Chapter 12, Problem 12.86QE

(a)

Interpretation Introduction

Interpretation:

The mole fraction of 2-methylpentane and cyclohexane in liquid phase has to be given.

Concept Introduction:

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,

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

Answer to Problem 12.86QE

The mole fraction of cyclohexane is $\text{0}\text{.36781}\text{.}$

The mole fraction of 2-methylpentane is $\text{0}\text{.63219}\text{.}$

Explanation of Solution

Given,

Weight of cyclohexane        =$\text{25}\text{.0g}$

Weight of 2-methylpentane        =$\text{44g}$

Vapor pressure of cyclohexane    =$\text{150}\text{torr}$

Vapor pressure of 2-methylpentane    =$\text{313 torr}$

The moles of 2-methylpentane and cyclohexane are calculated from their molar masses.

  Moles of cyclohexane=$\text{25g×}\frac{\text{1}\text{mol}}{\text{84}\text{.16g}}\text{=0}\text{.2970mole}$

  Moles of 2-methylpentane=$\text{44g×}\frac{\text{1}\text{mol}}{\text{86}\text{.18g}}\text{=0}\text{.5105}\text{mole}$

The mole fraction of cyclohexane in the liquid phase is calculated as,

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.2970}\text{mole}}{\text{0}\text{.2970}\text{mole+0}\text{.5105}\text{mole}}$

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.2970}\text{mole}}{\text{0}\text{.8075}\text{mole}}$

  Mole fraction of cyclohexane=$\text{0}\text{.36781}$

The mole fraction of 2-methylpentane is calculated from the mole fraction of cyclohexane.

  Mole fraction of 2-methylpentane=$\text{1-0}\text{.36781}$

  Mole fraction of 2-methylpentane=$\text{0}\text{.63219}$

The mole fraction of cyclohexane is $\text{0}\text{.36781}\text{.}$

The mole fraction of 2-methylpentane is $\text{0}\text{.63219}\text{.}$

(b)

Interpretation Introduction

Interpretation:

The vapor pressures of cyclohexane and 2-methylpentane above the solution have to be given.

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 $\left({\text{P}}^{\text{o}}{}_{\text{solv}}\right).$

Answer to Problem 12.86QE

The vapor pressure of cyclohexane is $\text{55}\text{.17 torr}\text{.}$

The vapor pressure of 2-methylpentane is $\text{197}\text{.87}\text{torr}\text{.}$

Explanation of Solution

Given,

Weight of cyclohexane        =$\text{25}\text{.0g}$

Weight of 2-methylpentane        =$\text{44g}$

Vapor pressure of cyclohexane    =$\text{150}\text{torr}$

Vapor pressure of 2-methylpentane    =$\text{313 torr}$

The moles of cyclohexane are calculated from its molar mass.

  Moles of cyclohexane=$\text{25g×}\frac{\text{1}\text{mol}}{\text{84}\text{.16g}}\text{=0}\text{.2970mole}$

The mole fraction of cyclohexane in the liquid phase is calculated as,

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.2970}\text{mole}}{\text{0}\text{.2970}\text{mole+0}\text{.5105}\text{mole}}$

  Mole fraction of cyclohexane=$\frac{\text{0}\text{.2970}\text{mole}}{\text{0}\text{.8075}\text{mole}}$

  Mole fraction of cyclohexane=$\text{0}\text{.36781}$

The vapor pressure of cyclohexane is given as,

  Vapor pressure of cyclohexane=${\text{χ}}_{\text{cyclohexane}}{\text{P}}^{\text{o}}{}_{\text{cyclohexane}}$

  Vapor pressure of cyclohexane=$\text{0}\text{.36781×150torr}$

  Vapor pressure of cyclohexane=$\text{55}\text{.17}\text{torr}$

The moles of 2-methylpentane are calculated from its molar mass.

  Moles of 2-methylpentane=$\text{44g×}\frac{\text{1}\text{mol}}{\text{86}\text{.18g}}\text{=0}\text{.5105}\text{mole}$

The mole fraction of 2-methylpentane is calculated from the mole fraction of cyclohexane.

  Mole fraction of 2-methylpentane=$\text{1-0}\text{.36781}$

  Mole fraction of 2-methylpentane=$\text{0}\text{.63219}$

The vapor pressure of 2-methylpentane is given as,

  Vapor pressure of 2-methylpentane=${\text{χ}}_{\text{2-methylpentane}}{\text{P}}^{\text{o}}{}_{\text{2-methylpentane}}$

  Vapor pressure of 2-methylpentane=$\text{0}\text{.63219×313}\text{torr}$

  Vapor pressure of 2-methylpentane=$\text{197}\text{.87 torr}$

The vapor pressure of cyclohexane is $\text{55}\text{.17 torr}\text{.}$

The vapor pressure of 2-methylpentane is $\text{197}\text{.87}\text{torr}\text{.}$

(c)

Interpretation Introduction

Interpretation:

The mole fraction of 2-methylpentane and cyclohexane in vapor phase has to be given.

Concept Introduction:

Refer to part (a) and (b).

Answer to Problem 12.86QE

The mole fraction of cyclohexane in vapor phase is $\text{0}\text{.2181}\text{.}$

The mole fraction of 2-methylpentane in vapor phase is $\text{0}\text{.7819}\text{.}$

Explanation of Solution

The vapor pressure of cyclohexane is $\text{55}\text{.17 torr}\text{.}$

The vapor pressure of 2-methylpentane is $\text{197}\text{.87}\text{torr}\text{.}$

The total vapor pressure is $\text{253}\text{.04 torr}\text{.}$

The mole fraction of cyclohexane and 2-methylpentane are calculated as,

  Mole fraction=$\frac{\text{Vapor}\text{pressure}\text{of}\text{cyclohexane/2-methylpentane}\text{(in}\text{torr)}}{\text{Total}\text{vapor}\text{pressure}\text{(in}\text{torr)}}$

  Mole fraction of cyclohexane=$\frac{55.17\text{torr}}{253.04\text{torr}}=0.2181$

  Mole fraction of 2-methylpentane=$\frac{197.87\text{torr}}{253.04\text{torr}}=0.7819$

The mole fraction of cyclohexane in vapor pressure is $\text{0}\text{.2181}\text{.}$

The mole fraction of 2-methylpentane in vapor pressure is $\text{0}\text{.7819}\text{.}$