Question

A polyatomic gas ($\gamma = 4/3$) is compressed to $\left(\frac{1}{8}\right)^{\text{th}}$ of its volume adiabatically. If its initial pressure is $P_0$, its new pressure will be

• A. $2P_0$
• B. $8P_0$
• C. $6P_0$
• D. $16P_0$

Hint:

For adiabatic volume compressions, if the volume drops by a factor of $x$, the pressure increases by a factor of $x^\gamma$. Here, the compression factor is 8 and $\gamma = 4/3$. Taking the cube root of 8 first gives 2, and then raising 2 to the $4^{\text{th}}$ power yields 16 instantly. Splitting the fractional exponent into root-then-power makes the calculation simple to perform mentally!

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
An ideal polyatomic gas undergoes adiabatic compression from volume V to V/8. Find final pressure in terms of initial P₀, given γ = 4/3.

Step 2: Key Formula or Approach:
For an adiabatic process, PV^γ = constant. Thus P₂ = P₁(V₁/V₂)^γ.

Step 3: Detailed Explanation:
Substitute: P₂ = P₀(V/(V/8))^(4/3) = P₀(8)^(4/3). Write 8 = 2³: P₂ = P₀(2³)^(4/3) = P₀·2^(3×4/3) = P₀·2⁴ = 16P₀.

Step 4: Final Answer:
The final pressure is 16P₀, option (D).