RMS Speed Ratio Calculation for Argon and Chlorine

The root mean square (RMS) speed of gas molecules is defined by the formula:

$V_{rms} = \sqrt{\frac{3RT}{M}}$

where $R$ is the ideal gas constant, $T$ is the absolute temperature, and $M$ is the molar mass of the gas.

Ratio of RMS Speeds

For a mixture containing two gases at the same temperature, the ratio of their RMS speeds depends only on their molar masses:

$\frac{V_{rms}^{Argon}}{V_{rms}^{Chlorine}} = \frac{\sqrt{\frac{3RT}{M_{Ar}}}}{\sqrt{\frac{3RT}{M_{Cl}}}}$

Since $3R$ and $T$ are constant for both gases, they cancel out:

$\frac{V_{rms}^{Ar}}{V_{rms}^{Cl}} = \sqrt{\frac{M_{Cl}}{M_{Ar}}}$

Molar Masses Provided

The problem provides the following molar masses:

• Molar mass of Argon ($M_{Ar}$) = $40 \text{ u}$
• Molar mass of Chlorine ($M_{Cl}$) = $70 \text{ u}$

Note: The mass ratio ($2:1$) and the temperature ($27^\circ\text{C}$) are not required for calculating the ratio of RMS speeds.

Calculating the Final Ratio

Substitute the molar masses into the ratio formula:

$\frac{V_{rms}^{Ar}}{V_{rms}^{Cl}} = \sqrt{\frac{70}{40}}$

Simplify the fraction inside the square root:

$\frac{V_{rms}^{Ar}}{V_{rms}^{Cl}} = \sqrt{\frac{7}{4}}$

Calculate the square root:

$\frac{V_{rms}^{Ar}}{V_{rms}^{Cl}} = \frac{\sqrt{7}}{\sqrt{4}} = \frac{\sqrt{7}}{2}$