Chapter 12, Problem 12.83QE

(a)

Interpretation Introduction

Interpretation:

The mole fraction of heptane and hexane 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,

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

Explanation of Solution

Given,

Weight of hexane        =$\text{15}\text{.0g}$

Weight of heptane        =$\text{20}\text{.0g}$

Vapor pressure of hexane    =$\text{278}\text{torr}$

Vapor pressure of heptane    =$\text{92}\text{.3}\text{torr}$

The moles of heptane and hexane are calculated from their molar masses respectively.

  Moles of hexane=$\text{15g×}\frac{\text{1}\text{mol}}{\text{86}\text{.18g}}\text{=0}\text{.1740mole}$

  Moles of heptane=$\text{20g×}\frac{\text{1}\text{mol}}{\text{100}\text{.21g}}\text{=0}\text{.1995}\text{mole}$

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

  Mole fraction of hexane=$\frac{\text{0}\text{.1740}\text{mole}}{\text{0}\text{.1740}\text{mole+0}\text{.1995}\text{mole}}$

  Mole fraction of hexane=$\frac{\text{0}\text{.1740}\text{mole}}{\text{0}\text{.3735}\text{mole}}$

  Mole fraction of hexane=$\text{0}\text{.4658}$

The mole fraction of heptane is calculated from the mole fraction of hexane.

  Mole fraction of heptane=$\text{1-0}\text{.4658}$

  Mole fraction of heptane=$\text{0}\text{.5342}$

The mole fraction of hexane is $\text{0}\text{.4658}\text{.}$

The mole fraction of heptane is $\text{0}\text{.5342}\text{.}$

(b)

Interpretation Introduction

Interpretation:

The vapor pressures of hexane and heptane 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 $\left({\text{P}}^{\text{o}}{}_{\text{solv}}\right).$

Explanation of Solution

Given,

Weight of hexane        =$\text{15}\text{.0g}$

Weight of heptane        =$\text{20}\text{.0g}$

Vapor pressure of hexane    =$\text{278}\text{torr}$

Vapor pressure of heptane    =$\text{92}\text{.3}\text{torr}$

The moles of hexane are calculated from its molar mass.

  Moles of hexane=$\text{15g×}\frac{\text{1}\text{mol}}{\text{86}\text{.18g}}\text{=0}\text{.1740mole}$

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

  Mole fraction of hexane=$\frac{\text{0}\text{.1740}\text{mole}}{\text{0}\text{.1740}\text{mole+0}\text{.1995}\text{mole}}$

  Mole fraction of hexane=$\frac{\text{0}\text{.1740}\text{mole}}{\text{0}\text{.3735}\text{mole}}$

  Mole fraction of hexane=$\text{0}\text{.4658}$

The vapor pressure of hexane is given as,

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

  Vapor pressure of hexane=$\text{0}\text{.4658×278}\text{torr}$

  Vapor pressure of hexane=$\text{129}\text{.5}\text{torr}$

The moles of heptane are calculated from its molar mass.

  Moles of heptane=$\text{20g×}\frac{\text{1}\text{mol}}{\text{100}\text{.21g}}\text{=0}\text{.1995}\text{mole}$

The mole fraction of heptane is calculated from the mole fraction of hexane.

  Mole fraction of heptane=$\text{1-0}\text{.4658}$

  Mole fraction of heptane=$\text{0}\text{.5342}$

The vapor pressure of heptane is given as,

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

  Vapor pressure of heptane=$\text{0}\text{.5342×92}\text{.3}\text{torr}$

  Vapor pressure of heptane=$\text{49}\text{.3066}\text{torr}$

The vapor pressure of hexane is $\text{129}\text{.5}\text{torr}\text{.}$

The vapor pressure of heptane is $\text{49}\text{.3066 torr}\text{.}$

(c)

Interpretation Introduction

Interpretation:

The mole fraction of heptane and hexane in vapor phase has to be given.

Concept Introduction:

Refer to part (a) and (b).

Explanation of Solution

The vapor pressure of hexane $\text{129}\text{.5}\text{torr}\text{.}$

The vapor pressure of heptane is $\text{49}\text{.30 torr}\text{.}$

The total vapor pressure is $\text{178}\text{.80}\text{torr}\text{.}$

The mole fraction of hexane and hexane are calculated as,

  Mole fraction=$\frac{\text{Vapor}\text{pressure}\text{of}\text{heptane/hexane}\text{(in}\text{torr)}}{\text{Total}\text{vapor}\text{pressure}\text{(in}\text{torr)}}$

  Mole fraction of hexane=$\frac{\text{129}\text{.5}\text{torr}}{\text{178}\text{.80}\text{torr}}=0.7242$

  Mole fraction of heptane=$\frac{49.30\text{torr}}{\text{178}\text{.80}\text{torr}}=0.2757$

The mole fraction of hexane in vapor pressure is $\text{0}\text{.7242}\text{.}$

The mole fraction of heptane in vapor pressure is $\text{0}\text{.2757}\text{.}$