Thermodynamics & Statistical Mechanics

Key Concepts & Laws

The science of energy, heat, and the statistical behavior of many-particle systems.

Zeroth & First Law

Zeroth: Thermal equilibrium is transitive.
First law: $$\Delta U = Q - W$$ Internal energy is a state function; heat and work are not.

Ideal Gas Law

$$PV = nRT = Nk_B T$$ $R = 8.314\;\text{J/(mol}\!\cdot\!\text{K)}$
$k_B = 1.381 \times 10^{-23}\;\text{J/K}$

Thermodynamic Processes

Isothermal: $$W = nRT\ln\!\left(\frac{V_2}{V_1}\right)$$ Adiabatic: $PV^{\gamma} = \text{const}$
Isobaric: $W = P\Delta V$ Isochoric: $W = 0$

Second Law & Entropy

Entropy of universe never decreases: $\Delta S_{\text{univ}} \ge 0$
Reversible process: $dS = \frac{\delta Q_{\text{rev}}}{T}$
Boltzmann: $$S = k_B \ln \Omega$$

Heat Engines & Carnot Cycle

Efficiency: $$\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H}$$ Carnot (maximum): $$\eta_C = 1 - \frac{T_C}{T_H}$$ No engine is more efficient than Carnot.

Kinetic Theory

Average KE: $\tfrac{3}{2}k_B T$ per molecule
RMS speed: $$v_{\text{rms}} = \sqrt{\frac{3k_B T}{m}}$$ Pressure: $P = \tfrac{1}{3}\frac{N}{V}m\langle v^2 \rangle$

Phase Transitions

Latent heat: $Q = mL$
Clausius-Clapeyron: $$\frac{dP}{dT} = \frac{L}{T\,\Delta v}$$ Triple point of water: 273.16 K, 611.73 Pa

Boltzmann Distribution

Probability of state with energy $E$:
$$P(E) \propto e^{-E/(k_B T)}$$ Partition function: $Z = \sum e^{-E_i/(k_B T)}$

Problems

12 problems from ideal gases to statistical mechanics.

Easy

A sealed container holds 2 moles of an ideal gas at 300 K and 1 atm. The gas is heated at constant volume to 600 K. (a) What is the final pressure? (b) How much heat is added if $C_v = \tfrac{3}{2}R$? (c) What is the change in internal energy?

Show Hint

At constant volume, $\frac{P_1}{T_1} = \frac{P_2}{T_2}$. Heat added: $Q = nC_v \Delta T$. Since $W = 0$ (constant volume), $\Delta U = Q$.

Medium

One mole of an ideal monatomic gas undergoes an isothermal expansion at 400 K from volume $V_1 = 10\;\text{L}$ to $V_2 = 30\;\text{L}$. (a) Calculate the work done by the gas. (b) How much heat is absorbed? (c) What is the entropy change of the gas?

Show Hint

Isothermal: $\Delta U = 0$, so $Q = W = nRT\ln(V_2/V_1)$. Entropy change: $\Delta S = Q/T = nR\ln(V_2/V_1)$.

Medium

An ideal diatomic gas ($\gamma = 7/5$) at 300 K and 1 atm is compressed adiabatically to $1/8$ of its original volume. Find (a) the final temperature, (b) the final pressure, and (c) the work done on the gas per mole.

Show Hint

Adiabatic relations: $TV^{\gamma-1} = \text{const}$ and $PV^{\gamma} = \text{const}$. Work done on gas: $W = nC_v(T_2 - T_1) = \Delta U$ since $Q = 0$.

Hard

A Carnot engine operates between a hot reservoir at 800 K and a cold reservoir at 300 K. It absorbs 2000 J per cycle from the hot reservoir. (a) Find the efficiency. (b) Find the work output per cycle. (c) If the engine runs at 25 cycles per second, what is the power output? (d) How much entropy is produced per cycle?

Show Hint

$\eta = 1 - T_C/T_H$. Work: $W = \eta \cdot Q_H$. Power: $P = W \times \text{frequency}$. For a reversible Carnot engine, the total entropy production per cycle is zero — $Q_H/T_H = Q_C/T_C$.

Medium

A refrigerator removes 500 J of heat from a cold space at 260 K and expels heat to a room at 300 K. (a) What is the minimum work input required (Carnot COP)? (b) If the actual COP is 4.5, how much work is actually needed? (c) How much heat is expelled to the room?

Show Hint

Carnot COP (refrigerator): $\text{COP} = \frac{T_C}{T_H - T_C}$. Work: $W = \frac{Q_C}{\text{COP}}$. Heat expelled: $Q_H = Q_C + W$.

