where H is the heat or enthalpy. The temperature dependence of the enthalpy is related to heat capacity, Cp, by

capacity data are provided from first principles calculations or experimentally, for example, from differential scanning calorimetry, solution calorime­try, or thermogravimetric measurements, is irrelevant as long as the information is accurate and applicable. The situation is similar for phase equilibria, that is, what phases form under what conditions. The devel­oped phase diagrams provide information that can be used to fit prospective thermochemical models. This data, together with current computational methods that facilitate development of accurate representa­tions for systems reproducing observed behavior, define the CALPHAD methodology. The results ide­ally are databases for specific components that may also be used in the construction of systems with yet larger numbers of constituents. A schematic of the CALPHAD approach can be seen in Figure 1.

The CALPHAD approach assumes that the sys­tems being assessed are in equilibrium, that is, the lowest energy state under given conditions of temper­ature, pressure, and composition. The previous section describes the mathematical relationships that govern minimization of the total free energy. Traditionally, one determined the minimum free energy state by writing competing reactions related with equilibrium constants, with the phase assemblage from the reaction that yielded the most negative Gibbs free energy state being the most stable.3,4 A more generalized approach was developed in the 1950s by White et al5 using Lagrangian multipliers. Zeleznik and Gordon6 inves­tigated the major approaches to computing equilibrium states, which led to their development of a computer code for computing equilibrium at NASA. The tech­niques were further developed by van Zeggeren and Storey7,8 through the 1960s. Ultimately, Eriksson9-11 developed an approach that was generally applicable to a wide variety of systems and included solution phases that could be nonideal. This led to the widely used code SOLGASMIX,11 whose equilibrium calculational methodology remains central to many contemporary software packages. While SOLGASMIX appears to be the first, other codes for equilibrium calculations such as those noted in Section 1.17.6 had similar develop­mental histories.

H

[3]

and to entropy by

[4]

S

The temperature dependence for Cp is expressed as a polynomial from which it is possible to generate what is termed the Gibbs free energy function, which is usually expressed as

G = A = BT + CTlnT + DT2 + ET3 + F/T [5]

The Gibbs free energy function is a very convenient form to work with, particularly for free energy mini­mization software that computes an equilibrium state. That is defined as Gibbs free energy of a system that is at its minimum value, or dG = 0. A very useful value to use when working with complex systems is the chemi­cal potential, m, which is the partial derivative of the Gibbs free energy with respect to the moles or mole fraction of a constituent. Thus, at constant temperature and pressure

Bi — {pG/dnj )t ;p [6]

where n is the number of moles of the constituent.

For constant temperature and pressure

@G = ^2 Bi dni [7]

i

A system’s equilibrium state is therefore computed by minimizing the total free energy expressed as the sum of the various Gibbs free energy functions constrained by the mass balance with a resulting assemblage of phases and their amounts.