Animated Solution for Chemistry - States of Matter: 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 is

Methane vs Oxygen

Imagine a closed, empty container. We pump in two gases: methane (${\text{CH}}_4$) and oxygen (${\text{O}}_2$). The catch? We pump in exactly the same mass of both gases. The temperature is kept constant at $25^{\circ }\text{C}$. A battle for pressure begins inside the container. Our mission is to find out what fraction of the total pressure is exerted by the oxygen gas.

Dalton's Law

To solve this, we need a powerful tool: Dalton's Law of Partial Pressures. This law states that in a mixture of non-reacting gases, the partial pressure of any individual gas is directly proportional to its mole fraction.

Mathematically, it is expressed as:

If we rearrange this, the fraction of the total pressure exerted by oxygen is simply its mole fraction:

So, our entire problem boils down to finding the mole fraction of oxygen!

We know the masses are equal. Let's assume the mass of each gas is $w$ grams. To find the mole fraction, we first need the number of moles of each gas.

The molar mass of methane (${\text{CH}}_4$) is $16\text{ g/mol}$, and the molar mass of oxygen (${\text{O}}_2$) is $32\text{ g/mol}$.

Notice something interesting? Even though their masses are equal, there are twice as many moles of methane as there are of oxygen because methane is a lighter molecule.

Now, let's plug these moles into the mole fraction formula for oxygen:

Substituting our values:

We can factor out $w$ from the numerator and denominator, which beautifully cancels out. To make the addition easier, let's write $\frac{1}{16}$ as $\frac{2}{32}$:

The $32$ in the denominators cancel out, leaving us with our final answer:

Conclusion: Oxygen exerts exactly $\frac{1}{3}$ of the total pressure in the container. And what about the $25^{\circ }\text{C}$ mentioned in the problem? It was a classic distractor! The mole fraction, and hence the pressure fraction, depends purely on the relative number of moles, completely independent of the temperature.