Consider a completely miscible two-component system whose overall composition is x, at a temperature where liquid and gas phases coexist. The composition of the gas phase at this temperature is $x_a$and the composition of the liquid phase is $x_b$. Prove the lever rule, which says that the proportion of liquid to gas is $\left(x-x_a\right)/\left(x_b-x\right)$. Interpret this rule graphically on a phase diagram.

Consider a completely miscible two-component system whose overall composition is x, at a temperature where liquid and gas phases coexist. The composition of the gas phase at this temperature is $x_a$ and the composition of the liquid phase is x_b .

Consider a miscible two-component system with liquid and gas phases, where the gas phase composition is $x_a$and the liquid phase composition is $x_b$. The composition of the gas multiplied by the number of molecules in the gas phase equals the number of molecules in the gas phase:

$N_{gas}=x_a{N}_g$

The composition of the liquid multiplied by the number of molecules in the liquid phase equals the number of molecules in the liquid phase:

$N_{liq}=x_b{N}_l$

Total number of molecules in the system is equal to:

$N_{sys}=x\left(N_g+N_l\right)\left(1\right)$

Moreover, the number of molecules in the system equals the sum of the number of molecules in the gas and the number of molecules in the liquid, resulting in:

$$\begin{gathered} N_{sys}=N_{gas}+N_{liq} \\ {N}_{sys}=x_a{N}_g+x_b{N}_l\left(2\right) \end{gathered}$$

Combining equation (1) and (2), we get

$x\left(N_g+N_l\right)=x_a{N}_g+x_b{N}_l$

We need to solve this equation for $N_l/N_g$ therefore divide both sides by N_g.

$$\begin{gathered} x\left(1+\frac{N_l}{N_g}\right)=x_a+x_b\left(\frac{N_l}{N_g}\right) \\ x+x\left(\frac{N_l}{N_g}\right)=x_a+x_b\left(\frac{N_l}{N_g}\right) \\ x-x_a=\left(x_b-x\right)\left(\frac{N_l}{N_g}\right) \\ \frac{N_l}{N_g}=\frac{x-x_a}{x_b-x} \end{gathered}$$

The graphical interpretation of the phase diagram is given below