Question

A monoatomic gas is suddenly compressed to (1/8)th of its initial volume adiabatically. The ratio of the final pressure to the initial pressure is

• A. 32
• B. 16
• C. 8
• D. 64

Hint:

The word "suddenly" is a keyword in thermodynamics that almost always signifies an adiabatic process because there is no time for heat exchange ($Q=0$). Remember the $\gamma$ values: monoatomic is 5/3, diatomic is 7/5.

Answer 1

The Correct Option is A

Step 1: Note the process.
A monoatomic gas is suddenly (adiabatically) squeezed to one eighth of its volume. We want the ratio of final pressure to initial pressure.

Step 2: Recall the adiabatic law.
For an adiabatic change, $PV^\gamma=\text{constant}$, where $\gamma$ is the ratio of specific heats.

Step 3: Use the right $\gamma$.
For a monoatomic gas, $\gamma=\frac{5}{3}$.

Step 4: Write the two-state relation.
\[ P_1V_1^\gamma=P_2V_2^\gamma, \] with $V_2=\frac{V_1}{8}$.

Step 5: Rearrange for the pressure ratio.
\[ \frac{P_2}{P_1}=\left(\frac{V_1}{V_2}\right)^\gamma=(8)^{5/3}. \]

Step 6: Simplify the power.
Since $8=2^3$, \[ (2^3)^{5/3}=2^{5}=32. \]

Step 7: State the result.
The pressure ratio is $32$, which is option (1).
\[ \boxed{32} \]