Question

The temperatures of source and sink of a Carnot heat engine are $127^\circ C$ and $27^\circ C$ respectively. If the working substance is 2 moles of a rigid diatomic gas, then the decrease in internal energy during adiabatic expansion is: (R = 8.31 J mol$^{-1}$K$^{-1}$)

• A. 2493 J
• B. 4986 J
• C. 3324 J
• D. 4155 J

Hint:

Use $\Delta U = nC_V\Delta T$ for ideal gases in adiabatic processes.

Answer 1

The Correct Option is B

Step 1: Convert temperatures to Kelvin: \[ T_1 = 127 + 273 = 400K,\quad T_2 = 27 + 273 = 300K \]
Step 2: For a rigid diatomic gas: \[ C_V = \frac{5}{2}R \]
Step 3: During adiabatic expansion from $T_1$ to $T_2$, change in internal energy: \[ \Delta U = nC_V (T_2 - T_1) \]
Step 4: Substitute values: \[ \Delta U = 2 \times \frac{5}{2} \times 8.31 \times (300 - 400) \]
Step 5: \[ \Delta U = 5 \times 8.31 \times (-100) \]
Step 6: \[ \Delta U = -4155\ \text{J} \]
Step 7: Magnitude of decrease: \[ 4155\ \text{J} \]