Question

An ideal gas of 5 moles has \( C_p = 8 \, \text{cal/mol}^\circ \text{C} \). If its temperature changes from 10°C to 20°C, then calculate the change in its internal energy (in cal).

Hint:

For an ideal gas, the change in internal energy depends only on the temperature change and the molar heat capacity at constant volume \( C_V \).

Answer 1

Correct Answer: 300

Step 1: Understanding the change in internal energy.
For an ideal gas, the change in internal energy (\( \Delta U \)) depends only on the change in temperature and the number of moles. The formula for the change in internal energy is given by: \[ \Delta U = n C_V \Delta T \] where:
- \( n \) is the number of moles,
- \( C_V \) is the molar heat capacity at constant volume,
- \( \Delta T \) is the change in temperature.
Since we are given the value of \( C_p \), the specific heat at constant pressure, we can relate \( C_p \) and \( C_V \) for an ideal gas by the relation: \[ C_p - C_V = R \] where \( R \) is the gas constant. For this case, since the unit of \( C_p \) is given in calories, we will assume \( R = 2 \, \text{cal/mol}^\circ \text{C} \) for an ideal gas. Therefore: \[ C_V = C_p - R = 8 - 2 = 6 \, \text{cal/mol}^\circ \text{C} \]
Step 2: Calculating the change in internal energy.
Now, we can calculate the change in internal energy: \[ \Delta U = n C_V \Delta T \] Substituting the given values: \[ \Delta U = 5 \times 6 \times (20 - 10) = 5 \times 6 \times 10 = 300 \, \text{cal} \]