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It takes $157 \mathrm{~s}$ for an unknown gas to effuse through a porous wall and $96 \mathrm{~s}$ for the same volume of $\mathrm{N}_{2}$ gas to effuse at the same temperature and pressure. What is the molar mass of the unknown gas?
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∻Solution∻ ‖ 1 ⋮ **Use Graham's Law of Effusion**: Graham's Law states that the rate of effusion of a gas is inversely proportional to the square root of its molar mass. ‖ ‖ 2 ⋮ **Set up the equation**: According to Graham's Law, $\frac{ ext{Rate of effusion of gas 1}}{ ext{Rate of effusion of gas 2}} = \sqrt{\frac{ ext{Molar mass of gas 2}}{ ext{Molar mass of gas 1}}}$. Here, gas 1 is the unknown gas and gas 2 is $\mathrm{N_2}$. ‖ ‖ 3 ⋮ **Calculate the rates of effusion**: The rate of effusion is inversely proportional to the time taken, so $ ext{Rate of effusion of unknown gas} = \frac{1}{157 \mathrm{~s}}$ and $ ext{Rate of effusion of N_2} = \frac{1}{96 \mathrm{~s}}$. ‖ ‖ 4 ⋮ **Insert the values into the equation**: $\frac{1/157}{1/96} = \sqrt{\frac{28 \mathrm{~g/mol}}{M}}$, where $M$ is the molar mass of the unknown gas. ‖ ‖ 5 ⋮ **Solve for the molar mass of the unknown gas**: $\frac{96}{157} = \sqrt{\frac{28}{M}}$ leads to $M = \frac{28 imes (157)^2}{(96)^2}$. ‖ ‖ 6 ⋮ **Calculate and find the molar mass**: After solving the equation, we find the molar mass $M$ of the unknown gas. ‖ ∻Answer∻ ⚹ $M = \frac{28 imes (157)^2}{(96)^2} \approx 44.1 \mathrm{~g/mol}$ ⚹ ∻Key Concept∻ ⚹ Graham's Law of Effusion ⚹ ∻Explanation∻ ⚹ Graham's Law relates the rates of effusion of two gases to their molar masses, allowing us to calculate the molar mass of an unknown gas when compared to a known gas. ⚹◊What is the equation used to calculate the molar mass of a gas based on its effusion rate?⍭ Generate me a similar question◊

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