Question

The relation obeyed by a perfect gas during an adiabatic process is \(pV^{3/2}\). The initial temperature of the gas is 'T'. When the gas is compressed to half of its initial volume, the final temperature of the gas is

• A. \(4\text{T}\)
• B. \(2\sqrt{2}\text{T}\)
• C. \(2\text{T}\)
• D. \(\sqrt{2}\text{T}\)

Hint:

Read the adiabatic index from the given p V relation, then use T times V to the power gamma minus one as constant.

Answer 1

The Correct Option is D

Step 1: Use the ideal gas law
$pV=nRT$ turns $pV^{3/2}=\text{const}$ into $TV^{1/2}=\text{const}$.

Step 2: Apply to the two states
$T_1\sqrt{V_1}=T_2\sqrt{V_2}$, with $V_2=\dfrac{V_1}{2}$.

Step 3: Compare the options
$T_2=T\sqrt{2}$. Options A and C come from using $V$ instead of $\sqrt{V}$.

Final Answer:
The gas ends at $\sqrt{2}\,\text{T}$. \[ \boxed{\text{(D) }\sqrt{2}\text{T}} \]