Physics 4311: Thermal Physics – Homework 4

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

The atoms of an interstellar cloud (in thermal equilibrium at temperature $T$) emit light. The emission frequency of a particular element is ${\omega }_0$ if the atom is a rest. Due to the thermal motion of the atoms, the observed frequencies are shifted (Doppler effect) to

where $v_x$ is the velocity component of the atom away from the detector.

a)

Calculate the average observed frequency $⟨\omega ⟩$.

b)

Find the width of the spectral line, i.e., the standard deviation ${\sigma }_{\omega }$ of the observed frequency.

Problem 2: Mean free paths (9 points)

A diffuse cloud of neutral hydrogen atoms in space has a number density of 500 cm${}^{-3}$ and a temperature of 30K. (The hydrogen atom can be approximated as a sphere of radius 0.53 Å)

a)

Find the collision cross section.

b)

Estimate the mean collision time.

c)

Estimate the mean free path.

Problem 3: Space walk (8 points)

An astronaut goes for a space walk, and her space suit is pressurized to 1 atm. Unfortunately, a tiny piece of space dust punctures her suit, and it develops a small hole of radius 10 $\mu$m. What force does she feel due to the effusing gas?

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)

Derive a differential equation for the pressure $p$ in the box as a function of the time $t$ after opening the hole.

b)

Solve the differential equation and find $p(t)$.