# Long-Range Order

• N. A. Gokcen

## Abstract

Chapter 2 dealt with the phase equilibria in alloys involving only the first-order phase transitions. The Gibbs energy of a given multicomponent system is a function of its variables of state P, T, n 1 , n 2,..., n c , i.e., G = G (P, T, n 1, n 2,..., n c ). This function is continuous for the first-order phase transitions but its derivative, with respect to one of its variables, becomes discontinuous upon a first-order transition. The variable of the greatest importance is the temperature; therefore, we limit our discussion to the derivatives of G with respect to T. A first-order transition is accompanied with a discontinuity in the first derivatives of G; thus,
$$\frac{{\partial G}}{{\partial T}} =- S;\quad or\;\frac{{\partial (G/T)}}{{\partial (1/T)}} = H$$
(5.1)
would show a discontinuity in the entropy or enthalpy when these properties are measured from a reference temperature such as T = 0, or often more conveniently, T = 298.15 K. Condensation and freezing of pure components provide some of the most elementary examples of first-order transitions. At a second-order transition, the second derivatives of G exhibit a discontinuity; i.e.,
$$\frac{{\partial ^2G}}{{\partial T}} = - \frac{{{C_P}}}{T};\quad or\;\frac{{{\partial ^2}(G/T)}}{{\partial (1/T)\partial T}} = {C_P}$$
(5.2)
In summary, S, or H, is discontinuous for the first-order phase transitions and C p is discontinuous for the second-order phase transitions.

## Keywords

Gibbs Energy Critical Temperature Regular Solution Zeroth Approximation Equiatomic Composition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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