The velocities of sound in an ideal gas at temperature `T_(1) and T_(2)` K are found to be `V_(1) and V_(2)` respectively. If ther.m.s velocities of the molecules of the same gas at the same temperatures `T_(1) and T_(2)` are `v_(1) and v_(2)` respectively then

Options

A. `v_2 = v_1 (V_1/V_2)`
B. `v_2 = v_1 (V_2/V_1)`
C. `v_2 =v_1 sqrt(V_2/V_1)`
D. `v_2 = v_1 sqrt(V_1/V_2)`

Answer

To solve the problem, we need to establish the relationship between the velocities of sound in an ideal gas and the root mean square (rms) velocities of the gas molecules at different temperatures.

### Step-by-Step Solution:

1. **Understanding the Velocities**:
- The velocity of sound in an ideal gas is given by the formula:
\[ V = \sqrt{\frac{\gamma R T}{M}} \]
where \( \gamma \) is the adiabatic index (ratio of specific heats), \( R \) is the universal gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

2. **Applying the Formula for Two Temperatures**:
- For temperature \( T_1 \):
\[ V_1 = \sqrt{\frac{\gamma R T_1}{M}} \]
- For temperature \( T_2 \):
\[ V_2 = \sqrt{\frac{\gamma R T_2}{M}} \]

3. **Root Mean Square (rms) Velocity**:
- The rms velocity of gas molecules is given by:
\[ v = \sqrt{\frac{3RT}{M}} \]
- For temperature \( T_1 \):
\[ v_1 = \sqrt{\frac{3RT_1}{M}} \]
- For temperature \( T_2 \):
\[ v_2 = \sqrt{\frac{3RT_2}{M}} \]

4. **Establishing Proportional Relationships**:
- From the equations for \( V_1 \) and \( V_2 \), we can see that:
\[ V_1 \propto \sqrt{T_1} \quad \text{and} \quad V_2 \propto \sqrt{T_2} \]
- Similarly, for the rms velocities:
\[ v_1 \propto \sqrt{T_1} \quad \text{and} \quad v_2 \propto \sqrt{T_2} \]

5. **Setting Up the Ratio**:
- We can set up the following ratios based on the proportional relationships:
\[ \frac{V_1}{v_1} = \frac{V_2}{v_2} \]

6. **Rearranging the Equation**:
- From the ratio, we can express \( v_2 \) in terms of \( V_2 \) and \( V_1 \):
\[ v_2 = \frac{V_2}{V_1} v_1 \]

### Final Result:
Thus, the correct relationship between the velocities is:
\[ v_2 = \frac{V_2}{V_1} v_1 \]