Question

An ideal gas, \(\overline{C_V} = \frac{5}{2} R\), is expanded adiabatically against a constant pressure of 1 atm until it doubles in volume.
If the initial temperature and pressure are \(298 \, \text{K}\) and \(5 \, \text{atm}\), respectively, then the final temperature is ______ \( \, \text{K} \) (nearest integer).
Given: \(\overline{C_V}\) is the molar heat capacity at constant volume.

Answer 1

Correct Answer: 274

For an adiabatic process:
\[\Delta U = q + w, \quad q = 0 \implies \Delta U = w\]
Step 1: Using the first law of thermodynamics:
\[n C_V \Delta T = -P_{\text{ext}} (V_2 - V_1)\]
Since $V_2 = 2V_1$, substitute and simplify:
\[nR \frac{T_2}{P_2} = 2nR \frac{T_1}{P_1}.\]
Step 2: Relation between $P_2$ and $T_2$:
\[P_2 = \frac{5T_2}{2 \times 298}.\]
Step 3: Using $C_V$:
\[\frac{5}{2} nR (T_2 - T_1) = -nR T_1 \left( \frac{P_2}{P_1} - 1 \right).\]
Step 4: Substitute and solve:
\[T_2 = \frac{5T_2}{2 \times 298}.\]
From the equation:
\[T_2 \approx 274.16 \, \text{K}.\]
Nearest integer:
\[T_2 \approx 274 \, \text{K}.\]