If at same temperature and pressure the densities fro two diatomic gases are respectively `d_(1) and d_(2)`, then the ratio of velocity of sound in these gases will be

Options

A. `sqrt((d_(2)//d_(1)))`
B. `sqrt((d_(1)//d_(2)))`
C. `d_(1)d_(2)`
D. `sqrt(d_(1)d_(2))`

Explanation

To find the ratio of the velocities of sound in two diatomic gases at the same temperature and pressure, we can follow these steps:

### Step-by-step Solution:

1. **Understand the Formula for Velocity of Sound:**
The velocity of sound in a gas is given by the formula:
\[ v = \sqrt{\frac{\gamma P}{d}} \]
where \( v \) is the velocity of sound, \( \gamma \) is the adiabatic index (ratio of specific heats), \( P \) is the pressure, and \( d \) is the density of the gas.

2. **Identify Constants:**
Since both gases are diatomic and are at the same temperature and pressure, \( \gamma \) and \( P \) will be the same for both gases. Let:
- For gas 1: density = \( d_1 \)
- For gas 2: density = \( d_2 \)

3. **Write the Equations for Each Gas:**
For gas 1:
\[ v_1 = \sqrt{\frac{\gamma P}{d_1}} \]
For gas 2:
\[ v_2 = \sqrt{\frac{\gamma P}{d_2}} \]

4. **Find the Ratio of Velocities:**
To find the ratio of the velocities \( \frac{v_1}{v_2} \):
\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{\gamma P}{d_1}}}{\sqrt{\frac{\gamma P}{d_2}}} \]

5. **Simplify the Ratio:**
This can be simplified as follows:
\[ \frac{v_1}{v_2} = \sqrt{\frac{\gamma P}{d_1}} \cdot \sqrt{\frac{d_2}{\gamma P}} = \sqrt{\frac{d_2}{d_1}} \]
Here, the \( \gamma P \) terms cancel out.

6. **Final Result:**
Thus, the ratio of the velocities of sound in the two gases is:
\[ \frac{v_1}{v_2} = \sqrt{\frac{d_2}{d_1}} \]

### Conclusion:
The ratio of the velocities of sound in the two diatomic gases is given by:
\[ \frac{v_1}{v_2} = \sqrt{\frac{d_2}{d_1}} \]