Small Systems in Solution

When small systems exist in a solution, they are open to C components in the solution, and temperature, pressure, and chemical potential /if become the environmental variables for the small systems. Here the following mass balances are set up  

Equation (5.21) is the condition for a closed ensemble at equilibrium. Equations (5.14) and (5.21), together with (5.17), lead to the following equation, which is the same as that for an ordinary macroscopic system  

This is the same as (5.2) for the case where f = 0. For a closed ensemble at equilibrium, C + 2 variables (T, P, and C) determine the state of the entire thermodynamic system, including its size. Because of the intensive properties are independent of the size, they are specified by C + 1 independent intensive variables T, P, and C - 1 mole fractions of components. This is easily understood by the Gibbs-Duhem equation from (5.2 ) any intensive property (T, P, ju,) is a function of C + 1 intensive variables. The number of degrees of freedom of the system is therefore identical to that of the ususal macroscopic system, that is, C + 1 for C components in one phase. 

Let us consider a macroscopic system at T and P composed of N molecules that have chemical fj, , where the system contains no small system (or micelle). The Gibbs-Duhem equation for the system becomes 

On the other hand, fi can be regarded as a function of T, P, N, and C through the following relation  

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