Question

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 \(\frac{c_p}{c_v} = \gamma = 1.4\)]

• A. The final pressure after the adiabatic expansion is 1.4 times greater than the final pressure after the isothermal expansion.
• B. The final temperature after the adiabatic expansion is 1.4 times less than the final temperature after the isothermal expansion.
• C. Pressure remains same in both adiabatic and isothermal expansion
• D. The final pressure after the adiabatic expansion is less than the final pressure after the isothermal expansion.

Hint:

Adiabatic expansion causes temperature drop, hence pressure falls more rapidly than in isothermal expansion.

Answer 1

The Correct Option is D

Step 1: Write relation for isothermal expansion.
For isothermal process:
\[ PV = \text{constant} \]
\[ P_1 V_1 = P_2 V_2 \]
Given \(V_2 = 6V_1\),
\[ P_2 = \frac{P_1}{6} \]

Step 2: Write relation for adiabatic expansion.
For adiabatic process:
\[ PV^\gamma = \text{constant} \]
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \]

Step 3: Substitute given expansion.
\[ P_2 = P_1 \left(\frac{V_1}{V_2}\right)^\gamma \]
\[ P_2 = P_1 \left(\frac{1}{6}\right)^{1.4} \]

Step 4: Compare pressures.
Since,
\[ \left(\frac{1}{6}\right)^{1.4} < \frac{1}{6} \]
Thus,
\[ P_{\text{adiabatic}} < P_{\text{isothermal}} \]

Step 5: Analyze options.
(A) Incorrect: Pressure is not greater in adiabatic case.
(B) Incorrect: Temperature changes differently in adiabatic process but not by factor \(1.4\).
(C) Incorrect: Pressures are not equal.
(D) Correct: Adiabatic pressure is less than isothermal pressure.

Step 6: Physical reasoning.
In adiabatic expansion, temperature decreases, reducing pressure more than in isothermal case where temperature remains constant.

Step 7: Final answer.
\[ \boxed{\text{The final pressure after the adiabatic expansion is less than the final pressure after the isothermal expansion}} \]