If at the same temperature and pressure, the densities of two diatomic gases are `d_(1)` and `d_(2)` respectively. The ratio of mean kinetic energy permolecule of gasses will be

Options

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

Explanation

To solve the problem of finding the ratio of mean kinetic energy per molecule of two diatomic gases with densities \( d_1 \) and \( d_2 \) at the same temperature and pressure, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Mean Kinetic Energy Formula**:
The mean kinetic energy per molecule of an ideal gas is given by the formula:
\[ E = \frac{3}{2} k T \]
where \( E \) is the mean kinetic energy, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature.

2. **Identify the Variables**:
In this problem, we are given that both gases are at the same temperature \( T \) and pressure. The densities of the gases are \( d_1 \) and \( d_2 \).

3. **Recognize Independence from Density**:
The mean kinetic energy per molecule depends only on the temperature \( T \) and the Boltzmann constant \( k \). It does not depend on the density of the gas. Therefore, even though the densities are different, the mean kinetic energy remains the same for both gases at the same temperature.

4. **Calculate the Ratio of Mean Kinetic Energies**:
Since the mean kinetic energy for both gases is the same, we can express this as:
\[ E_1 = \frac{3}{2} k T \quad \text{and} \quad E_2 = \frac{3}{2} k T \]
Thus, the ratio of the mean kinetic energies \( E_1 \) and \( E_2 \) is:
\[ \frac{E_1}{E_2} = \frac{\frac{3}{2} k T}{\frac{3}{2} k T} = 1 \]

5. **Conclusion**:
Therefore, the ratio of the mean kinetic energy per molecule of the two diatomic gases is:
\[ \frac{E_1}{E_2} = 1:1 \]

### Final Answer:
The ratio of mean kinetic energy per molecule of the two gases is \( 1:1 \).