Physics 4311: Thermal Physics – Final Exam

Friday, May 15, 2026

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Problem 1: Short questions (10 points each = 30 points)

a): A system with five states is in equilibrium at temperature $T$ . The energies fulfill $E_{1} = E_{2} > E_{3} = E_{4} = E_{5}$ . What values does the system’s entropy take for $T \to 0$ and $T \to \infty$ ?
b): A box of volume $V$ contains an ideal gas in equilibrium at temperature $T$ . The gas is now isothermally expanded to volume $2 V$ . Find the ratio $⟨ v_{f} ⟩ ∕ ⟨ v_{i} ⟩$ between the average speeds of the particles before and after the expansion.
c): You have two classical ideal gases, one consisting of $N$ atoms, the other consisting of $N$ diatomic molecules. Which gas has the larger heat capacity? Give a short explanation for your answer.

Problem 2: Polymer chain (60 points)

A model of a polymer is formed by connecting $N$ ellipsoid-shaped molecules into a one-dimensional chain. Each molecule has two ways of connecting to the polymer (see figure). It can align either its long axis (length $ℓ + a$ ) or its short axis (length $ℓ - a$ ) with the direction of the chain. One end of the chain is fixed at the ceiling, while the other holds a mass M (see figure). Gravity is acting along the negative z direction.

Neglecting the mass of the monomers, the energy of a given state of the polymer can be written as

E = - 𝑀𝑔𝐿 = - 𝑀𝑔 \sum_{i = 1}^{N} L_{i}

where $L_{i}$ is either $ℓ + a$ or $ℓ - a$ , depending on the orientation of monomer $i$ . The system is in equilibrium at temperature $T$ .

a): Calculate the single-monomer partition function $Z_{1}$ .
b): Find the partition function of the polymer chain, its free energy $F$ , and its internal energy $U$ .
c): Compute the average vertical distance $⟨ L ⟩$ between the fixed end of the polymer and the mass $M$ as a function of $M$ , $g$ , $ℓ$ , $a$ , and $T$ .
d): Analyze your result for $⟨ L ⟩$ in the limits of $T \to 0$ and $T \to \infty$ .

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Problem 3: Rubber band heat engine (70 points)

A rubber band has the equation of state $f = 𝛼𝐿𝑇$ where $L$ is its length, $f$ is the tension force, $T$ is temperature, and $α$ is a constant. The internal energy is given by $U = C_{L} T$ where the heat capacity $C_{L}$ at fixed length is a constant.

Consider a heat engine using this rubber band. The cycle consists of an isothermal shrinking (A $\to$ B), a cooling at constant length (B $\to$ C), an isothermal stretching (C $\to$ D), and a heating a constant length (D $\to$ A).

a): From the figure, find the temperatures at points A, B, C, and D and express them in terms of $α$ , $L_{0}$ , and $f_{0}$ .
b): Find the work $W_{𝐴𝐵}$ and heat $Q_{𝐴𝐵}$ during the process A $\to$ B in terms of $α$ , $L_{0}$ , $f_{0}$ , and $C_{L}$ .
c): Find the work $W_{𝐵𝐶}$ and heat $Q_{𝐵𝐶}$ during the process B $\to$ C in terms of $α$ , $L_{0}$ , $f_{0}$ , and $C_{L}$ .
d): Find the work $W_{𝐶𝐷}$ and heat $Q_{𝐶𝐷}$ during the process C $\to$ D in terms of $α$ , $L_{0}$ , $f_{0}$ , and $C_{L}$ .
e): Find the work $W_{𝐷𝐴}$ and heat $Q_{𝐷𝐴}$ during the process D $\to$ A in terms of $α$ , $L_{0}$ , $f_{0}$ , and $C_{L}$ .
f): Find the total work $W$ done on the engine during one cycle.
g): During which of the four processes does the system absorb heat and during which processes does it release heat?
h): The efficiency $η$ of the cycle is defined as work done by the system in one cycle divided by the absorbed heat in one cycle. Compute $η$ and express it in terms of $α$ , $L_{0}$ , and $C_{L}$ .

[Hint: For parts b), c), d), and e) carefully think about what quantities change during the process, and which stay constant.]

Problem 4: Nonharmonic classical ideal gas (40 points)

Consider a gas of $N$ non-interacting, indistinguishable, classical particles in a cubic box of linear size $L$ at tenmperature $T$ . The particles have non-harmonic energy-momentum relation; the Hamiltonian reads

H = \sum_{i = 1}^{N} A | {\vec{p}}_{i} |^{s} (A, s > 0)

a): Calculate the partition function and the free energy [Hint: Work in spherical coordinates.]
b): Calculate the internal energy $U$ and the specific heat $C_{V}$ at constant volume as functions of $T$ .
c): Calculate the pressure $p$ and find the equation of state (relation between $p, V, T$ ).

Hint: $\int_{0}^{\infty} 𝑑𝑥 x^{2} e^{- a x^{s}} = \frac{1}{s} a^{- 3 ∕ s} Γ (3 ∕ s) (Γ denotes the Gamma function.)$