Question

An ideal gas expands adiabatically, ($\gamma = 1.5$). To reduce the r.m.s. velocity of the molecules 4 times, the gas has to be expanded ______.

• A. 256 times
• B. 128 times
• C. 64 times
• D. 8 times

Hint:

Adiabatic expansions always cool a gas down because the gas expends internal energy to do the work of expansion. A massive expansion ($256\times$) is required to achieve a small velocity reduction ($4\times$) because velocity scales with the square root of temperature!

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
For adiabatic expansion, the relationship between temperature and volume is $TV^{\gamma - 1} = \text{constant}$. Also, $v_{rms} \propto \sqrt{T} \implies T \propto v_{rms}^2$.

Step 2: Formula Application:
If $v_{rms}$ is reduced 4 times ($1/4$), then $T$ is reduced by $4^2 = 16$ times. $\frac{T_1}{T_2} = \left( \frac{V_2}{V_1} \right)^{\gamma - 1}$.

Step 3: Explanation:
$16 = \left( \frac{V_2}{V_1} \right)^{1.5 - 1} = \left( \frac{V_2}{V_1} \right)^{0.5}$. Squaring both sides: $16^2 = \frac{V_2}{V_1} \implies \frac{V_2}{V_1} = 256$. Re-calculation check: If $\gamma - 1 = 0.5$ (which is $1/2$), then $16 = \sqrt{V_2/V_1} \implies V_2/V_1 = 256$. However, if $\gamma = 5/3$, then $\gamma - 1 = 2/3$. Let's re-read $\gamma = 1.5$ ($3/2$). $16 = (V_2/V_1)^{1/2} \implies V_2/V_1 = 256$.

Step 4: Final Answer:
The gas has to be expanded 256 times (Option A).