Calculation of Freezing-Point Depression and the Eutectic Point Compute the freezing-point temperature versus composition curve for an ethyl benzene–toluene mixture. The physical properties for this system are given in the table.

Question

Calculation of Freezing-Point Depression and the Eutectic Point

Compute the freezing-point temperature versus composition curve for an ethyl benzene–toluene mixture. The physical properties for this system are given in the table.

	Toluene	Ethyl Benzene
[latex]\delta( cal / cc )^{1 / 2} ext { at } 25^{\circ} C[/latex]	8.9	8.8
[latex]\underline{V}^{ L }( cc / mol ) ext { at } 25^{\circ} C[/latex]	107	123
Normal melting point (K)	178.16	178.2
Heat of fusion (J/mol)	6610.7	9070.9
[latex]C_{ P }^{ S }( J / mol K )[/latex]	87.0	105.9
[latex]C_{ P }^{ L }( J / mol K )[/latex]	135.6	157.4

Accepted Answer

The solubility parameters for toluene and ethyl benzene are nearly equal, so that the liquid mixture can be considered to be ideal [latex]\left(\gamma_{1}, \gamma_{2}=1\right)[/latex]. Since we do not know a priori whether pure toluene or pure ethyl benzene will freeze out as the solid component from a given mixture, the calculational procedure we will follow is to first assume that toluene appears as the solid component and compute the freezing-point temperature [latex]T_{f}[/latex] of all toluene–ethyl benzene mixtures.

The calculations will then be repeated assuming ethyl benzene appears as the solid phase. The freezing point of a given mixture, and the solid that appears, will then be determined by noting which of the two calculations leads to the higher freezing point, as it is that solid that will freeze out first.

If we assume that toluene freezes out, [latex]T_{f}[/latex] is found from the solution of

[latex]\ln x_{1}=-\frac{6610.7}{8.314}\left(\frac{178.16-T_{f}}{178.16 T_{f}}\right)-\frac{48.6}{8.314}\left(1-\frac{178.16}{T_{f}}+\ln \frac{178.16}{T_{f}}\right)[/latex]

where x is the mole fraction of toluene. Since [latex]\Delta C_{ P }[/latex] is so large, none of the terms in this equation can be neglected. The results are given in the following table. Similarly, if we assume that the solid precipitate is ethyl benzene, the mixture freezing point is computed from

[latex]\ln x_{1}=-\frac{9070.9}{8.314}\left(\frac{178.2-T_{f}}{178.2 T_{f}}\right)-\frac{51.5}{8.314}\left(1-\frac{178.2}{T_{f}}+\ln \frac{178.2}{T_{f}}\right)[/latex]

where x is now the mole fraction of ethyl benzene. The results of this calculation are also given in the following table. Both sets of freezing points are plotted in Fig. 12.3-1.

	[latex]T_{f}( K )[/latex]
[latex]x_{ T }[/latex]	Toluene as the Solid	Ethyl Benzene as the Solid
1.0	178.16
0.9	174.0	122.2
0.8	169.4	137.0
0.7	164.3	146.4
0.6	158.6	153.4
0.5	151.9	159.1
0.4	144.0	163.9
0.3	134.2	168.1
0.2	121.0	171.8
0.1	99.8	175.1
0		178.2

Question 12.3.2: Calculation of Freezing-Point Depression and the Eutectic Po...

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