Question

A monoatomic ideal gas of \( n \) moles heated from temperature \( T_1 \) to \( T_2 \) under two different conditions (i) at constant pressure, (ii) at constant volume. The change in internal energy of the gas is:

• A. more in process (ii)
• B. more in process (i)
• C. same in both the processes
• D. zero
• E. proportional to \( \frac{T_1 + T_2}{2} \)

Hint:

For an ideal gas, the change in internal energy is independent of the process type (constant pressure or constant volume), it only depends on the temperature change.

Answer 1

The Correct Option is C

Step 1: The change in internal energy \( \Delta U \) for an ideal gas depends only on the change in temperature and 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,
- \( \Delta T = T_2 - T_1 \) is the change in temperature.
Step 2: For a monoatomic ideal gas, the molar heat capacity at constant volume is \( C_V = \frac{3}{2} R \). The change in internal energy is: \[ \Delta U = n \left( \frac{3}{2} R \right) (T_2 - T_1) \] Step 3: The change in internal energy depends only on the temperature change, not on whether the process is at constant pressure or constant volume. Thus, the change in internal energy is the same in both processes.