Question

If ${X_m},{X_p}\& {X_v}$ are mole fraction, pressure fraction and volume fraction respectively for a gaseous component in mixture. Select the correct relationship between them.
A) ${X_m} = {X_p} = {X_v}$
B) ${X_p} = \dfrac{1}{{{X_v}}}$
C) ${X_m} = \dfrac{1}{{{X_p}}} = \dfrac{1}{{{X_v}}}$
D) ${X_v} = \dfrac{1}{{{X_p}}} = \dfrac{1}{{{X_m}}}$

Answer 1

Application of the concepts of partial pressures and partial volumes along with ideal gas equation to one gas and mixture as a whole. Finally, taking their ratios to know the required relationships.

Complete step by step answer:
Let the partial pressure of one of the gases in the mixture be ${P_1}$ and the total pressure be $P$
Ideal gas equation for one gas is: ${P_1}V = {n_1}RT$---------- (i)
where, ${n_1}$is the number of moles of the gas and $V$ and $T$ represent the total volume and the common temperature of the mixture of gases.
Ideal gas equation for the entire mixture of gases: $PV = {n_t}RT$-----------(ii)
Where $P$ is the total pressure, $V$ is the total volume, ${n_t}$ is the total number of moles of all gases.
Dividing (i) by (ii), we get,
$\dfrac{{{p_1}.V}}{{PV}} = \dfrac{{{n_1}RT}}{{{n_t}RT}}$
$\Rightarrow \dfrac{{{p_1}}}{P} = \dfrac{{{n_1}}}{{{n_t}}}$
$\Rightarrow {X_p} = {X_m}$
If the pressure of a single gas is kept the same as the total pressure of the mixture, then the volume occupied is called partial volume. In that case, the ideal gas equation for a single gas can be written as $P{v_1} = {n_1}RT$ ……………………………………(iii)
Dividing (iii) by (ii), we get
$\dfrac{{P{v_1}}}{{PV}} = \dfrac{{{n_1}RT}}{{{n_t}RT}}$
$\Rightarrow \dfrac{{{v_1}}}{V} = \dfrac{{{n_1}}}{{{n_t}}}$
$\Rightarrow {X_v} = {X_m}$
Therefore, from step 5 and step 3, we can write, ${X_m} = {X_p} = {X_v}$

Hence, the option A is the correct answer.

Note: The temperature and the Universal gas constant remain the same throughout. When taking partial pressure, the volume remains the same and when taking the partial volume, the pressure remains the same.