Chapter 6, Problem 6.95P

Derive Equation 6.5−4 for the boiling-point elevation of a dilute solution of a nonvolatile solute with mole fraction x in a solvent that has a pure-component vapor pressure $p_{\text{s}}^{*}\left(T\right)$. To do so, suppose that when the pressure is P₀, the pure solvent boils at temperature T_b0[so that $p_{\text{s}}^{*}\left(T_{\text{b}0}\right)$] and when the solvent is in the solution ii boils at $T_{\text{bs}}>T_{\text{b}0}$. Further suppose that at temperature T_b0the effective vapor pressure of the solvent $P_{\text{s}}={\left(p_{\text{s}}^{*}\right)}_{\text{e}}\left(T_{\text{b}0}\right)<P_0$. (See diagram.)

The procedure is as follows.

(a) Write the Clausius−Clapeyron equation (Equation 6.1-3) for P (the effective solvent vapor pressure at T_b0) and then for P₀(the effective solvent vapor pressure at T_bs), assuming that at the low solute concentrations in question the heat of vaporization is the same at both temperatures.

Subtract the two equations. Simplify the equation algebraically, assuming that T_b0and T_bsare close enough together to say that $T_{\text{b}0}{T}_{\text{bs}}\approx T_{\text{b0}}^2$.

(b) Substitute the Raoult’s law expression (Equation 6.5−2) for $P_{\text{s}}={\left(p_{\text{s}}^{*}\right)}_{\text{e}}\left(T_{\text{b}0}\right)$. Observe that if $x\ll 1$(which it is for highly dilute solutions), then $\text{In}\left(1-x\right)\approx -x$. The desired result follows.