Physics 4311: Thermal Physics – Homework 3

due date: Wednesday, Feb 18, 2026

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Problem 1: Root-mean-square velocity of gas molecules (6 points)

Calculate the root-mean-square velocities of an N $_{2}$ molecule at room temperature, 293 K, and at a temperature of 1 K.

Problem 2: Velocity components (6 points)

Consider an ideal gas in equilibrium. Using symmetry arguments, find which of the following averages involving velocity components $v_{x}, v_{y}, v_{z}$ are zero: $⟨ v_{x} ⟩$ , $⟨ v_{y}^{2} ⟩$ , $⟨ v_{z}^{3} ⟩$ , $⟨ v_{x} v_{z} ⟩$ , $⟨ v_{y}^{2} v_{z} ⟩$ , $⟨ v_{x}^{2} v_{y}^{4} ⟩$ .

Problem 3: Two-dimensional Maxwell distribution (12 points)

Consider an ideal gas of molecules of mass $m$ at temperature $T$ , restricted to move in two dimenisons.

a): Write down the two-dimensional Maxwell velocity distribution $P (v_{x}, v_{y})$
b): Starting from this Maxwell distribution, derive the (properly normalized) probability density $f (v)$ for the speed $v = | \vec{v} | = \sqrt{v_{x}^{2} + v_{y}^{2}}$ . (Hint: Go to polar coordinates and follow the steps used in class for the three-dimensional case.)

Problem 4: Low-speed molecules (16 points)

Consider a three-dimensional ideal gas of molecules of mass $m$ at temperature $T$ . The goal of this problem is to estimate the fraction of molecules whose kinetic energy is lower than $0.05 k_{B} T$ .

a): Find the speed $v_{\max}$ of a molecule of kinetic energy $0.05 k_{B} T$ in terms of $T$ and $m$ .
b): Write down the three-dimensional Maxwell-Boltzmann velocity distribution $P (v_{x}, v_{y}, v_{z})$
c): Write down an integral for the probability of a molecule to have a speed below $v_{\max}$ . Transform to spherical coordinates.
d): Solve the integral for the probability of a molecule to have a speed below $v_{\max}$ .
(Hint: The integral over $v$ cannot be solved in closed form by elementary means. As the energy is below $0.05 k_{B} T$ , the Boltzmann factor can be approximated, $e^{- m v^{2} ∕ (2 k_{B} T)} \approx 1$ .)