Question

What is the vapor pressure of the pure solvent if the vapor pressure of a solution of $10.0g$ of sucrose, ${C_{12}}{H_{22}}{O_{11}}$ , in $100.0g$ of ethanol is $55.0mmHg$ ?

Answer 1

The vapor pressure of solution and mass of ethanol and sucrose were given. From the molar mass and mass of ethanol and sucrose the number of moles can be calculated. Mole fraction of ethanol can be calculated from the moles of ethanol and moles of sucrose. Substituting the mole fraction of ethanol and vapor pressure of solution in Raoult’s law the vapor pressure of pure solvent can be obtained.
${P_{solution}} = {\chi _{solvent}} \times {P^0}_{solvent}$
${P_{solution}}$ is vapor pressure of solution
${\chi _{solvent}}$ is mole fraction of solvent which is ethanol
${P^0}_{solvent}$ is the vapor pressure of the solvent.

Complete answer:
The mass of ethanol and sucrose are given as $100g$ and $10g$ . The number of moles can be calculated from the mass and molar mass. The molar mass of sucrose is $342.3gmo{l^{ - 1}}$ and molar mass of ethanol is $46gmo{l^{ - 1}}$
${n_{sucrose}} = \dfrac{{10}}{{342.3}} = 0.0292moles$
${n_{ethanol}} = \dfrac{{100}}{{46}} = 2.171moles$
The total number of moles of sucrose and ethanol is ${n_{total}} = 0.0292 + 2.171 = 2.20moles$
The vapor pressure of solution is given as $55.0mmHg$
The mole fraction of each component can be written as the ratio of moles of a component to the total moles of all the components in mixture
Substitute the mole fraction of ethanol which is a solvent and vapor pressure of solution in the above formula to obtain the vapor pressure of solvent.
${P^0}_{solvent} = \dfrac{{55.0}}{{\dfrac{{2.171}}{{2.20}}}} = 55.7mmHg$
Thus, the vapor pressure of the solvent is $55.7mmHg$ .

Note:
In the given solution, ethanol is a solvent and sucrose is a solute. Thus, while calculating the vapor pressure of solvent the mole fraction of solvent only must be considered but not the mole fraction of solute. The vapor pressure of a pure solvent is always more than the vapor pressure of the solution.