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Equal masses of methane and oxygen are mixed in an empty container at ${25^ \circ }{\text{C}}$. The fraction of the total pressure exerted by oxygen will be:
(A) $\dfrac{1}{2}$
(B) $\dfrac{2}{3}$
(C) $\dfrac{1}{3} \times \dfrac{{273}}{{298}}$
(D) $\dfrac{1}{3}$

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Last updated date: 15th Jun 2024
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Answer
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Hint: To solve this first calculate the number of moles of both the gases. Then calculate the mole fraction of oxygen. The mole fraction can then be related to the fraction of the total pressure exerted by oxygen.

Complete Step by step answer: We are given that equal masses of methane and oxygen are mixed in an empty container at ${25^ \circ }{\text{C}}$.
Consider that \[32{\text{ g}}\] of oxygen and \[32{\text{ g}}\] of methane are mixed. This is because it makes calculation for the number of moles easier.
Calculate the number of moles of oxygen in \[32{\text{ g}}\] of oxygen as follows:
${\text{Number of moles of oxygen}} = \dfrac{{{\text{Mass of oxygen}}}}{{{\text{Molar mass of oxygen}}}}$
Substitute \[32{\text{ g}}\] for the mass of oxygen, $32{\text{ g/mol}}$ for the molar mass of oxygen and solve for the number of moles of oxygen. Thus,
${\text{Number of moles of oxygen}} = \dfrac{{32{\text{ g}}}}{{32{\text{ g/mol}}}}$
$\Rightarrow {\text{Number of moles of oxygen}} = 1{\text{ mol}}$
Thus, the number of moles of oxygen in \[32{\text{ g}}\] of oxygen are $1{\text{ mol}}$.
Calculate the number of moles of methane in \[32{\text{ g}}\] of oxygen as follows:
${\text{Number of moles of methane}} = \dfrac{{{\text{Mass of methane}}}}{{{\text{Molar mass of methane}}}}$
Substitute \[32{\text{ g}}\] for the mass of methane, ${\text{16 g/mol}}$ for the molar mass of methane and solve for the number of moles of methane. Thus,
${\text{Number of moles of methane}} = \dfrac{{32{\text{ g}}}}{{{\text{16 g/mol}}}}$
$\Rightarrow {\text{Number of moles of methane}} = 2{\text{ mol}}$
Thus, the number of moles of methane in \[32{\text{ g}}\] of methane are ${\text{2 mol}}$.
Now, we will calculate the total number of moles in the container as follows:
The total number of moles is the sum of the number of moles of oxygen and the number of moles of methane. Thus,
${\text{Total number of moles}} = {\text{Number of moles of oxygen}} + {\text{Number of moles of methane}}$
${\text{Total number of moles}} = {\text{1 mol}} + {\text{2 mol}}$
$\Rightarrow {\text{Total number of moles}} = 3{\text{ mol}}$
Thus, the total number of moles in the container are $3{\text{ mol}}$
Now, we will calculate the mole fraction of oxygen in the container as follows:
The mole fraction is the number of moles of one component divided by the total number of moles. Thus,
${\text{Mole fraction of oxygen}} = \dfrac{{{\text{Number of moles of oxygen}}}}{{{\text{Total number of moles}}}}$
Substitute $1{\text{ mol}}$ for the number of moles of oxygen and $3{\text{ mol}}$ for the total number of moles. Thus,
${\text{Mole fraction of oxygen}} = \dfrac{{{\text{1 mol}}}}{{{\text{3 mol}}}}$
$\Rightarrow {\text{Mole fraction of oxygen}} = \dfrac{{\text{1}}}{{\text{3}}}$
Thus, the mole fraction of oxygen in the container is $\dfrac{{\text{1}}}{{\text{3}}}$.
Now, we know that the mole fraction is proportional to the fraction of the total pressure exerted. Thus, the fraction of the total pressure exerted by oxygen is $\dfrac{{\text{1}}}{{\text{3}}}$.

Thus, the correct option is (D) $\dfrac{{\text{1}}}{{\text{3}}}$.

Note: The fraction of pressure exerted is directly proportional to the mole fraction. Thus, the fraction of the total pressure exerted by methane is $\dfrac{2}{3}$. While solving such questions one should keep in mind that the values of the terms must be put down in the proper unit.