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 Concept:
A "sudden" compression indicates an adiabatic process, where no heat exchange occurs with the surroundings. For an adiabatic process, the relationship between pressure and volume is \(PV^\gamma = \text{constant}\).

Step 2: Key Formula or Approach:
\[ P_1 V_1^\gamma = P_2 V_2^\gamma \implies P_2 = P_1 \left( \frac{V_1}{V_2} \right)^\gamma \]
Step 3: Detailed Explanation:
Given:
Initial Pressure \(P_1 = P\).
Initial Volume \(V_1 = 360\) cc.
Final Volume \(V_2 = 90\) cc.
Adiabatic index \(\gamma = 5/2\).
1. Calculate the volume ratio:
\[ \frac{V_1}{V_2} = \frac{360}{90} = 4 \] 2. Substitute into the formula:
\[ P_2 = P \cdot (4)^{5/2} \] 3. Solve the power:
\((4)^{5/2} = (4^{1/2})^5 = 2^5 = 32\).
\[ P_2 = 32 P \]
Step 4: Final Answer:
The final pressure is \(32 P\).