$10-14.$ Continue the calculations in Example $10-3$ to obtain $y_2$ as a function of $x_2$ by varying $x_2$ from $0$ to $1.$ Plot your result.

$10-15.$ Use your results from Problem $10-14$ to construct the pressure$-$composition diagram in Figure $10\mathrm{.}4.$

EXAMPLE $10-3$

$1-$propanol and $2-$propanol form essentially an ideal solution at all concentrations at $25C.$ Letting the subscripts $1$ and $2$ denote $1-$propanol and $2-$propanol, respectively, and given that $P_1^{**}=20\mathrm{.}9$ torr and $P_2^{**}=45\mathrm{.}2$ torr at $25C,$ calculate the total vapor pressure and the composition of the vapor phase at $x_2=0\mathrm{.}75.$

SOLUTION: We use Equation $10\mathrm{.}18$:

$)=(0\mathrm{.}75$

Let $y_j$ denote the mole fraction of each component in the vapor phase. Then, by Dalton's law of partial pressures,

$y_1=\frac{P_1}{P_{\text{t}\text{o}\text{t}\text{a}\text{l}}}=\frac{x_1{P}_1^{**}}{P_{\text{t}\text{o}\text{t}\text{a}\text{l}}}=\frac{(0\mathrm{.}25)(20\mathrm{.}9\text{t}\text{o}\text{r}\text{r})}{39\mathrm{.}1\text{t}\text{o}\text{r}\text{r}}=0\mathrm{.}13$

Similarly,

$y_2=\frac{P_2}{P_{\text{t}\text{o}\text{t}\text{a}\text{l}}}=\frac{x_2{P}_2^{**}}{P_{\text{t}\text{o}\text{t}\text{a}\text{l}}}=\frac{(0\mathrm{.}75)(45\mathrm{.}2\text{t}\text{o}\text{r}\text{r})}{39\mathrm{.}1\text{t}\text{o}\text{r}\text{r}}=0\mathrm{.}87$

Note that $y_1+y_2=1.$ Also note that the vapor is richer than the solution in the more volatile component.