Читать книгу Non-equilibrium Thermodynamics of Heterogeneous Systems - Signe Kjelstrup - Страница 24

The starting point for our derivations is the Gibbs equation

(3.43)

By integrating the Gibbs equation with constant composition, temperature, pressure and displacement field, we obtain

(3.44)

The Gibbs energy can then be defined:

(3.45)

By using again the Gibbs equation, we have

(3.46)

Two equivalent definitions are obtained from these equations for the chemical potential, the partial energy change that follows when we add a particular component to a system:

(3.47)

We note that μ_j = Gj. By using Maxwell relations for Eq. (3.46), we find the following expressions for the partial molar volume, entropy and polarization:

(3.48)

This results in the following expression for a change in the chemical potential

(3.49)

A frequently used combination of terms is

(3.50)

where we used the partial molar entropy. In order to find dμj,T, we differentiate dμj at constant temperature. Equivalent expressions for unpolarized systems are

(3.51)

The partial molar volume, the partial molar entropy and the partial molar polarization for the i-phase are, respectively:

(3.52)

Furthermore, we have

(3.53)

With these partial molar quantities, we can define the partial molar internal energy and enthalpy

(3.54)

The Uj and Hj are functions of p, T, ck and E_eq.