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: Understanding the Question:
We need to calculate the pressure ratio for a monoatomic ideal gas that undergoes a rapid (adiabatic) compression.

Step 2: Key Formula or Approach:
For an adiabatic process, the relationship between pressure and volume is governed by the equation:
$$PV^\gamma = \text{constant}$$ For a monoatomic gas, the ratio of specific heats ($\gamma$) is $\frac{5}{3}$.

Step 3: Detailed Explanation:
Let the initial state be $P_1$ and $V_1$.
The final volume is $V_2 = \frac{V_1}{8}$.
Using the adiabatic equation:
$$P_1 V_1^\gamma = P_2 V_2^\gamma$$ Rearrange to find the ratio of final pressure to initial pressure:
$$\frac{P_2}{P_1} = \left(\frac{V_1}{V_2}\right)^\gamma$$ Substitute the volume ratio ($V_1 / V_2 = 8$) and $\gamma = 5/3$:
$$\frac{P_2}{P_1} = (8)^{5/3}$$ To solve this easily, recognize that $8 = 2^3$:
$$\frac{P_2}{P_1} = (2^3)^{5/3} = 2^5$$ $$2^5 = 32$$

Step 4: Final Answer:
The ratio of final to initial pressure is 32, matching option (A).