Question

A gas having (\gamma = \frac{5}{2}) and volume (360 c.c.) is suddenly compressed to (90 c.c.) If initial pressure is (P), then final pressure will be

• A. (\frac{P}{4})
• B. (8 P)
• C. (16 P)
• D. (32 P)

Hint:

Sudden changes in volume are always considered adiabatic.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
"Suddenly compressed" implies that the process is adiabatic, meaning no heat is exchanged with the surroundings.
Step 2: Key Formula or Approach:
For an adiabatic process:
\[ P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \]
Step 3: Detailed Explanation:
Given:
Initial pressure \(P_1 = P\)
Initial volume \(V_1 = 360 \text{ c.c.}\)
Final volume \(V_2 = 90 \text{ c.c.}\)
Adiabatic index \(\gamma = 5/2\)
Using the adiabatic equation:
\[ P_2 = P_1 \left( \frac{V_1}{V_2} \right)^{\gamma} \]
\[ P_2 = P \left( \frac{360}{90} \right)^{5/2} \]
\[ P_2 = P (4)^{5/2} \]
We can rewrite \(4\) as \(2^2\):
\[ P_2 = P (2^2)^{5/2} \]
\[ P_2 = P (2)^{2 \times \frac{5}{2}} \]
\[ P_2 = P \cdot 2^5 \]
\[ P_2 = 32 P \]
Step 4: Final Answer:
The final pressure will be \(32 P\).