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 must result in an increase in the average kinetic energy of the molecules of a sample of nitrogen gas (N₂(g)), we can follow these steps:

### Step 1: Understand the formula for average kinetic energy
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,
- \( T \) is the absolute temperature in Kelvin.

### Step 2: Identify the factors affecting average kinetic energy
From the formula, we can see that the average kinetic energy is directly proportional to the temperature (T). This means that any change that increases the temperature will also increase the average kinetic energy.

### Step 3: Analyze the options
1. **Pressure changes**: Changing the pressure of a gas does not directly affect the average kinetic energy since it does not change the temperature.
2. **Volume changes**: Similar to pressure, changing the volume of a gas does not directly affect the average kinetic energy unless it results in a temperature change.
3. **Temperature changes**: An increase in temperature will directly increase the average kinetic energy of the gas molecules. For example, changing the temperature from 20°C to 30°C (which is an increase) will increase the average kinetic energy.
4. **Density changes**: Changes in density do not directly affect the average kinetic energy unless they are associated with changes in temperature.

### Step 4: Conclusion
The only change that must result in an increase in the average kinetic energy of the molecules of a sample of nitrogen gas is an increase in temperature.

### Final Answer
The correct answer is that an increase in temperature (from 20°C to 30°C) must result in an increase in the average kinetic energy of the molecules of N₂(g).

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