It is desirable to have empirical expressions for activity coefficients as funct

Question

It is desirable to have empirical expressions for activity coefficients as functions of composition. The following expressions have been proposed for an isothermal binary solution where the standard states are the pure components at the same pressure and temperature as the solution:

(a) gA = AxA ; gB = BxB where A and B are constants.

(b) gA =1+ AxB ; gB =1+ BxA where A and B are constants.

(c) gA = AxnB ; gB = BxAn where A, B, and n are constants.

Criticize the acceptability of each of the above suggestions. Support your conclusions quantitatively.

g is actually supposed to be the greek gamma.

Explanation / Answer

SOLUTION:

If component B of a binary liquid mixture has low volatility, it is not practical to use its fugacity in a gas phase to evaluate its activity coefficient. If, however, component A is volatile enough for fugacity measurements over a range of liquid composition, we can instead use the Gibbs–Duhem equation for this purpose.

    Consider a binary mixture of two liquids that mix in all proportions. We assume that only component A is appreciably volatile. By measuring the fugacity of A in a gas phase equilibrated with the binary mixture, we can evaluate its activity coefficient based on a pure-liquid reference state: A =f A /(x A f A ) We wish to use the same fugacity measurements to determine the activity coefficient of the nonvolatile component, B.

    The Gibbs–Duhem equation for a binary liquid mixture in the form given by,

x A d A +x B d B =0

where d A   and d B   are the chemical potential changes accompanying a change of composition at constant T and p . Taking the differential at constant T and p of A = A +RTln( A x A ) we obtain.

d A =RTdln A +RTdlnx A =RTdln A +RTx A  dx A

For component B, we obtain in the same way

d B =RTdln B +RTx B  dx B =RTdln B RTx B  dx A

Substituting these expressions for d A   and d B   in Eq. x A d A +x B d B =0 and solving for dln B   , we obtain the following relation:

dln B =x A x B  dln A

Integration from x B =1 , where B   equals 1 and ln B   equals 0 , to composition x B   gives

(binary mixture,constant T and p)

This equation allows us to evaluate the activity coefficient of the nonvolatile component, B, at any given liquid composition from knowledge of the activity coefficient of the volatile component A as a function of composition.

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