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

Options

A. `d_(1)d_(2)`
B. `sqrt(d_(2)//d_(1))`
C. `sqrt(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 1: Understand the formula for the velocity of sound in a gas
The velocity of sound \( V \) in a gas is given by the formula:
\[ V = \sqrt{\frac{\gamma RT}{M}} \]
where:
- \( \gamma \) is the adiabatic index (ratio of specific heats),
- \( R \) is the universal gas constant,
- \( T \) is the absolute temperature,
- \( M \) is the molar mass of the gas.

### Step 2: Express the velocity of sound for two gases
Let \( V_1 \) be the velocity of sound in gas 1 and \( V_2 \) be the velocity of sound in gas 2. We can write:
\[ V_1 = \sqrt{\frac{\gamma_1 RT}{M_1}} \quad \text{and} \quad V_2 = \sqrt{\frac{\gamma_2 RT}{M_2}} \]

### Step 3: Take the ratio of the velocities
The ratio of the velocities of sound in the two gases can be expressed as:
\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{\gamma_1 RT}{M_1}}}{\sqrt{\frac{\gamma_2 RT}{M_2}}} \]

### Step 4: Simplify the ratio
Since both gases are diatomic and are at the same temperature and pressure, we can assume:
- \( \gamma_1 = \gamma_2 \) (because both are diatomic gases),
- \( R \) is the same for both gases.

Thus, the equation simplifies to:
\[ \frac{V_1}{V_2} = \sqrt{\frac{M_2}{M_1}} \]

### Step 5: Relate molar mass to density
Using the relationship between density \( d \) and molar mass \( M \):
\[ d = \frac{M}{V} \implies M = d \cdot V \]
Since the volume is constant for both gases at the same conditions, we can express the molar mass in terms of density:
\[ M_1 = d_1 \quad \text{and} \quad M_2 = d_2 \]

### Step 6: Substitute the densities into the ratio
Substituting \( M_1 \) and \( M_2 \) into the ratio gives:
\[ \frac{V_1}{V_2} = \sqrt{\frac{d_2}{d_1}} \]

### Conclusion
Thus, the ratio of the velocities of sound in the two diatomic gases is:
\[ \frac{V_1}{V_2} = \sqrt{\frac{d_2}{d_1}} \]

### Final Answer
The ratio of the velocities of sound in the two diatomic gases is \( \sqrt{\frac{d_2}{d_1}} \).

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