Question

A gas at N.T.P. is suddenly compressed to $(\frac{1}{4})^{th}$ of its original volume. The final pressure in atmosphere is (Given $\gamma = \frac{3}{2}$ and $P = $ original pressure)

• A. 4P
• B. $\frac{3}{2}P$
• C. 8P
• D. $\frac{1}{4}P$

Hint:

Logic Tip: In adiabatic compression, the pressure rises more sharply than in isothermal compression due to the simultaneous rise in temperature.

Answer 1

The Correct Option is C

Concept:
Sudden compression is an adiabatic process where $PV^\gamma = \text{constant}$.
Step 1: Set up the adiabatic relation.
The new pressure $P_{new}$ is found using: $$\frac{P_{new{P} = \left( \frac{V}{V_{new \right)^\gamma$$
Step 2: Substitute the values.
Given $V_{new} = \frac{V}{4}$ and $\gamma = \frac{3}{2}$: $$\frac{P_{new{P} = \left( \frac{V}{V/4} \right)^{3/2} = (4)^{3/2}$$
Step 3: Calculate final result.
$$(4)^{3/2} = (\sqrt{4})^3 = 2^3 = 8$$ $$P_{new} = 8P$$