What is the ratio of gas speeds for CO_(2) to He? 11m/s 11^1/2m/s 1/11m/s 1/11^1/2m/s

To determine the ratio of gas speeds for \( CO_2 \) to He, we can use the root-mean-square (RMS) speed of gases. The RMS speed of a gas is given by the formula:<br /><br />\[ v_{\text{rms}} = \sqrt{\frac{3RT}{M}} \]<br /><br />where:<br />- \( R \) is the universal gas constant,<br />- \( T \) is the temperature in Kelvin,<br />- \( M \) is the molar mass of the gas.<br /><br />Given that the temperature and gas constant are the same for both gases, the ratio of their speeds will be inversely proportional to the square root of their molar masses.<br /><br />The molar mass of \( CO_2 \) is \( 44 \, \text{g/mol} \) and the molar mass of He is \( 4 \, \text{g/mol} \).<br /><br />Thus, the ratio of the speeds is:<br /><br />\[ \sqrt{\frac{M_{\text{He}}}{M_{\text{CO}_2}}} = \sqrt{\frac{4}{44}} = \sqrt{\frac{1}{11}} = \frac{1}{\sqrt{11}} \]<br /><br />Therefore, the correct answer is:<br /><br />\[ \frac{1}{\sqrt{11}} \, \text{m/s} \]<br /><br />So, the ratio of gas speeds for \( CO_2 \) to He is \( \frac{1}{\sqrt{11}} \, \text{m/s} \).