Define Relative Volatility and derive the expression XA = YA/α - (α - 1)YA from the ratio of partial pressure to mole fraction. (b) Compute the vapour-liquid equilibria at constant pressure of 750mmHg for a mixture of n-heptane and n-octane which may be expected to form ideal solutions. Complete the table below and plot (i) vapour composition y as a function of liquid composition x at constant pressure. (ii) temperature composition diagram

Part (a)

Step 1: Definition of relative volatility.
The relative volatility $\alpha_{xy}$ (n-heptane = x, n-octane = y) is
$\alpha_{xy} = \frac{y_x / x_x}{y_y / x_y}$
where $x_x, y_x$ are mole fractions of n-heptane in liquid and vapor.

Step 2: Derivation for ideal solution (Raoult's law).
Partial pressure: $p_x = y_x P = x_x P_x^\circ$, so
$\frac{y_x}{x_x} = \frac{P_x^\circ}{P}.$
Similarly,
$\frac{y_y}{x_y} = \frac{P_y^\circ}{P}.$
Thus,
$\alpha_{xy} = \frac{P_x^\circ / P}{P_y^\circ / P} = \frac{P_x^\circ}{P_y^\circ}.$

Part (b)
For ideal VLE at constant $P = 760$ mmHg:
Bubble point:
$x_x P_x^\circ + (1 - x_x) P_y^\circ = 760.$
Formula first:
$x_x = \frac{760 - P_y^\circ}{P_x^\circ - P_y^\circ}.$
Then
$y_x = x_x \frac{P_x^\circ}{760}, \quad \alpha_{xy} = \frac{P_x^\circ}{P_y^\circ}.$

Step 3: $T = 95^\circ$C, $P_x^\circ = 750$ mmHg, $P_y^\circ = 320$ mmHg.
$\alpha_{xy} = \frac{750}{320} = 2.34375 \approx 2.34.$
$x_x = \frac{760 - 320}{750 - 320} = \frac{440}{430} \approx 1.02 \approx 1.0.$
$y_x = 1.0 \times \frac{750}{760} \approx 0.99 \approx 1.0.$

Step 4: $T = 115^\circ$C, $P_x^\circ = 1150$ mmHg, $P_y^\circ = 650$ mmHg.
$\alpha_{xy} = \frac{1150}{650} = 1.769 \approx 1.77.$
$x_x = \frac{760 - 650}{1150 - 650} = \frac{110}{500} = 0.22.$
$y_x = 0.22 \times \frac{1150}{760} = 0.22 \times 1.513 = 0.333.$

Step 5: $T = 123^\circ$C, $P_x^\circ = 1230$ mmHg, $P_y^\circ = 750$ mmHg.
$\alpha_{xy} = \frac{1230}{750} = 1.64.$
$x_x = \frac{760 - 750}{1230 - 750} = \frac{10}{480} = 0.0208 \approx 0.021.$
$y_x = 0.0208 \times \frac{1230}{760} = 0.0208 \times 1.618 = 0.0337 \approx 0.034.$

Completed table:

\begin{array{|c|c|c|c|c|c|} \hline 95 & 750 & 320 & 1.0 & 1.0 & 2.34 \ 115 & 1150 & 650 & 0.22 & 0.33 & 1.77 \ 123 & 1230 & 750 & 0.021 & 0.034 & 1.64 \ \hline \end{array} }