Question

An ideal monotonic gas containing five moles at $27^\circ\text{C}$ and one atm pressure is expanded adiabatically to thrice of its volume. The change in internal energy is:

• A. \(+9007 \text{ J} \)
• B. \(-9007 \text{ J} \)
• C. \(+9711 \text{ J} \)
• D. \(-9711 \text{ J} \)

Hint:

Adiabatic expansion always leads to cooling of the gas ($T_2 < T_1$), meaning the change in internal energy $\Delta U$ must be negative.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:

• Five moles of an ideal monatomic gas at \( 27^\circ C \) (300 K) expand adiabatically to three times the volume. We need \( \Delta U \).

Step 2: Detailed Explanation:

• For an adiabatic process \( T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \), with \( \gamma = 5/3 \).

• \( T_2 = 300 \times (1/3)^{2/3} \approx 300 \times 0.4807 \approx 144.2 \text{ K} \).

• \( \Delta U = n C_v \Delta T = 5 \times \tfrac{3}{2}(8.314)(144.2 - 300) \approx -9711 \text{ J} \).

• The sign is negative because the gas does work at the expense of its internal energy.

Step 3: Final Answer:
The correct option is (D) \( -9711 \text{ J} \).