Practice Exam Problems

Heat, Work, and Ideal Gas (in HW 4)

One mole of an ideal diatomic gas at temperature $T$ is expanded isothermally (at constant temperature) from volume $V_{1}$ to final volume $V_{2}$ .

a) Plot this process on a PV diagram, assuming it is done slowly enough that the gas remains in equilibrium throughout. What is the functional form, i.e. how does P depend on V?

b) What is the final pressure?

c) Compute the heat, work and change in internal energy during this process? In writing down an equation for work, you must specify whether it's work done by the gas or on the gas.

d) Sketch a path from $V_{2}$ to $V_{1}$ that, if done immediately after the process described above, would result in the gas having done net positive work. Explain why? How is the net positive work related to the heat flowing into or out of the gas?

Two-state system (in HW 4)

Consider a collection of $N$ two-state particles. Each can be in a state with energies ${0, ϵ}$ . The total energy is the macroscopic variable:

U = \sum_{i = 1}^{N} ϵ s_{i}

where $s_{i} \in {0, 1}$ , i.e. each

What is the multiplicity of microscopic configurations at a fixed energy $U$ ?
Assume both $U$ and $N$ are very large, and comparable in size. Use the Stirling approximation to write the entropy to leading order in these large variables. How many terms do you need to keep in the Stirling approximation to get a nontrivial result?
Define the energy density $U / N = ϵ ρ$ , and rewrite the entropy as a function of $ρ$ . At what value of $ρ$ is the entropy maximal? What is the value of the maximal entropy?
From the entropy you found in Part 3, compute the temperature. Then invert this equation to find the density $ρ (T)$ as a function of temperature. Note that $\partial S / \partial U$ = $(\partial S / \partial ρ) \partial ρ / \partial U = (\partial S / \partial ρ) (1 / ϵ N)$

Entropy of 3-state spin (in HW 4)

Consider a collection of spins which can take three values: $s = {- 1, 0, + 1}$ in an external magnetic field $B$ .

The macroscopic variables for this system are the magnetization:

M = \sum_{i = 1}^{N} s_{i}

And the total energy

U = - μ B \sum_{i} s_{i}

a) Take N = 2. Find all the possible macrostates, and enumerate their microstates.

b) Plot the entropy vs. U for this small system. Sketch what you think it will look like for larger systems (large $N$ )? Sketch the temperature as a function of energy.

Challenge:

How many microstates exist for arbitrary $N$ at a given energy $U$ ?

Entropy and Heat

An Einstein solid with $q >> N$ , and total energy $U = ϵ q$ , has an entropy (Schroeder Eq. 3.9)

S = N k [\ln (\frac{U}{ϵ N}) + 1]

a) How is the energy related to the temperature? Compare it to the equipartition theorem, and see if the number of degrees of freedom makes sense.
b) Consider a constant volume process which changes the temperature from $T_{0}$ to $2 T_{0}$ . What is the change in entropy over this process? You may compute this in any way you like.

Multiple Paths (Challenging)

Consider a 2D grid world, where you are allowed to only move up, down, left, and right on a 2D lattice of points.

Assume someone starts at point $(x, y) = (0, 0)$ . They take a journey out into grid world, but ultimately come back home. Assume the total length of their journey is fixed. How many possible paths could they have taken at a fixed length $L$ .

PV process and First Law

Below are two paths denoted 1 and 2, which take the system reversibly from $A$ to $B$ . Along which path does the gas do more work? Along with path does more heat flow into the system?

Screenshot 2025-09-25 at 11.47.07 AM.png

Adiabatic and Isothermal processes

Two identical gases start initially at the same pressure and volume, and end at the same pressure. Assume one process is isothermal, and the other is adiabatic. Sketch these processes on a P-V diagram under two conditions: 1) the gases are compressed, and 2) the gases are expanded. What is the difference in final volume between the two gases in each of these scenarios? (c.f. Problem 1.38)