Physics 4311: Thermal Physics – HW solution 4

Problem 1: Doppler broadening of spectral lines (8 points)

a)

$$⟨\omega ⟩=⟨{\omega }_0\left(1-\frac{v_x}{c}\right)⟩={\omega }_0\left(1-\frac{⟨v_x⟩}{c}\right)={\omega }_0$$

b)

$$\begin{array}{rcll}⟨{\omega }^2⟩& =& ⟨{\omega }_0^2{\left(1-\frac{v_x}{c}\right)}^2⟩={\omega }_0^2-2{\omega }_0^2⟨v_x⟩∕c+{\omega }_0^2⟨v_x^2∕c^2⟩& \\ ⟨{\omega }^2⟩& =& {\omega }_0^2+{\omega }_0^2{k}_{B}T∕(m{c}^2)& \\ {\sigma }_{\omega }^2& =& ⟨{\omega }^2⟩-{⟨\omega ⟩}^2={\omega }_0^2{k}_{B}T∕(m{c}^2)& \end{array}$$

Problem 2: Mean free paths (9 points)

a)

$$\sigma =\pi {(a_1+a_2)}^2=4a^2=3.53\times 10^{-20}\mathrm{\text{m}}$$

b)

$$\begin{array}{rcll}⟨v⟩=\sqrt{8k_{B}T∕(\mathit{𝜋𝑚})}=797\mathrm{\text{m/s}}& & & \\ ⟨v_r⟩\approx \sqrt{2}⟨v⟩=1124\mathrm{\text{m/s}}& & & \\ \tau =\frac{1}{\mathit{𝑛𝜎}{v}_r}=5.05\times 10^7\mathrm{\text{s}}& & & \end{array}$$

c)

$$\lambda =\frac{1}{\sqrt{2}\mathit{𝑛𝜎}}=4.02\times 10^{10}\mathrm{\text{m}}$$

Problem 3: Space walk (8 points)

Momentum conservation: momentum transfer to astronaut is negative of the momentum carried away by the gas. Momentum carried out through hole is half of what would have been transferred to the wall upon reflection.

Problem 4: Pressure change due to effusion (15 points)

A box of volume $V$ contains an ideal gas that is kept at temperature $T$ by a thermostat. Its initial pressure is $p_0$. At time $t=0$, a small hole of cross section area $A$ is opened in the box, and gas starts escaping. (Assume that the hole is sufficiently small so that the gas in the box remains in equilibrium during the effusion process.)

a)

$$\frac{\mathit{𝑑𝑁}}{\mathit{𝑑𝑡}}=-\mathit{𝛷𝐴}=-\frac{1}{4}⟨v⟩\mathit{𝑛𝐴}=-\frac{1}{4}⟨v⟩\frac{N}{V}A$$

now use ideal gas law $p=N{k}_{B}T∕V$

$$\frac{\mathit{𝑑𝑝}}{\mathit{𝑑𝑡}}=-\frac{1}{4}\frac{A}{V}⟨v⟩p$$

b)

$$p=p_0\mathit{𝑒𝑥𝑝}\left(-\frac{1}{4}\frac{A}{V}⟨v⟩t\right)=p_{0}exp\left(-\frac{A}{V}t\sqrt{\frac{k_{B}T}{2\mathit{𝜋𝑚}}}\right)$$