A jar has a mixture of hydrogen and oxygen gas in the ratio of 1 : 5. The ratio of mean kinetic energies of hydrogen and oxygen molecules is

Options

A. `1:16`
B. `1:4`
C. `1:5`
D. `1:1`

Answer

To find the ratio of mean kinetic energies of hydrogen and oxygen molecules in a jar with a mixture of gases in the ratio of 1:5, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Mean Kinetic Energy**:
The mean kinetic energy (KE) of one molecule of a gas can be expressed using the formula:
\[ KE = \frac{F}{2} k T \]
where \( F \) is the degrees of freedom, \( k \) is the Boltzmann constant, and \( T \) is the absolute temperature.

2. **Degrees of Freedom**:
For diatomic gases like hydrogen (H₂) and oxygen (O₂), the degrees of freedom \( F \) is typically 5 (3 translational and 2 rotational).

3. **Mean Kinetic Energy for Hydrogen**:
For hydrogen molecules:
\[ KE_{H_2} = \frac{5}{2} k T \]

4. **Mean Kinetic Energy for Oxygen**:
For oxygen molecules:
\[ KE_{O_2} = \frac{5}{2} k T \]

5. **Calculating the Ratio**:
To find the ratio of mean kinetic energies of hydrogen to oxygen, we can set up the following ratio:
\[ \frac{KE_{H_2}}{KE_{O_2}} = \frac{\frac{5}{2} k T}{\frac{5}{2} k T} \]
Simplifying this gives:
\[ \frac{KE_{H_2}}{KE_{O_2}} = 1 \]

6. **Conclusion**:
Therefore, the ratio of mean kinetic energies of hydrogen and oxygen molecules is:
\[ \text{Ratio} = 1 : 1 \]

### Final Answer:
The ratio of mean kinetic energies of hydrogen and oxygen molecules is \( 1 : 1 \).

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