At the same temperature, the mean kinetic energies of the molecules of hydrogen and oxygen are in the ratio

Options

A. `1 : 16`
B. `8 : 1`
C. `16 : 1`
D. `1 : 1`

Explanation

To find the ratio of the mean kinetic energies of the molecules of hydrogen and oxygen at the same temperature, we can use the formula for mean kinetic energy of gas molecules.

### Step-by-Step Solution:

1. **Understand the Formula for Mean Kinetic Energy**:
The mean kinetic energy (KE) of a gas is given by the formula:
\[ KE = \frac{3}{2} k T \]
where \( k \) is the Boltzmann constant and \( T \) is the absolute temperature in Kelvin.

2. **Identify the Key Variables**:
In this case, we are comparing two different gases (hydrogen and oxygen) at the same temperature. Therefore, the temperature \( T \) is constant for both gases.

3. **Mean Kinetic Energy for Hydrogen and Oxygen**:
Since the mean kinetic energy depends only on the temperature and not on the type of gas, we can express the mean kinetic energy for both gases as:
\[ KE_{H_2} = \frac{3}{2} k T \quad \text{(for Hydrogen)} \]
\[ KE_{O_2} = \frac{3}{2} k T \quad \text{(for Oxygen)} \]

4. **Calculate the Ratio**:
Since both expressions for mean kinetic energy are equal at the same temperature, we can write:
\[ \text{Ratio} = \frac{KE_{H_2}}{KE_{O_2}} = \frac{\frac{3}{2} k T}{\frac{3}{2} k T} = 1 \]

5. **Conclusion**:
Therefore, the mean kinetic energies of the molecules of hydrogen and oxygen at the same temperature are in the ratio:
\[ \text{Ratio} = 1:1 \]