Chapter 5, Problem 5C.11E

(a)

Interpretation Introduction

Interpretation:

The total vapor pressure of the solutions at ${\text{25}}^{\text{o}}\text{C}$ and mole fraction of each substance above the solution in the vapor phase for $\text{1}\text{.50}\text{mol}$ of benzene and $\text{0}\text{.50}\text{mol}$ of toluene has to be calculated.

Concept Introduction:

The equilibrium between a liquid and its vapor produces a characteristic vapor pressure for each substance that depends on the temperature.  The lowering of the vapor pressure is caused by a lesser ability of the solvent to evaporate, so equilibrium is reached with a smaller concentration of the solvent in the gas phase.  The vapor pressure of a solution is expressed using Raoult’s law:

  ${\text{P}}_{\text{solv}}{\text{=χ}}_{\text{solv}}{\text{P}}^{\text{o}}{}_{\text{solv}}$

The vapor pressure of the solvent $\left({\text{P}}_{\text{solv}}\right)$ above a dilute solution is equal to the mole fraction of the solvent ${\text{(χ}}_{\text{solv}}\text{)}$ times the vapor pressure of the pure solvent $\left({\text{P}}^{\text{o}}{}_{\text{solv}}\right).$

Mole fraction:  Mole fraction of a substance in a solution is the number of moles of that substance divided by the total number of moles of all substances present.  The formula is,

  $\begin{array}{l}{\text{χ}}_{\text{A}}\text{=}\frac{\text{Moles}\text{of}\text{A (in}\text{mol)}}{\text{Moles}\text{of}\text{A (in mol) +Moles}\text{of}\text{B (in mol) +Moles}\text{of}\text{C (in mol) +}\mathrm{...}}\\ \\ {\text{χ}}_{\text{A}}\text{=}\frac{\text{Moles}\text{of}\text{A (in}\text{mol)}}{\text{Total}\text{number}\text{of}\text{moles}\text{of}\text{components}\text{(in}\text{mol)}}\end{array}$

Dalton’s Law:

The total pressure of a gas mixture is the sum of the partial pressures of its component gases.

  ${\text{P}}_{\text{total}}{\text{=P}}_{\text{a}}{\text{+P}}_{\text{b}}\text{+}\mathrm{...}$

Where ${\text{P}}_{\text{a,}}{\text{P}}_{\text{b}}$ is the partial pressure of the gases in the mixture.

The partial pressure of the gas can be obtained by multiplying the total pressure of the mixture with the percent of the gases present in the mixture.

Answer to Problem 5C.11E

The total vapor pressure of the solutions at ${\text{25}}^{\text{o}}\text{C}$ is $\text{78}\text{.2}\text{Torr}\text{.}$

The mole fraction of benzene in the vapor phase above the solution is $\text{0}\text{.91}\text{.}$

The mole fraction of toluene in the vapor phase above the solution is $\text{0}\text{.09}\text{.}$

Explanation of Solution

Total vapor pressure of the solution:

Given,

The vapor pressure of pure benzene at ${\text{25}}^{\text{o}}\text{C}$ is $\text{94}\text{.6}\text{Torr}\text{.}$

The vapor pressure of pure toluene at ${\text{25}}^{\text{o}}\text{C}$ is $29.1\text{Torr}\text{.}$

Moles of benzene is $\text{1}\text{.50}\text{mol}\text{.}$

Moles of toluene is $\text{0}\text{.50}\text{mol}\text{.}$

The mole fraction of benzene is calculated as,

  Mole fraction of benzene=$\frac{\text{Moles}\text{of}\text{benzene}}{\text{Moles}\text{of}\text{benzene+Moles}\text{of}\text{toluene}}$

  Mole fraction of benzene=$\frac{\text{1}\text{.50}\text{mol}}{\text{1}\text{.50}\text{mol+0}\text{.50}\text{mol}}$

  Mole fraction of benzene=$\text{0}\text{.75}$

The mole fraction of benzene is $\text{0}\text{.75}\text{.}$

The vapor pressure of benzene is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{benzene}}{\text{=χ}}_{\text{benzene}}{\text{P}}^{\text{o}}{}_{\text{benzene}}\\ {\text{P}}_{\text{benzene}}\text{=}\left(\text{0}\text{.75}\right)\left(\text{94}\text{.6}\text{Torr}\right)\\ {\text{P}}_{\text{benzene}}\text{=70}\text{.95}\text{Torr}\end{array}$