Hard

A 2 kg block of copper at 500 K is dropped into 5 kg of water at 290 K in an insulated container. Specific heats: copper $385\;\text{J/(kg}\!\cdot\!\text{K)}$, water $4186\;\text{J/(kg}\!\cdot\!\text{K)}$. (a) Find the final equilibrium temperature. (b) Calculate the entropy change of the copper, the water, and the universe. Verify that $\Delta S_{\text{univ}} > 0$.

Show Hint

Heat lost by copper = heat gained by water. For entropy: $\Delta S = mc\ln(T_f/T_i)$ for each substance (since specific heat is constant). The sum must be positive.

Easy

Calculate the RMS speed of nitrogen molecules ($\text{N}_2$, molar mass 28 g/mol) at room temperature (300 K). Compare this with the RMS speed of helium atoms (4 g/mol) at the same temperature. Why are there virtually no helium atoms in Earth's atmosphere?

Show Hint

$v_{\text{rms}} = \sqrt{3RT/M}$ where $M$ is molar mass in kg/mol. Compare with Earth's escape velocity (~11.2 km/s). The high-speed tail of the Maxwell-Boltzmann distribution matters here.

Hard

An Otto cycle (model for gasoline engines) has a compression ratio $r = 8$ and uses air ($\gamma = 1.4$) as the working substance. Starting at 300 K and 1 atm: (a) Find the temperature after adiabatic compression. (b) If 1800 J of heat is added at constant volume per mole, find the peak temperature. (c) Calculate the thermal efficiency.

Show Hint

Otto efficiency: $\eta = 1 - \frac{1}{r^{\gamma-1}}$. Temperature after compression: $T_2 = T_1 \cdot r^{\gamma-1}$. Peak temperature after heat addition: $T_3 = T_2 + \frac{Q}{nC_v}$.

Advanced

Using the Clausius-Clapeyron equation, estimate the boiling point of water at the top of Mount Everest (pressure ≈ 34 kPa). The enthalpy of vaporization of water is 40.7 kJ/mol, and normal boiling point is 373 K at 101.3 kPa.

Show Hint

Integrated form: $\ln\!\left(\frac{P_2}{P_1}\right) = -\frac{\Delta H_{\text{vap}}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)$. Solve for $T_2$. The answer should be around 70°C — this is why cooking at high altitude takes longer!

Advanced

Consider a two-state system where each particle can have energy 0 or $\varepsilon$. For $N = 100$ particles at temperature $T = \varepsilon/k_B$: (a) Find the partition function for a single particle. (b) Find the average energy per particle. (c) Find the heat capacity. (d) Sketch $C$ vs $T$ and explain the Schottky anomaly.

Show Hint

Single-particle $Z = 1 + e^{-\varepsilon/(k_B T)}$. Average energy: $\langle E \rangle = -\frac{\partial (\ln Z)}{\partial \beta}$ where $\beta = \frac{1}{k_B T}$. At $T = \varepsilon/k_B$, evaluate $e^{-1}$. The Schottky peak occurs when $k_B T \sim \varepsilon$.

Hard

500 g of ice at $-10$°C is added to 1 kg of water at 50°C in an insulated container. Specific heat of ice: $2090\;\text{J/(kg}\!\cdot\!\text{K)}$, water: $4186\;\text{J/(kg}\!\cdot\!\text{K)}$, latent heat of fusion: $334{,}000\;\text{J/kg}$. (a) Does all the ice melt? (b) Find the final equilibrium temperature. (c) Calculate the total entropy change.

Show Hint

First check if the heat available from cooling water to 0°C exceeds the heat needed to warm ice to 0°C and then melt it all. If yes, all ice melts and the final $T > 0$°C. Track entropy contributions from each stage.

Advanced

Thought Experiment: Maxwell's demon sits at a trapdoor between two chambers of gas at equal temperature. It opens the door only for fast molecules going right and slow molecules going left, seemingly decreasing entropy without work. (a) Explain why this doesn't violate the second law. (b) What is the minimum energy cost of erasing one bit of information (Landauer's principle)? (c) Calculate this energy at room temperature.

Show Hint

The demon must store information about each molecule's speed. Landauer's principle: erasing one bit of information dissipates at least $k_B T \ln 2$ of heat. At 300 K: $E_{\min} = 1.381 \times 10^{-23} \times 300 \times \ln 2 \approx 2.87 \times 10^{-21}\;\text{J}$.