Question

A monoatomic gas at pressure P having volume V expands isothermally to a volume 3V and then adiabatically to a volume 81 V. The final pressure of the gas is (\(γ = 5/3\)).

• A. \(\frac{\text{P}}{3}\)
• B. \(\frac{\text{P}}{(3)^2}\)
• C. \(\frac{\text{P}}{(3)^4}\)
• D. \(\frac{\text{P}}{(3)^6}\)

Hint:

Use Boyle's law for the isothermal step and PV^gamma for the adiabatic step.

Answer 1

The Correct Option is D

Step 1: Combine ratios:
Pressure after step 1 is $P/3$. Volume ratio in step 2 is $81V/3V = 27$.

Step 2: Adiabatic factor:
Pressure falls by $27^{\gamma} = 27^{5/3} = 3^5 = 243$.

Step 3: Final:
$P_{final} = \dfrac{P}{3\cdot243} = \dfrac{P}{729} = \dfrac{P}{3^6}$, option (D).

Final Answer:
The final pressure is P / 3^6. \[ \boxed{\text{(D) }\dfrac{P}{3^6}} \]