At constant pressure average kinetic energy of gas molecules increases 3 times if-

Options

A. temperature is increased by 3 times
B. no change in temperature
C. temperature is halved
D. none of above

Answer

To solve the question, we need to understand the relationship between the average kinetic energy of gas molecules and temperature. The average kinetic energy (KE) of gas molecules can be expressed using the formula:

\[ KE = \frac{3}{2} k T \]

where:
- \( KE \) is the average kinetic energy,
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature in Kelvin.

### Step-by-Step Solution:

1. **Understand the relationship**: The average kinetic energy of gas molecules is directly proportional to the temperature of the gas. This means that if the temperature increases, the average kinetic energy also increases.

2. **Set up the equation**: If the average kinetic energy increases by a factor of 3, we can express this mathematically. Let the initial average kinetic energy be \( KE_1 = \frac{3}{2} k T_1 \) and the final average kinetic energy be \( KE_2 = \frac{3}{2} k T_2 \). According to the problem, we have:

\[ KE_2 = 3 \times KE_1 \]

3. **Substitute the expressions**: Substitute the expressions for kinetic energy into the equation:

\[ \frac{3}{2} k T_2 = 3 \times \left(\frac{3}{2} k T_1\right) \]

4. **Simplify the equation**: The \( \frac{3}{2} k \) cancels out from both sides:

\[ T_2 = 3 T_1 \]

5. **Conclusion**: This means that the temperature must increase to three times its initial value to achieve an increase in the average kinetic energy of the gas molecules by a factor of 3 at constant pressure.

### Final Answer:
At constant pressure, the average kinetic energy of gas molecules increases 3 times if the temperature is increased to three times its initial value.

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