The vapor pressure of benzene is $\text{70}\text{.95 Torr}\text{.}$

The mole fraction of toluene is calculated as,

  Mole fraction of toluene=$\frac{\text{Moles}\text{of}\text{toluene}}{\text{Moles}\text{of}\text{benzene+Moles}\text{of}\text{toluene}}$

  Mole fraction of toluene =$\frac{\text{0}\text{.50}\text{mol}}{\text{1}\text{.50}\text{mol+0}\text{.50 mol}}$

  Mole fraction of toluene =$\text{0}\text{.25}$

The mole fraction of toluene is $\text{0}\text{.25}\text{.}$

The vapor pressure of toluene is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{toluene}}{\text{=χ}}_{\text{toluene}}{\text{P}}^{\text{o}}{}_{\text{toluene}}\\ {\text{P}}_{\text{toluene}}\text{=}\left(\text{0}\text{.25}\right)\left(\text{29}\text{.1}\text{Torr}\right)\\ {\text{P}}_{\text{toluene}}\text{=7}\text{.275}\text{Torr}\end{array}$

The vapor pressure of toluene is $\text{7}\text{.275}\text{Torr}\text{.}$

The total vapor pressure is calculated using Dalton’s law.

The vapor pressure of benzene $\left({\text{P}}_{\text{a}}\right)$ is $\text{70}\text{.95 Torr}\text{.}$

The vapor pressure of toluene $\left({\text{P}}_{\text{b}}\right)$ is $\text{7}\text{.275}\text{Torr}\text{.}$

  $\begin{array}{l}{\text{P}}_{\text{total}}{\text{=P}}_{\text{a}}{\text{+P}}_{\text{b}}\\ {\text{P}}_{\text{total}}\text{=70}\text{.95}\text{Torr+7}\text{.27}\text{Torr}\\ {\text{P}}_{\text{total}}\text{=78}\text{.22}\text{Torr}\end{array}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{78}\text{.2}\text{Torr}\text{.}$

Mole fraction of benzene in vapor phase:

The vapor pressure of benzene is $\text{70}\text{.95 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{78}\text{.2}\text{Torr}\text{.}$

The mole fraction of benzene in the vapor phase is calculated as

  ${\text{χ}}_{\text{benzene}}\text{=}$$\frac{\text{70}\text{.95}\text{Torr}}{\text{78}\text{.2}\text{Torr}}$

  ${\text{χ}}_{\text{benzene}}\text{=0}\text{.907}$

The mole fraction of benzene in the vapor phase above the solution is $\text{0}\text{.91}\text{.}$

Mole fraction of toluene in vapor phase:

The vapor pressure of benzene is $\text{70}\text{.95 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{78}\text{.2}\text{Torr}\text{.}$

The mole fraction of toluene in the vapor phase is calculated as

  $\begin{array}{l}{\text{χ}}_{\text{toluene}}{\text{=1-χ}}_{\text{benzene}}\\ {\text{χ}}_{\text{toluene}}\text{=1-0}\text{.91}\\ {\text{χ}}_{\text{toluene}}\text{=0}\text{.09}\end{array}$

The mole fraction of toluene in the vapor phase above the solution is $\text{0}\text{.09}\text{.}$

(b)

Interpretation Introduction

Interpretation:

The total vapor pressure of the solutions at ${\text{25}}^{\text{o}}\text{C}$ and mole fraction of each substance above the solution in the vapor phase for $\text{15}\text{.0}\text{g}$ of benzene and $\text{64}\text{.3}\text{g}$ of toluene has to be calculated.

Concept Introduction:

Refer to part (a).

Answer to Problem 5C.11E

The total vapor pressure of the solutions at ${\text{25}}^{\text{o}}\text{C}$ is $\text{43}\text{.2}\text{Torr}\text{.}$

The mole fraction of benzene in the vapor phase above the solution is $\text{0}\text{.473}\text{.}$

The mole fraction of toluene in the vapor phase above the solution is $\text{0}\text{.527}\text{.}$

Explanation of Solution

Given,

The vapor pressure of pure benzene at ${\text{25}}^{\text{o}}\text{C}$ is $\text{94}\text{.6}\text{Torr}\text{.}$

The vapor pressure of pure toluene at ${\text{25}}^{\text{o}}\text{C}$ is $29.1\text{Torr}\text{.}$

Grams of benzene is $\text{15}\text{.0}\text{g}\text{.}$

Grams of toluene is $\text{64}\text{.3}\text{g}\text{.}$

The moles of benzene is calculated as,

  Moles of benzene=$\text{15}\text{.0}\text{g×}\frac{\text{1}\text{mol}}{\text{78}\text{.11}\text{g}}$

