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_(1)=v_(1)((V_(1))/(V_(2)))`
B. `v_(1)=v_(1)((V_(2))/(V_(1)))`
C. `v_(1)=v_(1) sqrt((V_(2))/(V_(1)))`
D. `v_(1)=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 two different temperatures \( T_1 \) and \( T_2 \).

### Step-by-Step Solution:

1. **Understanding the Formulas**:
- The RMS velocity of gas molecules at temperature \( T \) is given by:
\[ v_{\text{RMS}} = \sqrt{\frac{3RT}{M}} \]
where \( R \) is the universal gas constant and \( M \) is the molecular mass of the gas.
- The velocity of sound in an ideal gas at temperature \( T \) is given by:
\[ V_s = \sqrt{\frac{\gamma RT}{M}} \]
where \( \gamma = \frac{C_p}{C_v} \) is the ratio of specific heats.

2. **Establishing Proportional Relationships**:
- From the formulas, we can see that both \( v_{\text{RMS}} \) and \( V_s \) are directly proportional to \( \sqrt{T} \):
\[ v_{\text{RMS}} \propto \sqrt{T} \quad \text{and} \quad V_s \propto \sqrt{T} \]

3. **Setting Up the Ratios**:
- For two different temperatures \( T_1 \) and \( T_2 \), we can write the ratios:
\[ \frac{v_{\text{RMS1}}}{V_1} = \frac{v_{\text{RMS2}}}{V_2} \]

4. **Substituting Known Values**:
- It is given that \( v_{\text{RMS1}} = V_1 \) and \( v_{\text{RMS2}} = V_2 \). Therefore, we can substitute these into the ratio:
\[ \frac{V_1}{V_1} = \frac{V_2}{V_2} \]
- This simplifies to:
\[ 1 = 1 \]
- This indicates a consistent relationship.

5. **Final Relationship**:
- Rearranging the ratio gives us:
\[ \frac{V_1}{v_{\text{RMS1}}} = \frac{V_2}{v_{\text{RMS2}}} \]
- From this, we can express \( v_{\text{RMS2}} \) in terms of \( v_{\text{RMS1}} \):
\[ v_{\text{RMS2}} = v_{\text{RMS1}} \cdot \frac{V_2}{V_1} \]

### Final Answer:
Thus, the relationship can be summarized as:
\[ v_{\text{RMS2}} = v_{\text{RMS1}} \cdot \frac{V_2}{V_1} \]