Question

A flask contains argon and chlorine in the ratio of 2 : 1 by mass. The temperature of the mixture is 27°C. The ratio of root mean square speed of the molecules of the two gases ($v_{rms}(Ar) / v_{rms}(Cl_2)$) is: (Atomic mass of argon = 40.0 u and molecular mass of chlorine = 70.0 u) ____.

• A. 7/4
• B. $\sqrt{7}/2$
• C. $2/\sqrt{7}$
• D. 7/2

Hint:

In kinetic theory problems, always look for what stays constant. Since $T$ is constant, the lighter molecule will always have a higher $v_{rms}$. Since Argon (40) is lighter than Chlorine (70), its speed must be greater.

Answer 1

The Correct Option is B

Step 1: Understanding the Topic:
This question comes from the "Kinetic Theory of Gases." It explores the relationship between the thermal energy of gas particles and their microscopic speeds. A key takeaway is that at a specific temperature, different gas molecules have different average speeds based on their masses.
Step 2: Key Formulas and Approach:
The root mean square speed ($v_{rms}$) is defined as: \[ v_{rms} = \sqrt{\frac{3RT}{M}} \] Where:
$R$ is the universal gas constant.
$T$ is the absolute temperature (in Kelvin).
$M$ is the molar mass (or molecular mass) of the gas.

Step 3: Detailed Explanation:

Analyze the environment: Both gases are in the same flask, meaning they are in thermal equilibrium at the same temperature ($T = 27^\circ \text{C} = 300 \text{ K}$).
Establish the relationship: Since $3, R,$ and $T$ are constant for both gases, we can say $v_{rms} \propto \frac{1}{\sqrt{M}}$.
Identify masses: Molecular mass of Argon ($M_{Ar}$) = 40 u. Molecular mass of Chlorine ($M_{Cl_2}$) = 70 u. Note that the 2:1 mass ratio of the mixture is a "distractor" and does not affect the speed of individual molecules.
Formulate the ratio: \[ \frac{v_{rms}(Ar)}{v_{rms}(Cl_2)} = \frac{\sqrt{3RT/M_{Ar}}}{\sqrt{3RT/M_{Cl_2}}} = \sqrt{\frac{M_{Cl_2}}{M_{Ar}}} \]
Substitute and calculate: \[ \text{Ratio} = \sqrt{\frac{70}{40}} = \sqrt{\frac{7}{4}} = \frac{\sqrt{7}}{2} \]
Step 4: Final Answer:
The ratio of the rms speeds is $\sqrt{7}/2$.