Chapter 5, Problem 5C.12E

(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 $\text{0}\text{.25}\text{mol}$ of hexane and $\text{0}\text{.65}\text{mol}$ of cyclohexane 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.12E

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{111}\text{.33}\text{Torr}\text{.}$

The mole fraction of hexane in the vapor phase above the solution is $\text{0}\text{.3662}\text{.}$

The mole fraction of cyclohexane in the vapor phase above the solution is $\text{0}\text{.6338}\text{.}$

Explanation of Solution

Total vapor pressure of the solution:

Given,

The vapor pressure of pure hexane at ${\text{25}}^{\text{o}}\text{C}$ is $\text{151}\text{Torr}\text{.}$

The vapor pressure of pure cyclohexane at ${\text{25}}^{\text{o}}\text{C}$ is $98\text{Torr}\text{.}$

Moles of hexane is $\text{0}\text{.25}\text{mol}\text{.}$

Moles of cyclohexane is $\text{0}\text{.65}\text{mol}\text{.}$

The mole fraction of hexane is calculated as,

  Mole fraction of hexane=$\frac{\text{Moles}\text{of}\text{hexane}}{\text{Moles}\text{of}\text{hexane+Moles}\text{of}\text{cyclohexane}}$

  Mole fraction of hexane=$\frac{\text{0}\text{.25}\text{mol}}{\text{0}\text{.25}\text{mol+0}\text{.65}\text{mol}}$

  Mole fraction of hexane=$\text{0}\text{.27}$

The mole fraction of hexane is $\text{0}\text{.27}\text{.}$

The vapor pressure of hexane is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{hexane}}{\text{=χ}}_{\text{hexane}}{\text{P}}^{\text{o}}{}_{\text{hexane}}\\ {\text{P}}_{\text{hexane}}\text{=}\left(\text{0}\text{.27}\right)\left(\text{151}\text{Torr}\right)\\ {\text{P}}_{\text{hexane}}\text{=40}\text{.77}\text{Torr}\end{array}$

The vapor pressure of hexane is $\text{40}\text{.77 Torr}\text{.}$

The mole fraction of cyclohexane is calculated as,

  Mole fraction of cyclohexane=$\frac{\text{Moles}\text{of}\text{cyclohexane}}{\text{Moles}\text{of}\text{hexane+Moles}\text{of}\text{cyclohexane}}$

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.65}\text{mol}}{\text{0}\text{.25}\text{mol+0}\text{.65 mol}}$

  Mole fraction of cyclohexane=$\text{0}\text{.72}$

The mole fraction of cyclohexane is $\text{0}\text{.72}\text{.}$

The vapor pressure of cyclohexane is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{cyclohexane}}{\text{=χ}}_{\text{cyclohexane}}{\text{P}}^{\text{o}}{}_{\text{cyclohexane}}\\ {\text{P}}_{\text{cyclohexane}}\text{=}\left(\text{0}\text{.72}\right)\left(\text{98}\text{Torr}\right)\\ {\text{P}}_{\text{cyclohexane}}\text{=70}\text{.56}\text{Torr}\end{array}$

The vapor pressure of cyclohexane is $\text{70}\text{.56}\text{Torr}\text{.}$

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

The vapor pressure of hexane $\left({\text{P}}_{\text{a}}\right)$ is $\text{40}\text{.77 Torr}\text{.}$

The vapor pressure of cyclohexane $\left({\text{P}}_{\text{b}}\right)$ is $\text{70}\text{.56}\text{Torr}\text{.}$

  $\begin{array}{l}{\text{P}}_{\text{total}}{\text{=P}}_{\text{a}}{\text{+P}}_{\text{b}}\\ {\text{P}}_{\text{total}}\text{=40}\text{.77}\text{Torr+70}\text{.56}\text{Torr}\\ {\text{P}}_{\text{total}}\text{=111}\text{.33}\text{Torr}\end{array}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{111}\text{.33}\text{Torr}\text{.}$

Mole fraction of hexane in vapor phase:

The vapor pressure of hexane is $\text{40}\text{.77 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{111}\text{.33}\text{Torr}\text{.}$

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

  ${\text{χ}}_{\text{hexane}}\text{=}$$\frac{\text{40}\text{.77}\text{Torr}}{\text{111}\text{.33}\text{Torr}}$

  ${\text{χ}}_{\text{hexane}}\text{=}$$\text{0}\text{.3662}$

The mole fraction of hexane in the vapor phase above the solution is $\text{0}\text{.3662}\text{.}$

Mole fraction of cyclohexane in vapor phase:

The vapor pressure of hexane is $\text{40}\text{.77 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{111}\text{.33}\text{Torr}\text{.}$

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

  $\begin{array}{l}{\text{χ}}_{\text{cyclohexane}}{\text{=1-χ}}_{\text{hexane}}\\ {\text{χ}}_{\text{cyclohexane}}\text{=1-0}\text{.3662}\\ {\text{χ}}_{\text{cyclohexane}}\text{=0}\text{.6338}\end{array}$

The mole fraction of cyclohexane in the vapor phase above the solution is $\text{0}\text{.6338}\text{.}$

(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 $\text{10}\text{.0}\text{g}$ of hexane and $\text{10}\text{.3}\text{g}$ of cyclohexane has to be calculated.

Concept Introduction:

Refer to part (a).

