Question

When a monoatomic gas at a pressure $P$ is suddenly compressed to $\frac{1}{8}$ of its original volume, the pressure of the gas becomes}

• A. 8P
• B. 5P
• C. 3P
• D. 32P
• E. 16P

Hint:

The word "suddenly" is a standard keyword in physics problems to indicate an adiabatic process ($PV^\gamma = \text{const}$), while "slowly" indicates an isothermal process ($PV = \text{const}$).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A "sudden" process is an adiabatic process where there is no time for heat exchange with the surroundings. For an adiabatic process, the relationship between pressure and volume is $PV^\gamma = \text{constant}$.
Key Formula or Approach:
For a monoatomic gas, $\gamma = \frac{5}{3}$.
The relation is $P_1 V_1^\gamma = P_2 V_2^\gamma$.

Step 2: Detailed Explanation:
Given:
Initial pressure $P_1 = P$.
Initial volume $V_1 = V$.
Final volume $V_2 = \frac{V}{8}$.
Using the adiabatic relation:
\[ P \cdot V^{5/3} = P_2 \cdot \left(\frac{V}{8}\right)^{5/3} \]
\[ P_2 = P \cdot \left(\frac{V}{V/8}\right)^{5/3} = P \cdot (8)^{5/3} \]
Calculate $(8)^{5/3}$:
\[ (8)^{5/3} = (2^3)^{5/3} = 2^5 = 32 \]
Thus, $P_2 = 32P$.

Step 3: Final Answer:
The final pressure of the gas is 32P.