PHYSICAL CHEMISTRY

M. USANOVICH

ON THE EQUATIONS OF THE LIQUIDUS AND SOLIDUS OF IDEAL SYSTEMS

(Presented by Academician V. A. Kargin, 31 VIII 1956)

From the Clapeyron–Clausius equation, written in the form

\[ \ln \frac{P_1}{P_2}=\frac{l\,(T_1-T_2)}{RT_1T_2}, \]

and the equation for the vapor-pressure isotherm of a binary system (Raoult’s law), it follows that

\[ \ln \frac{Y_A}{X_A}=\frac{L_A(T_A-T)}{RT_AT}; \tag{1} \]

\[ \ln \frac{Y_B}{X_B}=\frac{L_B(T_B-T)}{RT_BT}, \tag{1a} \]

where \(Y_A\) is the mole fraction of component A in the solid phase; \(X_A\) is the mole fraction of A in the liquid phase in equilibrium with the solid phase at temperature \(T\); \(L_A\) and \(T_A\) are, respectively, the heat and temperature of melting of A; \(R\) is the universal gas constant; \(Y_B\), \(X_B\), \(L_B\), and \(T_B\) are the corresponding quantities for component B.

Equations (1) and (1a) were derived by Van Laar \((^1)\) for a continuous series of solid solutions. In fact, as we shall see, they are not restricted to the case of a continuous series of solid solutions, but represent, in the most general form, the equation of state of ideal systems, combining the Clapeyron–Clausius equation with Raoult’s law. However, they are not equations of the liquidus \(f(x,T)=0\) or of the solidus \(f(y,T)=0\).

The latter can be derived from (1) and (1a) when additional conditions are varied. In the case when A and B do not form solid solutions, i.e. \(Y_A=1\) and \(Y_B=1\), each of the two equations becomes the well-known Schröder logarithmic equation (the liquidus equation)

\[ -\ln X_A=\frac{L_A(T_A-T)}{RT_AT}; \tag{2} \]

\[ -\ln X_B=\frac{L_B(T_B-T)}{RT_BT}. \tag{2a} \]

Solving (2) and (2a) jointly for the eutectic point \((T=T_E)\), at which \(X_A+Y_B=1\), we obtain

\[ \exp\left(-\frac{L_A(T_A-T_E)}{RT_AT_E}\right) + \exp\left(-\frac{L_B(T_B-T_E)}{RT_BT_E}\right) =1 \tag{3} \]

From the point of intersection of the curves expressing the dependences

\[ 1-\exp\left(-\frac{L_A(T_A-T)}{RT_AT}\right) \quad\text{and}\quad \exp\left(-\frac{L_B(T_B-T)}{RT_BT}\right) \]

on \(T\), we find \(T_E\). Knowing \(T_E\), we can easily find the composition of the eutectic.

Equation (3) represents the solidus equation of an ideal system with a eutectic.

In the case where A and B form a continuous series of solid solutions without extrema*, \(X_A + X_B = 1\) and \(Y_A + Y_B = 1\). Combining these conditions with (1) and (1a), we obtain:

\[ X = \frac{ 1 - \exp \frac{L_B (T_B - T)}{R T_B T} }{ \exp \frac{L_A (T_A - T)}{R T_A T} - \exp \frac{L_B (T_B - T)}{R T_B T} }. \tag{4} \]

This equation was derived by Seltz² by a rather complicated route. It is the liquidus equation for a continuous series of solid solutions obeying Raoult’s law.

The solidus equation, likewise obtained by Seltz, is written in the form:

\[ Y_A = \frac{ 1 - \exp \frac{L_B (T_B - T)}{R T_B T} }{ 1 - \exp \left( \frac{L_B (T_B - T)}{R T_B T} - \frac{L_A (T_A - T)}{R T_A T} \right) }. \tag{5} \]

Thus, ideal systems (obeying Raoult’s law) correspond to two types of diagrams: the eutectic diagram and the Roseboom type I diagram. They are described by equations (2), (2a), (3), (4), and (5).

Institute of Chemical Sciences
Academy of Sciences of the Kazakh SSR

Received
31 VIII 1956

CITED LITERATURE

¹ V. Ya. Anosov, S. A. Pogodin, Basic Principles of Physicochemical Analysis, Moscow—Leningrad, 1947, pp. 319–320.
² H. Seltz, J. Am. Chem. Soc., 56, 307 (1934).

* Extrema contradict Raoult’s law.