The Differential Relationships of Thermodynamic Variables for the Equilibrium of Heterogeneous Substances

For the system without adiabatic walls, rigid walls or semi-permeable walls and without chemical reactions or without other restrictions except restrictions of phase equilibrium conditions, if the number of components of the system is k and the number of phases is φ, the degree of freedom of the system at equilibrium is f=k-φ+2. Because the degree of freedom is incapable of being negative, f=k-φ+2≥0, viz.φ≤k+2. For the heterogeneous equilibrium, the number of phases is at least 2, so φ=k+2-f≥2, viz. f≤k. Hence the range of change of φ and f is 2≤φ≤k+2,0≤f≤k, respectitvely. If φ=k+2, there are no independent variables in the system at equilibrium. If φ=k+1, there is one independent variable; if the temperature is selected as the independent variable, the other dependent variables can be expressed as the function of the temperature. If φ=k, there are two independent variables; if the temperature and pressure are selected as the independent variables, the other dependent variables can be expressed as the function of the temperature and pressure. If 2≤φ≤k-1, there are more than two independent variables; if the temperature, pressure and some concentrations are selected as independent variables, the other dependent variables can be expressed as the function of the temperature, pressure and these concentrations. The differential relationships of dependent variables and independent variables are educed out according to the principle of phase equilibriums for 2≤φ≤k-1. In any phase the number of the variables is(k+1), viz. temperature T, pressure p and (k-1) mole fractions x1, x2,…, xk-1. The temperature and pressure are common variables of every phase. The number of independent variables is at best k for the heterogeneous equilibriums of k components. The temperature, pressure and (k-2) concentrations are selected as independent variables. The independent concentration variables are selected entirely from the first phase and the concentration variables of the other phases all act as dependent variables. There is at least one dependent concentration variable in the first phase.