The velocities of sound at same temperature in two monoatomic gases densities `rho_(1)` and `rho_(2)` are `v_(1)` and `v_(2)` repectively , if `(rho_(1))/(rho_(2)) = 4 `, then the value of `(v_(1))/(v_(2))` will be

Options

A. `1//4`
B. 2
C. `1//2`
D. 4

Explanation

To solve the problem, we need to find the ratio of the velocities of sound in two monoatomic gases given their densities. Let's denote the densities and velocities as follows:

- \( \rho_1 \) and \( \rho_2 \) are the densities of the two gases.
- \( v_1 \) and \( v_2 \) are the velocities of sound in the two gases.

We are given that:

\[ \frac{\rho_1}{\rho_2} = 4 \]

We need to find the value of:

\[ \frac{v_1}{v_2} \]

### Step 1: Recall the formula for the velocity of sound in a gas

The velocity of sound \( v \) in a gas can be expressed as:

\[ v = \sqrt{\frac{P}{\rho}} \]

Where:
- \( P \) is the pressure of the gas.
- \( \rho \) is the density of the gas.

### Step 2: Establish the relationship for both gases

For the two gases, we can write:

\[ v_1 = \sqrt{\frac{P_1}{\rho_1}} \quad \text{and} \quad v_2 = \sqrt{\frac{P_2}{\rho_2}} \]

### Step 3: Consider the conditions given in the problem

Since both gases are at the same temperature, we can assume that the pressures are equal (\( P_1 = P_2 = P \)). Also, since both gases are monoatomic, their specific heat ratios (\( \gamma \)) are the same.

### Step 4: Simplify the ratio of velocities

Using the above relationships, we can write the ratio of the velocities as:

\[ \frac{v_1}{v_2} = \frac{\sqrt{\frac{P}{\rho_1}}}{\sqrt{\frac{P}{\rho_2}}} = \sqrt{\frac{\rho_2}{\rho_1}} \]

### Step 5: Substitute the given density ratio

From the problem, we know:

\[ \frac{\rho_1}{\rho_2} = 4 \implies \frac{\rho_2}{\rho_1} = \frac{1}{4} \]

Now substituting this into the velocity ratio:

\[ \frac{v_1}{v_2} = \sqrt{\frac{1}{4}} = \frac{1}{2} \]

### Conclusion

Thus, the value of \( \frac{v_1}{v_2} \) is:

\[ \frac{v_1}{v_2} = \frac{1}{2} \]

### Final Answer

The ratio of the velocities of sound in the two gases is \( \frac{1}{2} \).

---