Q151 / 180 · COMEDK 2025 (10 May Shift 3) · Physics
A given volume of gas at NTP is allowed to expand 6 times of its original volume, first under isothermal condition and then under adiabatic condition. Which of the given statement is correct?
[Given $\dfrac{C_p}{C_v}=\gamma=1.4$ ]
- Correct answerAThe final pressure after the adiabatic expansion is less than the final pressure after the isothermal expansion.
- BPressure remains same in both adiabatic and isothermal expansion
- CThe final temperature after the adiabatic expansion is 1.4 times less than the final temperature after the isothermal expansion.
- DThe final pressure after the adiabatic expansion is 1.4 times greater than the final pressure after the isothermal expansion.
Solution
Let the initial pressure be $P_0$, initial volume be $V_0$, and initial temperature be $T_0$. The gas expands to a final volume $V_f = 6V_0$.
For isothermal expansion, the final pressure $P_{iso}$ is given by $P_0 V_0 = P_{iso} (6V_0)$, which gives $P_{iso} = \dfrac{P_0}{6}$.
For adiabatic expansion, the final pressure $P_{adia}$ is given by $P_0 V_0^{\gamma} = P_{adia} (6V_0)^{\gamma}$, which gives $P_{adia} = P_0 \left( \dfrac{1}{6} \right)^{\gamma}$.
Given $\gamma = 1.4$, we have $P_{adia} = P_0 \left( \dfrac{1}{6} \right)^{1.4}$.
Comparing $P_{iso} = P_0 (6)^{-1}$ and $P_{adia} = P_0 (6)^{-1.4}$, since $6^{1.4} > 6^1$, it follows that $\dfrac{1}{6^{1.4}} < \dfrac{1}{6^1}$.
Therefore, $P_{adia} < P_{iso}$. The final pressure after adiabatic expansion is less than the final pressure after isothermal expansion.
Answer: The final pressure after the adiabatic expansion is less than the final pressure after the isothermal expansion.
For isothermal expansion, the final pressure $P_{iso}$ is given by $P_0 V_0 = P_{iso} (6V_0)$, which gives $P_{iso} = \dfrac{P_0}{6}$.
For adiabatic expansion, the final pressure $P_{adia}$ is given by $P_0 V_0^{\gamma} = P_{adia} (6V_0)^{\gamma}$, which gives $P_{adia} = P_0 \left( \dfrac{1}{6} \right)^{\gamma}$.
Given $\gamma = 1.4$, we have $P_{adia} = P_0 \left( \dfrac{1}{6} \right)^{1.4}$.
Comparing $P_{iso} = P_0 (6)^{-1}$ and $P_{adia} = P_0 (6)^{-1.4}$, since $6^{1.4} > 6^1$, it follows that $\dfrac{1}{6^{1.4}} < \dfrac{1}{6^1}$.
Therefore, $P_{adia} < P_{iso}$. The final pressure after adiabatic expansion is less than the final pressure after isothermal expansion.
Answer: The final pressure after the adiabatic expansion is less than the final pressure after the isothermal expansion.
Hence, the correct answer is option: 'A'.