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 use the relationship between the two velocities. Here’s a step-by-step solution:

### Step 1: Understand the relationship between velocities
The relationship between the most probable velocity (C_mp) and the RMS velocity (C_rms) is given by the formula:
\[ C_{rms} = C_{mp} \times \sqrt{\frac{3}{2}} \]

### Step 2: Identify the given values
From the problem, we know:
- Most probable velocity (C_mp) = 300 m/s

### Step 3: Substitute the values into the formula
Now we can substitute the value of C_mp into the equation:
\[ C_{rms} = 300 \, \text{m/s} \times \sqrt{\frac{3}{2}} \]

### Step 4: Calculate the square root
First, calculate \(\sqrt{\frac{3}{2}}\):
\[ \sqrt{\frac{3}{2}} \approx 1.2247 \]

### Step 5: Multiply to find C_rms
Now, multiply this value by 300 m/s:
\[ C_{rms} = 300 \, \text{m/s} \times 1.2247 \approx 367.41 \, \text{m/s} \]

### Step 6: Round to appropriate significant figures
Rounding this to three significant figures gives:
\[ C_{rms} \approx 367 \, \text{m/s} \]

### Final Answer
Thus, the RMS velocity of the gas molecule is approximately **367 m/s**.

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