Question

A gas having \(\gamma = \frac{5}{2}\) and volume 360 cc is suddenly compressed to 90 cc. If the initial pressure of the gas is \(P\), find the final pressure.

Hint:

The keyword "suddenly" always implies an adiabatic process. Use \(PV^\gamma = \text{const}\). If the compression were "slow", it would be isothermal, and you would use \(PV = \text{const}\), which would give a final pressure of only \(4P\).

Answer 1

Step 1: Understanding the Question:
"Suddenly" implies an adiabatic process.
Step 2: Key Formula or Approach:
\( P_1 V_1^{\gamma} = P_2 V_2^{\gamma} \).
Step 3: Detailed Explanation:
\( P_2 = P_1 \left( \frac{V_1}{V_2} \right)^{\gamma} \).
Given \( V_1 = 360 \), \( V_2 = 90 \), \( \gamma = 5/2 \).
\( V_1/V_2 = 360/90 = 4 \).
\[ P_2 = P \cdot (4)^{5/2} = P \cdot (\sqrt{4})^5 = P \cdot 2^5 = 32P \]
Step 4: Final Answer:
The final pressure is \( 32P \).