Long-Range Order

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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.