Answer to Problem 5C.12E

The total vapor pressure of the solutions at ${\text{25}}^{\text{o}}\text{C}$ is $\text{124}\text{.16}\text{Torr}\text{.}$

The mole fraction of hexane in the vapor phase above the solution is $\text{0}\text{.600}\text{.}$

The mole fraction of cyclohexane in the vapor phase above the solution is $\text{0}\text{.400}\text{.}$

Explanation of Solution

Given,

The vapor pressure of pure hexane at ${\text{25}}^{\text{o}}\text{C}$ is $\text{151}\text{Torr}\text{.}$

The vapor pressure of pure cyclohexane at ${\text{25}}^{\text{o}}\text{C}$ is $98\text{Torr}\text{.}$

Grams of hexane is $\text{10}\text{.0}\text{g}\text{.}$

Grams of cyclohexane is $\text{10}\text{.0}\text{g}\text{.}$

The moles of hexane is calculated as,

  Moles of hexane=$\text{10}\text{.0}\text{g×}\frac{\text{1}\text{mol}}{\text{86}\text{.18}\text{g}}$

  Moles of hexane=$\text{0}\text{.1160}\text{mol}$

The moles of cyclohexane is calculated as,

  Moles of cyclohexane=$\text{10}\text{.0}\text{g×}\frac{\text{1}\text{mol}}{84.16\text{g}}$

  Moles of cyclohexane=$\text{0}\text{.1188}\text{mol}$

The mole fraction of hexane is calculated as,

  Mole fraction of hexane=$\frac{\text{Moles}\text{of}\text{hexane}}{\text{Moles}\text{of}\text{hexane+Moles}\text{of}\text{cyclohexane}}$

  Mole fraction of hexane=$\frac{\text{0}\text{.1160}\text{mol}}{\text{0}\text{.1160}\text{mol+0}\text{.1188}\text{mol}}$

  Mole fraction of hexane=$\text{0}\text{.4940}$

The mole fraction of hexane is $\text{0}\text{.4940}\text{.}$

The vapor pressure of hexane is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{hexane}}{\text{=χ}}_{\text{hexane}}{\text{P}}^{\text{o}}{}_{\text{hexane}}\\ {\text{P}}_{\text{hexane}}\text{=}\left(\text{0}\text{.4940}\right)\left(\text{151}\text{Torr}\right)\\ {\text{P}}_{\text{hexane}}\text{=74}\text{.594}\text{Torr}\end{array}$

The vapor pressure of hexane is $\text{74}\text{.59 Torr}\text{.}$

The mole fraction of cyclohexane is calculated as,

  Mole fraction of cyclohexane=$\frac{\text{Moles}\text{of}\text{cyclohexane}}{\text{Moles}\text{of}\text{hexane+Moles}\text{of}\text{cyclohexane}}$

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.1188}\text{mol}}{\text{0}\text{.1188 mol+0}\text{.1160 mol}}$

  Mole fraction of cyclohexane=$\text{0}\text{.5059}$

The mole fraction of cyclohexane is $\text{0}\text{.5059}\text{.}$

The vapor pressure of cyclohexane is calculated as,

  $\begin{array}{l}{\text{P}}_{\text{cyclohexane}}{\text{=χ}}_{\text{cyclohexane}}{\text{P}}^{\text{o}}{}_{\text{cyclohexane}}\\ {\text{P}}_{\text{cyclohexane}}\text{=}\left(\text{0}\text{.5059}\right)\left(\text{98}\text{Torr}\right)\\ {\text{P}}_{\text{cyclohexane}}\text{=49}\text{.57}\text{Torr}\end{array}$

The vapor pressure of cyclohexane is $\text{49}\text{.57 Torr}\text{.}$

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

The vapor pressure of hexane $\left({\text{P}}_{\text{a}}\right)$ is $\text{74}\text{.59 Torr}\text{.}$

The vapor pressure of cyclohexane $\left({\text{P}}_{\text{b}}\right)$ is $\text{49}\text{.57 Torr}\text{.}$

  $\begin{array}{l}{\text{P}}_{\text{total}}{\text{=P}}_{\text{a}}{\text{+P}}_{\text{b}}\\ {\text{P}}_{\text{total}}\text{=74}\text{.59}\text{Torr+49}\text{.57}\text{Torr}\\ {\text{P}}_{\text{total}}\text{=124}\text{.16}\text{Torr}\end{array}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{124}\text{.16}\text{Torr}\text{.}$

Mole fraction of hexane in vapor phase:

The vapor pressure of hexane is $\text{74}\text{.59 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{124}\text{.16}\text{Torr}\text{.}$

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

  ${\text{χ}}_{\text{hexane}}\text{=}$$\frac{\text{74}\text{.59}\text{Torr}}{124.16\text{Torr}}$

  ${\text{χ}}_{\text{hexane}}\text{=}$$\text{0}\text{.600}$

The mole fraction of hexane in the vapor phase above the solution is $\text{0}\text{.600}\text{.}$

Mole fraction of cyclohexane in vapor phase:

The vapor pressure of hexane is $\text{74}\text{.59 Torr}\text{.}$

The total vapor pressure of the solution at ${\text{25}}^{\text{o}}\text{C}$ is $\text{124}\text{.16}\text{Torr}\text{.}$

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

  $\begin{array}{l}{\text{χ}}_{\text{cyclohexane}}{\text{=1-χ}}_{\text{hexane}}\\ {\text{χ}}_{\text{cyclohexane}}\text{=1-0}\text{.600}\\ {\text{χ}}_{\text{cyclohexane}}\text{=0}\text{.400}\end{array}$

The mole fraction of cyclohexane in the vapor phase above the solution is $\text{0}\text{.400}\text{.}$