Derive an equation for a solution that shows the relation between total pressure and the mole fraction of a volatile solute and a volatile solvent,and explain it by plotting a graph.

(N/A) Consider a binary solution containing volatile components $1$ and $2$. Let their mole fractions be $X_{1}$ and $X_{2}$,and their partial vapour pressures be $P_{1}$ and $P_{2}$,respectively.
According to Raoult's law,for a solution of volatile liquids,the partial vapour pressure of each component is directly proportional to its mole fraction in the solution.
For component $1$: $P_{1} = P_{1}^{0} X_{1}$,where $P_{1}^{0}$ is the vapour pressure of pure component $1$.
For component $2$: $P_{2} = P_{2}^{0} X_{2}$,where $P_{2}^{0}$ is the vapour pressure of pure component $2$.
According to Dalton's law of partial pressures,the total pressure $P_{\text{total}}$ is the sum of the partial pressures:
$P_{\text{total}} = P_{1} + P_{2} = P_{1}^{0} X_{1} + P_{2}^{0} X_{2}$.
Since $X_{1} + X_{2} = 1$,we have $X_{1} = 1 - X_{2}$.
Substituting this:
$P_{\text{total}} = P_{1}^{0} (1 - X_{2}) + P_{2}^{0} X_{2} = P_{1}^{0} + (P_{2}^{0} - P_{1}^{0}) X_{2}$.
Conclusions:
$(i)$ $P_{\text{total}}$ is linearly related to the mole fraction of any one component.
$(ii)$ The graph of $P_{\text{total}}$ versus $X_{2}$ is a straight line.
$(iii)$ $P_{\text{total}}$ varies linearly with $X_{2}$ between $P_{1}^{0}$ and $P_{2}^{0}$.