemical potential of the fcth component in a solution at the temperature T, pressure P, and composition x referred to the pure fcth component at the same T and P and in the same state of aggregation is equal to RT In xk. This difference is also known as the change of the chemical potential on mixing at constant temperature and pressure, so... 

The quantity ASM[T, E, x] is referred to as the change of entropy on mixing at constant temperature and pressures or, more briefly, the entropy of mixing. Similarly, we obtain... 

The change of enthalpy on mixing, AHM[T, P, x], at constant temperature and pressure is seen to be zero for an ideal solution. The change of the heat capacity on mixing at constant temperature and pressure is also zero for an ideal solution, as are all higher derivatives of AHM with respect to both the temperature and pressure at constant composition. Differentiation of Equations (8.57), (8.59), and (8.60) with respect to the pressure yields... 

Thus, the change of the volume on mixing at constant temperature and pressure, AEM[T, P, x], is also zero, as are all higher derivatives of AFM with respect to both the temperature and pressure at constant composition. 

For an ideal mixture, AinixLTldeal and AmixKideal are zero for mixing at constant P and T, and so AmixGideal is given by... 

Notice that AmixHideal is also zero and thus, with Equation 2.3 in mind, ideal mixing at constant P and T will take place without heat effects. 

Suppose that nA moles of pure A and nB moles of pure B are mixed to form a binary solution at constant T and P. The total free energy change, G, associated with the mixing may be obtained in the following manner ... 

Because of the simplicity of these relationships, we sometimes say that the natural variables of U are S and V, of // arc S and P, of A are V and T, of G are P and T, and of S are U and V. It is noteworthy that the natural variables of U are both extensive variables and those of G are both intensive variables the natural variables of H and A are mixed—one extensive and one intensive variable for each. Because Eqs. (20)-(24) only hold for systems at material equilibrium, they will become our criteria for material equilibrium under each set of conditions. For example, at constant T and P, for a system to be at material equilibrium for a process, dG must equal zero for the (infinitesimal) process. 

It remains for us to outline methods for calculating IJ and A/ for the different types of processes we have discussed. The methods for the first four (change in P at constant T. change in T at constant P, change in phase at constant T and P, and mixing or dissolving at constant T and P) are outlined in Sections 8.2-8.5 of this chapter, and methods for chemical reactions at constant T and P are given in Chapter 9. 

Thus hf is equal to the heat absorbed (per mole of 1) when a small quantity 8ui of substance 1 is dissolved in the solution at constant T and p. For this reason it is often called the heat of solution of component 1, differential heat of mixing or partial molar heat of mixing." The partial molar heats may be determined readily from the heat of mixing h by the Bakhuis-Rooseboom method described in 3 of chap. I. 

If mixing is carried out at constant T and V, the analogous criterion for miscilibity is that the change in Helmholtz free energy, AA ... 

Repeat the derivations of the previous problem for a mixing proce.ss in which both pure fluids, initially at a temperature T and pressure P, are mixed at constant temperature and the pressure then adjusted so that the final volume of the mixture is equal to the sum of the initial volumes of the pure components (i.e., there is no volume change on mixing). 

Equation (14-50) is the formula for electrostriction at constant T and p. In general, 8eldp)j is positive so that a system will contract when put into an electric field at constant T and p. It is interesting to note that this theory, when applied on the molecular scale in the consideration of electrostriction of solvent in the field of an ion, qualitatively accounts for the decrease in volume resulting when some solutions are mixed. 

Figure 8.15 Change of Gibbs energy on mixing for class II stability behavior at constant T and P. Both the mixture equation of state and the fugadty equation bifurcate, producing distinct branches in and a vapor-Uquid phase separation. However, no branch spans all x. Filled circles are phases in equilibrium long dashes metastable short dashes unstable. Curves computed using Redlich-Kwong equation.

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