Step 1: Calculate the number of moles of oxygen gas ($\text{O}_2$). The molar mass of $\text{O}_2$ is $2 \times 16.00 \text{ g/mol} = 32.00 \text{ g/mol}$. $$ n_{\text{O}_2} = \frac{\text{mass of O}_2}{\text{molar mass of O}_2} = \frac{40 \text{ g}}{32.00 \text{ g/mol}} = 1.25 \text{ mol} $$ Step 2: Calculate the number of moles of helium gas ($\text{He}$). The molar mass of $\text{He}$ is $4.00 \text{ g/mol}$. $$ n_{\text{He}} = \frac{\text{mass of He}}{\text{molar mass of He}} = \frac{2 \text{ g}}{4.00 \text{ g/mol}} = 0.50 \text{ mol} $$ Step 3: Calculate the total number of moles in the mixture. $$ n_{\text{total}} = n_{\text{O}_2} + n_{\text{He}} = 1.25 \text{ mol} + 0.50 \text{ mol} = 1.75 \text{ mol} $$ Step 4: Calculate the mole fraction of oxygen gas ($\chi_{\text{O}_2}$). The mole fraction of a component is the ratio of the moles of that component to the total moles in the mixture. $$ \chi_{\text{O}_2} = \frac{n_{\text{O}_2}}{n_{\text{total}}} = \frac{1.25 \text{ mol}}{1.75 \text{ mol}} $$ $$ \chi_{\text{O}_2} \approx 0.714 $$ The total pressure information (2.4 atm) is not needed to calculate the mole fraction. The mole fraction of oxygen gas in the mixture is approximately 0.714. $$ \boxed{0.714} $$