Which change must result in an increase in the average kinetic energy of the molecules of a sample of `N_2(g)` ?

Options

A. The pressure changes from 0.5 atmosphere to 1 atm
B. The volume changes from 1 L to 2 L
C. The temperature changes from `20^@C " to " 30^@C`
D. The density changes from 2.0 g/L to 25 g/L

Answer

To determine which change results in an increase in the average kinetic energy of the molecules of a sample of \(N_2(g)\), we need to analyze the relationship between kinetic energy and temperature.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy in Gases**:
- The average kinetic energy (\(KE\)) of gas molecules 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. **Identifying Factors Affecting Kinetic Energy**:
- From the formula, we can see that the average kinetic energy is directly proportional to the temperature (\(T\)). This means that if the temperature increases, the average kinetic energy of the gas molecules also increases.

3. **Analyzing the Given Changes**:
- **Change 1**: Increasing pressure from 0.5 to 1 atm.
- While pressure can influence the behavior of gas, it does not directly increase the average kinetic energy unless accompanied by a temperature change.
- **Change 2**: Changing volume from 1 L to 2 L.
- Changing the volume alone does not affect the average kinetic energy unless the temperature is also changed.
- **Change 3**: Increasing temperature from 20°C to 30°C.
- This change directly increases the temperature, thereby increasing the average kinetic energy of the gas molecules.
- **Change 4**: Increasing density.
- Increasing density does not directly increase kinetic energy; it may even imply a decrease in temperature if the gas is compressed.

4. **Conclusion**:
- The only change that results in an increase in the average kinetic energy of the molecules of \(N_2(g)\) is the increase in temperature from 20°C to 30°C.

### Final Answer:
The change that must result in an increase in the average kinetic energy of the molecules of a sample of \(N_2(g)\) is **increasing the temperature**.

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