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

Options

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

Explanation

To find the ratio of the mean kinetic energies of the molecules of hydrogen and oxygen at the same temperature, we can follow these steps:

### Step-by-Step Solution:

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

2. **Apply the formula to both gases**:
Since we are comparing hydrogen (H₂) and oxygen (O₂) at the same temperature:
- Mean kinetic energy of hydrogen:
\[ KE_{H_2} = \frac{3}{2} k_B T \]
- Mean kinetic energy of oxygen:
\[ KE_{O_2} = \frac{3}{2} k_B T \]

3. **Set up the ratio of the mean kinetic energies**:
To find the ratio of the mean kinetic energies of hydrogen to oxygen, we can write:
\[ \text{Ratio} = \frac{KE_{H_2}}{KE_{O_2}} = \frac{\frac{3}{2} k_B T}{\frac{3}{2} k_B T} \]

4. **Simplify the ratio**:
Since both the numerator and the denominator are the same, they cancel out:
\[ \text{Ratio} = 1 \]

5. **Express the ratio in standard form**:
The ratio can be expressed as:
\[ \text{Ratio} = 1:1 \]

### Final Answer:
The mean kinetic energies of the molecules of hydrogen and oxygen at the same temperature are in the ratio \( 1:1 \).

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