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In $2.00 \mathrm{~min}, 28.1 \mathrm{~mL}$ of He effuse through a small hole. Under the same conditions of pressure and temperature, $9.31 \mathrm{~mL}$ of a mixture of $\mathrm{CO}$ and $\mathrm{CO}_{2}$ effuse through the hole in the same amount of time. Calculate the percent composition by volume of the mixture. The effusion rate of a gas is proportional to its root-mean-square speed, which is related to its molar mass.
$\square$ $\% \mathrm{CO}$
$\square$ $\% \mathrm{CO}_{2}$

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∻Solution∻ ‖ 1 ⋮ **Apply 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. We can write this as $\frac{Rate_1}{Rate_2} = \sqrt{\frac{M_2}{M_1}}$, where $Rate_1$ and $Rate_2$ are the effusion rates of two gases, and $M_1$ and $M_2$ are their respective molar masses. ‖ ‖ 2 ⋮ **Calculate the molar mass of the unknown mixture**: Since the rates of effusion for He and the mixture are given by their volumes effused in the same time, we can set up the equation $\frac{28.1 ext{ mL He}}{9.31 ext{ mL mixture}} = \sqrt{\frac{M_{mixture}}{M_{He}}}$. We know the molar mass of He is approximately $4 ext{ g/mol}$. Solving for $M_{mixture}$ gives us the molar mass of the unknown mixture. ‖ ‖ 3 ⋮ **Determine the percent composition**: Let $x$ be the volume percent of CO in the mixture and $(100-x)$ be the volume percent of CO2. The average molar mass of the mixture $M_{mixture}$ can be expressed as $M_{mixture} = x \cdot M_{CO} + (100-x) \cdot M_{CO2}$. We know $M_{CO} \approx 28 ext{ g/mol}$ and $M_{CO2} \approx 44 ext{ g/mol}$. Using the molar mass of the mixture calculated in step 2, we can solve for $x$ to find the percent composition of CO and CO2. ‖ ∻Answer∻ ⚹ [Insert final answer here] ⚹ ∻Key Concept∻ ⚹ Graham's Law of Effusion and the relationship between effusion rate and molar mass ⚹ ∻Explanation∻ ⚹ By comparing the effusion rates of helium and the gas mixture, we can determine the molar mass of the mixture. With the molar masses of CO and CO2, we can then calculate the percent composition of the mixture. ⚹◊What is the relationship between effusion rate, root-mean-square speed, and molar mass in gas effusion?⍭ Generate me a similar question◊

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