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Detailed Solutions: Master Sheet (Kinetic Theory of Gases)
Student Name: ____________________________________
Class: 11th (CBSE/NEET/JEE)
Subject: Physics
Section A: Pressure & RMS Speed Solutions
1.
Find $v_{rms}$ of oxygen at $27^\circ\text{C}$ ($M = 32\text{ g/mol}$).
Sol:
$v_{rms} = \sqrt{\frac{3 R T}{M}} = \sqrt{\frac{3 \times 8.314 \times 300}{0.032}} = \sqrt{233831.25} \approx 483.56\text{ m/s}$.
2.
Temperature where $v_{rms}$ of $H_2$ doubles its value at $0^\circ\text{C}$ ($273\text{ K}$).
Sol:
$v_{rms} \propto \sqrt{T} \implies \frac{v'}{v} = \sqrt{\frac{T'}{273}} = 2 \implies T' = 4 \times 273 = 1092\text{ K} = 819^\circ\text{C}$.
Section B: Degrees of Freedom Solutions
3.
Find $\gamma = C_p/C_v$ for monatomic, diatomic, polyatomic gases.
Sol:
$\gamma = 1 + \frac{2}{f}$. Monatomic ($f=3$): $\gamma = 5/3 \approx 1.67$. Diatomic ($f=5$): $\gamma = 7/5 = 1.40$. Polyatomic ($f=6$): $\gamma = 4/3 \approx 1.33$.
4.
Mean free path $\lambda = \frac{1}{\sqrt{2}\pi n d^2}$.
Sol:
Average distance traveled by a gas molecule between successive collisions. Inversely proportional to number density $n$ and molecular diameter squared $d^2$.