  Moles of benzene=$\text{0}\text{.1920}\text{mol}$

The moles of toluene is calculated as,

  Moles of toluene=$\text{64}\text{.3}\text{g×}\frac{\text{1}\text{mol}}{\text{92}\text{.14}\text{g}}$

  Moles of toluene=$\text{0}\text{.6978}\text{mol}$

The mole fraction of benzene is calculated as,

  Mole fraction of benzene=$\frac{\text{Moles}\text{of}\text{benzene}}{\text{Moles}\text{of}\text{benzene+Moles}\text{of}\text{toluene}}$

  Mole fraction of benzene=$\frac{0.1920\text{mol}}{\text{0}\text{.1920}\text{mol+0}\text{.6978}\text{mol}}$

  Mole fraction of benzene=$\text{0}\text{.2157}$

The mole fraction of benzene is $\text{0}\text{.2157}\text{.}$

The vapor pressure of benzene is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{benzene}}{\text{=χ}}_{\text{benzene}}{\text{P}}^{\text{o}}{}_{\text{benzene}}\\ {\text{P}}_{\text{benzene}}\text{=}\left(\text{0}\text{.2157}\right)\left(\text{94}\text{.6}\text{Torr}\right)\\ {\text{P}}_{\text{benzene}}\text{=20}\text{.40}\text{Torr}\end{array}$

The vapor pressure of benzene is $\text{20}\text{.40 Torr}\text{.}$

The mole fraction of toluene is calculated as,

  Mole fraction of toluene=$\frac{\text{Moles}\text{of}\text{toluene}}{\text{Moles}\text{of}\text{benzene+Moles}\text{of}\text{toluene}}$

  Mole fraction of toluene =$\frac{\text{0}\text{.6978}\text{mol}}{\text{0}\text{.1920 mol+0}\text{.6978 mol}}$

  Mole fraction of toluene =$\text{0}\text{.7842}$

The mole fraction of toluene is $\text{0}\text{.7842}\text{.}$

The vapor pressure of toluene is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{toluene}}{\text{=χ}}_{\text{toluene}}{\text{P}}^{\text{o}}{}_{\text{toluene}}\\ {\text{P}}_{\text{toluene}}\text{=}\left(\text{0}\text{.7842}\right)\left(\text{29}\text{.1}\text{Torr}\right)\\ {\text{P}}_{\text{toluene}}\text{=22}\text{.82}\text{Torr}\end{array}$

The vapor pressure of toluene is $\text{22}\text{.82 Torr}\text{.}$

The total vapor pressure is calculated using Dalton’s law.

The vapor pressure of benzene $\left({\text{P}}_{\text{a}}\right)$ is $\text{20}\text{.40 Torr}\text{.}$

The vapor pressure of toluene $\left({\text{P}}_{\text{b}}\right)$ is $\text{22}\text{.82 Torr}\text{.}$

  $\begin{array}{l}{\text{P}}_{\text{total}}{\text{=P}}_{\text{a}}{\text{+P}}_{\text{b}}\\ {\text{P}}_{\text{total}}\text{=20}\text{.40}\text{Torr+22}\text{.82}\text{Torr}\\ {\text{P}}_{\text{total}}\text{=43}\text{.22}\text{Torr}\end{array}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{43}\text{.2}\text{Torr}\text{.}$

Mole fraction of benzene in vapor phase:

The vapor pressure of benzene is $\text{20}\text{.40 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{43}\text{.2}\text{Torr}\text{.}$

The mole fraction of benzene in the vapor phase is calculated as

  ${\text{χ}}_{\text{benzene}}\text{=}$$\frac{\text{20}\text{.40}\text{Torr}}{\text{43}\text{.2}\text{Torr}}$

  ${\text{χ}}_{\text{benzene}}\text{=0}\text{.473}$

The mole fraction of benzene in the vapor phase above the solution is $\text{0}\text{.473}\text{.}$

Mole fraction of toluene in vapor phase:

The vapor pressure of benzene is $\text{20}\text{.40 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{43}\text{.2}\text{Torr}\text{.}$

The mole fraction of toluene in the vapor phase is calculated as

  $\begin{array}{l}{\text{χ}}_{\text{toluene}}{\text{=1-χ}}_{\text{benzene}}\\ {\text{χ}}_{\text{toluene}}\text{=1-0}\text{.473}\\ {\text{χ}}_{\text{toluene}}\text{=0}\text{.527}\end{array}$

The mole fraction of toluene in the vapor phase above the solution is $\text{0}\text{.527}\text{.}$