# Difference between revisions of "The Phase Rule"

## Introduction

The Phase Rule was an incidental offshoot of the classic investigations in which J. Willard Gibbs laid out the foundations of chemical thermodynamics between 1875 and 1878. A reprint exists (Gibbs, 1961) together with two helpful commentaries (Donnan and Haas, 1936; Seeger, 1974). Connolly (1990) gives a thorough discussion of multivariable phase diagrams and includes a useful account of thermodynamic nomenclature. Regarding phase equilibria in aqueous systems under earth-surface conditions (Pankow, 1999) should be consulted.

Gibbs' Phase rule provides a theoretical basis for considering problems concerning the mineralogy of soils. In a deductive way, it serves as a check and a balance to the science of pedology, which like all the earth sciences is fundamentally inductive in nature. Since the Phase Rule has its derivation in classical thermodynamics it deals with systems at equilibrium. As soils are manifestly in a state of disequilibrium, application of the Phase Rule (or thermodynamics in general) needs some initial justification. First, the equilibrium state represents the state a system would achieve given the time and energy to get there. As such the equilibrium model lays fundamental constraints on any hypothesis of mineral genesis in soils. It indicates the direction of change, and places an end bracket on all states the system might pass through. In some cases, an equilibrium mineralogy may be approached closely, for example in microscopic systems ( Chesworth and Dejou, 1980) and as a result of long term weathering in the humid tropics. The model it provides is a rigorous one and although the natural, disequilibrium state can be expected to differ from the model, the differences are themselves instructive.

As a guide to what mineralogical equilibria are likely to be of interest in the present context, a brief review of the geochemistry of the zone of soil formation will be necessary. This follows a basic introduction to the terminology of thermodynamics and of phase equilibria.·

## Phase Rule

The phase rule is a method to count the number of degrees of freedom (how many independent variables are sufficient to specify a multi-component, multi-phase system.

The idea is from Gibbs and the derivation of the equation is:

 Consider the general case: C components and P phases. At equilibrium all pressures, temperatures, and each chemical potential is constant: \begin{align}  & p_{a}=p_{b}=\ldots =p_{P} \\ & T_{a}=T_{b}=\ldots =T_{P} \\ & \mu _{ia}=\mu _{ib}=\ldots =\mu _{iP};\text{ }i=1,2,\ldots C \\  \end{align} The number of unknowns in each phase is (C+1): p, T, and (C-1) mole fractions times the number of phases P or $=\left( C+1 \right)P$] The number of equations: $=\left( P-1 \right)\left( C+2 \right)$ Therefore the degrees of freedom are: $F=\left( C+1 \right)P-\left( P-1 \right)\left( C+2 \right)=C+2-P$ The phase rule: the number of degrees of freedom is: $F=C+2-P$