1.At what temperature will the rms speed of oxygen gas molecules ($M = 32\text{ g/mol}$) be equal to the escape speed from Earth ($11.2\text{ km/s}$)?
Sol: $v_{rms} = \sqrt{\frac{3RT}{M}} = 11200 \implies T = \frac{M v_e^2}{3 R} = \frac{0.032 \times (11200)^2}{3 \times 8.314} \approx 160,860\text{ K} \approx 1.61 \times 10^5\text{ K}$.
2.A mixture of 2 moles of Helium ($\text{He}$) and 3 moles of Oxygen ($\text{O}_2$) is kept at temperature $T$. Find total internal energy and ratio of specific heats $\gamma_{mix}$.
Sol: $U_{mix} = 2\left(\frac{3}{2}RT\right) + 3\left(\frac{5}{2}RT\right) = 3RT + 7.5RT = 10.5 RT$. $C_{v,mix} = \frac{10.5 R}{5} = 2.1 R$. $C_{p,mix} = 3.1 R \implies \gamma_{mix} = \frac{3.1}{2.1} \approx 1.48$.
