The root-mean square speeds of the molecules of different ideal gases, maintained at the same temperature are

Options

A. The same
B. Inversely proportional to the square root of the molecular weight
C. Directly proportional to the molecular weight
D. Inversely proportional to the molecular weight

Answer

To solve the question regarding the root-mean square (RMS) speeds of the molecules of different ideal gases maintained at the same temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the RMS Speed Formula**:
The root-mean square speed (v_rms) of gas molecules is given by the formula:
\[ v_{rms} = \sqrt{\frac{3RT}{M}} \]
where:
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass (molecular weight) of the gas.

2. **Identify Constants**:
Since the question states that the gases are maintained at the same temperature, \( T \) is constant for all gases. The gas constant \( R \) is also a constant.

3. **Analyze the Relationship**:
From the formula, we can see that the RMS speed is inversely proportional to the square root of the molar mass:
\[ v_{rms} \propto \frac{1}{\sqrt{M}} \]
This means that as the molar mass \( M \) increases, the RMS speed \( v_{rms} \) decreases.

4. **Conclusion**:
Therefore, we conclude that the root-mean square speeds of the molecules of different ideal gases, maintained at the same temperature, are inversely proportional to the square root of their molecular weights.

5. **Select the Correct Option**:
Based on the analysis, the correct answer is that the RMS speeds are inversely proportional to the square root of the molecular weight, which corresponds to option number 2.

### Final Answer:
The root-mean square speeds of the molecules of different ideal gases, maintained at the same temperature, are **inversely proportional to the square root of the molecular weight**.

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