If `c_(0)` and `c` denote the sound velocity and the rms velocity of the molecules in a gas, then

Options

A. `c_(s) lt c`
B. `c_(s) = c`
C. `c_(s) = c ((gamma)/(3))^(1//2)`
D. None of these

Answer

To solve the problem, we need to establish a relationship between the sound velocity \( c_0 \) and the root mean square (rms) velocity \( c \) of the molecules in a gas.

### Step-by-Step Solution:

1. **Define the formulas for sound velocity and rms velocity**:
- The sound velocity \( c_0 \) in a gas is given by:
\[ c_0 = \sqrt{\gamma \frac{RT}{M}} \]
where \( \gamma \) is the adiabatic constant, \( R \) is the gas constant, \( T \) is the absolute temperature, and \( M \) is the molar mass of the gas.

- The rms velocity \( c \) of the gas molecules is given by:
\[ c = \sqrt{\frac{RT}{M}} \]

2. **Relate the two velocities**:
- We can express \( c_0 \) in terms of \( c \) by substituting the expression for \( c \) into the equation for \( c_0 \):
\[ c_0 = \sqrt{\gamma \frac{RT}{M}} = \sqrt{\gamma} \cdot \sqrt{\frac{RT}{M}} = \sqrt{\gamma} \cdot c \]

3. **Express the relationship**:
- From the above equation, we can express the relationship between \( c_0 \) and \( c \):
\[ c_0 = c \cdot \sqrt{\gamma} \]

4. **Find the ratio of \( c_0 \) to \( c \)**:
- We can also express the ratio of the two velocities:
\[ \frac{c_0}{c} = \sqrt{\gamma} \]

5. **Final expression**:
- Thus, we conclude that:
\[ c_0 = c \cdot \sqrt{\gamma} \]

### Conclusion:
The relationship between the sound velocity \( c_0 \) and the rms velocity \( c \) of the molecules in a gas is given by:
\[ c_0 = c \cdot \sqrt{\gamma} \]