Consider the constant-T-and-P dilution process of adding nA,2 - nA,1 moles of solvent A to...

Consider the constant-T-and-P dilution process of adding nA,2 - nA,1 moles of solvent A to an ideally dilute solution (solution 1) that contains ni moles of solute i and nA,1 moles of A to give an ideally dilute solution of ni moles of i and nA,2moles of A. Experimental vapor-pressure data for highly dilute solutions show that ΔG for this process is given by

ΔG = niRT (ln xi,2 - ln xi,1) + RT(nA,2 ln xA,2-nA,1 ln xA,1)             (9.66)

where xi,2,xi,1, xA,2, and xA,1 are the final and initial mole fractions of the solute and the solvent in the solution. (a) Use Eq. (9.23) to show that for this process

  (9.67)

where μi,2,μA,1, μA,2 and μA,1are the final and initial chemical potentials of the solute and the solvent in the solution. Comparison of the coefficient of ni in (9.67) with that in (9.66) gives

μi,2-μi,1 = RT (ln xi,2 - ln xi,1) const. T, P  (9.68)

The only way (9.68) can hold is if

μi = RT ln xi + f(T, P)                     (9.69)

where fi(T, P) is some function of T and P, which cancels in μi,2 - μi,1 at constant T and P. (b) Use Eqs. (9.66) to (9.68) to show that

   (9.70)

The only way (9.70) can hold is if

    (9.71)