Chapter 15, Problem 15.1P

Interpretation Introduction

Interpretation:

To develop result for G^E and to find the corresponding bound for an equimolar mixture containing N species.

Concept introduction:

The definition of Gibbs Energy is given as,

  $G^E=\Delta G-RT{\displaystyle \sum _i{x}_i}\ln x_i$

For equimolar solution, x_i=1/ N

Answer to Problem 15.1P

The corresponding bound for an equimolar mixture containing N species

  $G^E{}_{\max }=RT\ln N$

Explanation of Solution

The definition of Gibbs Energy is given as,

  $G^E=\Delta G-RT{\displaystyle \sum _i{x}_i}\ln x_i$

For the upper bound case, ?G = 0

  $G^E=-RT{\displaystyle \sum _i{x}_i}\ln x_i$

For an equimolar solution x_i = 1/N when N is the number of species. Therefore,

Let the absolute upper bound be G^E_max

  $G^E{}_{\max }=-RT{\displaystyle \sum _i\frac{1}{N}}\ln \frac{1}{N}$

  $G^E{}_{\max }=RT{\displaystyle \sum _i\frac{1}{N}}\ln N$

  $G^E{}_{\max }=RT\ln N$

For the special case of a binary solution, N = 2, and

  $G^E{}_{\max }=RT\ln 2$

Conclusion

The corresponding bound for an equimolar mixture containing N species is

  $G^E{}_{\max }=RT\ln N$