Summary
Highlights
A system is in equilibrium when no changes occur when isolated. This includes mechanical (constant pressure), thermal (constant temperature), phase (no phase transition), and chemical (constant composition) equilibrium. The criteria for chemical and phase equilibrium can be complex. Equilibrium criteria for reacting systems are based on the second law of thermodynamics, specifically the increase of entropy principle for adiabatic systems, where entropy reaches a maximum at equilibrium. For systems with heat transfer, a more practical approach involves developing equilibrium criteria based on the properties of the reacting system only.
For a simple compressible system at specific temperature and pressure, the condition for equilibrium is expressed by dU + P dV - T dS ≤ 0. The differential of the Gibbs function (G = H - TS) at constant temperature and pressure is dG = dU + P dV - T dS. Therefore, chemical reactions at constant temperature and pressure proceed in the direction of decreasing Gibbs function, reaching equilibrium when G attains a minimum value (dG = 0 at constant T and P). If temperature or pressure changes, the system will find a new equilibrium state corresponding to a new minimum Gibbs function.
The phase rule, developed by Josiah Willard Gibbs, determines the number of degrees of freedom (f), which are the independent variables (temperature, pressure, composition) needed to uniquely define the intensive state of a system. The relationship is given by f = C - P + 2, where C is the number of component and P is the number of phases. This rule can be derived by considering the total number of variables and the number of independent equations arising from chemical potential equalities between phases.
For a homogeneous system of a single substance, C=1, P=1, so f=2, meaning two variables (e.g., temperature and pressure) must be specified. For pure liquid water in equilibrium with its vapor, C=1, P=2, so f=1, meaning only one intensive property (temperature or pressure) can be changed freely. If solid, liquid, and vapor are in equilibrium (triple point), C=1, P=3, so f=0, meaning no intensive property can be changed. For a binary liquid mixture in equilibrium with its vapor, C=2, P=2, so f=2, allowing two intensive properties to be changed freely.
Duhem's theorem helps establish the extensive state of a system, specifying that for any closed system formed from given masses of prescribed chemical species, the equilibrium state is completely determined when any two independent variables are fixed. These two variables can be intensive or extensive. The theorem accounts for the amount or mass of various phases present, recognizing that material balance equations reduce the number of independent variables to two, allowing for the comprehensive determination of the system's state.