Question

Use the Clausius-Clapeyron relation to derive equation 5.90 directly from Raoult's law. Be sure to explain the logic carefully.

Short answer

$T-T_{\circ }=\frac{n_{B}R{T^2}_{\circ }}{L}$IS PROVED

Step-by-step solution

Clausius-Clapeyron statement and further deduction

Let L be latent heat of fusion V be the volume T₀ be the initial boiling temperature of pure solvent and P₀ be the pressure at the pure solvent begins to boil then,

$$\begin{gathered} \frac{dP}{dT}=\frac{L}{T_0∆V} \\ \Rightarrow dT=\frac{T_0∆V}{L}dP \\ \Rightarrow T-T_0=\frac{T∆V}{L}\left(P-P_0\right) \end{gathered}$$

Use of Raoult's law in the above equation

Raoult's equation states that

$P-P_0=\frac{N_B}{N_A}{P}_0$

Substituting the value in previous equation we get

$$\begin{gathered} T-T_{\circ }=\left(\frac{T_{\circ }∆V}{L}\right)\left(\frac{N_B}{N_A}{P}_{\circ }\right) \\ T-T_{\circ }=\left(\frac{T_{\circ }{P}_{\circ }∆V}{L}\right)\left(\frac{N_B}{N_A}\right) \\ WeknowP∆V=N_{A}kT \\ T-T_{\circ }=\left(\frac{T_o{N}_{A}k{T}_o}{L}\right)\left(\frac{N_B}{N_A}\right) \\ T-T_{\circ }=\left(\frac{{T_o}^{2}k}{L}\right)\left(N_B\right) \end{gathered}$$

Hence Equation 5.90 is proved.