Validating Phase Diagram of a mixture

Question

In lab I did several different runs to create a phase diagram for the mixture between biphenyl and naphthalene. My phase diagram is attached, just for visual purposes. My question is I'm supposed to validate my phase diagram by using the following formula to calculate the enthalpies of fusion for both compounds.

$$\left(\frac{\mathrm dT}{\mathrm dX}\right)_{(T=T_\mathrm m)} = \frac{RT^2}{\Delta H_\text{fusion}}$$ I understand how to do the math, however I'm trying to figure out where that formula comes from to better understand the relationship between the enthalpy of fusion and the phase diagram. Thanks!

A side question is where could I find an accepted value of the eutectic temperature for this mixture to compare my results with?

Answer 1

From the Gibbs-Helmholtz equation we usually write $$\left (\frac{\partial \ln(X)}{\partial T}\right )_p = \frac {\Delta H^0_{fus}}{RT^2}$$ where X is the molec fraction of solute. Integrating this equation gives $$ \ln(X)=\frac{\Delta H^0_{fus}}{R}\left(\frac{1}{T_{fus}} - \frac{1}{T} \right) $$ thus a plot of log mole fraction vs reciprocal temperature should produce the enthalpy. The eutectic point is the point where the melting curves meet with the solid phase, its about 0.4 to 0.5 on your figure. On the left should be the melting curve of liquid with solid biphenyl as you plot T vs naphthalene mole fraction, and on the right, liquid plus solid naphthalene. Solid biphenyl + naphthalene should be observed below $\approx 30 $ C at all mole fraction and liquid above the blue dots.