The most probable velocity of a gas molecule at 298 K is 300 m/s. Its rms velocity, in m/s) is

Options

A. 420
B. 245
C. 402
D. 367

Answer

To find the root mean square (RMS) velocity of a gas molecule given its most probable velocity, we can follow these steps:

### Step 1: Understand the Given Information
We are given the most probable velocity (Vm) of a gas molecule at 298 K, which is 300 m/s.

### Step 2: Use the Relationship Between Most Probable Velocity and RMS Velocity
The formulas for the most probable velocity (Vm) and the root mean square (RMS) velocity (Vrms) are as follows:
- Most Probable Velocity:
\[ V_m = \sqrt{\frac{2RT}{M}} \]
- RMS Velocity:
\[ V_{rms} = \sqrt{\frac{3RT}{M}} \]

### Step 3: Relate RMS Velocity to Most Probable Velocity
From the above formulas, we can see that:
\[ V_{rms} = \sqrt{\frac{3}{2}} V_m \]
This means that the RMS velocity can be calculated using the most probable velocity.

### Step 4: Substitute the Given Value
Now, substituting the given value of Vm:
\[ V_{rms} = \sqrt{\frac{3}{2}} \times 300 \text{ m/s} \]

### Step 5: Calculate the Value
Calculating \(\sqrt{\frac{3}{2}}\):
\[ \sqrt{\frac{3}{2}} \approx 1.2247 \]
Now, multiply this by 300 m/s:
\[ V_{rms} \approx 1.2247 \times 300 \approx 367.41 \text{ m/s} \]

### Step 6: Final Answer
Rounding off, we can say:
\[ V_{rms} \approx 367 \text{ m/s} \]

### Conclusion
The RMS velocity of the gas molecule at 298 K is approximately **367 m/s**.

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