Chapter 10, Problem 1KSP

Determine the mole fraction of helium in a gaseous mixture consisting of 0.524 g He, $\text{0}\text{.275 g Ar}\text{. and 2}\text{.05 g}$ $\text{C}{\text{H}}_{\text{4}}\text{.}$

(a)

0.0069

(b)

0.0259

(c)

0.481

(d)

0.493

(e)

0.131

Interpretation Introduction

Interpretation:

The mole fraction of helium in a gaseous mixture is to be calculated.

Concept introduction:

The mole fraction of an individual gas, for a combination of gases, is the ratio of the moles of the individual gas to the total number of moles of the mixture.

${\chi }_i=\frac{n_i}{n_{\text{total}}}$

Here, ${\chi }_i$ is the mole fraction, $n_i$ is the mole fraction of individual gas, and $n_{\text{total}}$ is the total number of moles.

Also, the mole fraction of an individual gas, for a combination of gases, can be calculated from the ratio of the partial pressure of the individual gas to the total pressure of the combination.

${\chi }_i=\frac{P_i}{P_{\text{total}}}$

Here, ${\chi }_i$ is the mole fraction, $P_i$ is the partial pressure of individual gas, and $P_{\text{total}}$ is the total pressure.

The number of moles can be calculated as

$n=\frac{m}{M}$

Here, m represents the mass of the molecule, M represents the molar mass of the molecule, and $n$ is the number of moles.

Answer to Problem 1KSP

Solution: Option (d).

Explanation of Solution

Given information:

Mass of helium $m_{\text{He}}=0.524\text{ g}$

Mass of argon $m_{\text{Ar}}=0.275\text{ g}$

Mass of methane $m_{{\text{CH}}_4}=2.05\text{ g}$

Reason for the correct option:

The molar mass of helium is $4\text{ g}/\text{mol}$.

Calculate the number of moles of helium as follows:

$n_{\text{He}}=\frac{m_{\text{He}}}{M_{\text{He}}}$

Substitute $0.524\text{ g}$ for $n_{\text{He}}$ and $4\text{ g}/\text{mol}$ for $M_{\text{He}}$ in the above equation

$\begin{array}{c}{n}_{\text{He}}=\frac{0.524\text{ g}}{4\text{ g}/\text{mol}}\\ =0.131\text{ mol}\end{array}$

The molar mass of argon is $\text{39}\text{.9 g}/\text{mol}$.

Calculate the number of moles of argon as follows:

$n_{\text{Ar}}=\frac{m_{\text{Ar}}}{M_{\text{Ar}}}$

Substitute $0.275\text{ g}$ for $n_{\text{Ar}}$ and $\text{39}\text{.9 g}/\text{mol}$ for $M_{\text{Ar}}$ in the above equation

$\begin{array}{c}{n}_{\text{Ar}}=\frac{0.275\text{ g}}{39.9\text{ g}/\text{mol}}\\ =0.0069\text{ mol}\end{array}$

The molar mass of methane is $\text{16 g}/\text{mol}$.

Calculate the number of moles of methane as follows:

$n_{{\text{CH}}_4}=\frac{m_{{\text{CH}}_4}}{M_{{\text{CH}}_4}}$

Substitute $\text{2}\text{.05 g}$ for $n_{{\text{CH}}_4}$ and $\text{16 g}/\text{mol}$ for $M_{{\text{CH}}_4}$ in the above equation

$\begin{array}{c}{n}_{{\text{CH}}_4}=\frac{2.05\text{ g}}{16\text{ g}/\text{mol}}\\ =0.128\text{ mol}\end{array}$

From Dalton’s law for the combination of gases, the total number of moles can be calculated as

$\begin{array}{c}{n}_{\text{total}}={\displaystyle \sum n_i}\\ =n_{\text{He}}+n_{\text{Ar}}+n_{{\text{CH}}_4}\end{array}$

Substitute $0.0069\text{ mol}$ for $n_{\text{Ar}}$, $0.128\text{ mol}$ for $n_{{\text{CH}}_4}$, and $0.131\text{ mol}$ for $n_{\text{He}}$ in the above equation

$\begin{array}{c}{n}_{\text{total}}=0.131\text{ mol}+0.0069\text{ mol}+0.128\text{ mol}\\ =0.266\text{ mol}\end{array}$

Calculate the mole fraction for $\text{He}$ gas as follows:

${\chi }_{\text{He}}=\frac{n_{\text{He}}}{n_{\text{total}}}$

Substitute $0.266\text{ mol}$ for $n_{\text{total}}$ and $0.131\text{ mol}$ for $n_{\text{He}}$ in the above equation

$\begin{array}{c}{\chi }_{\text{He}}=\frac{0.131\text{ mol mol}}{0.266\text{ mol mol}}\\ =0.493\end{array}$

Thus, for $\text{He}$, the mole fraction is $0.493$.

Hence, option (d) is correct.

Reason for the incorrect options:

Option (a) is incorrect because the calculated value of mole fraction of heliumfor the given gaseous mixturedoes not match the given value in option (a).

Option (b) is incorrect because the calculated value of mole fraction of helium for the given gaseous mixturedoes not match the given value in option (b).

Option (c) is incorrect because the calculated value of mole fraction of helium for the given gaseous mixturedoes not match the given value in option (c).

Option (e) is incorrect because the calculated value of mole fraction of helium for the given gaseous mixturedoes not match the given value in option (e).

Hence, options(a), (b), (c), and (e) are incorrect.