# The composition of vapour over a binary ideal solution is determined by the composition of the liquid. If $X_A$ and $Y_A$ are the mole-fraction of A in the liquid and vapour respectively, find the value of $X_A$ for which $Y_A−X_A$ has a minimum. What is the value of the pressure at this composition?

$Y_A=\large\frac{X_AP_A^0}{P_B^0+(P_A^0-P_B^0)X_A}$
Subtracting $X_A$ from both the sides, we get
$Y_A-X_A=\large\frac{X_AP_A^0}{P_B^0+(P_A^0-P_B^0)X_A}$$-X_A Now differentiating w.r.t. X_A, we get \large\frac{d(Y_A-X_A)}{dX_A}-\frac{P_A^0}{(P_A^0-P_B^0)X_A+P_B^0}-\frac{X_AP_A^0(P_A^0-P_B^0)}{(P_B^0+(P_A^0-P_B^0)X_A)^2}$$-1$
The value of XA at which $Y_A − X_A$ has a minimum value can be obtained by putting the above derivative equal to zero. Thus we have
$\large\frac{P_A^0}{(P_A^0-P_B^0)X_A+P_B^0}-\frac{X_AP_A^0(P_A^0-P_B^0)}{(P_B^0+(P_A^0-P_B^0)X_A)^2}$$-1=0$
Solving for $X_A$, we get $X_A = \large\frac{\sqrt{P_A^0-P_B^0}-P_B^0}{P_a^0-P_B^0}$
Hence $P=\sqrt {P_A^0P_B^0}$