Thermodynamics

Thermodynamics is the branch of physics that studies heat, work, temperature and the macroscopic behaviour of matter in terms of bulk variables such as pressure, volume and internal energy. It emerged in the 19th century from attempts to improve steam engines and reached maturity through the work of Carnot, Clausius, Kelvin, Boltzmann and Gibbs.

The four laws

1. Zeroth law — if two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. This law defines temperature.
2. First law — energy is conserved: ΔU = Q − W, where Q is heat added to the system and W is the work done by the system on its surroundings.
3. Second law — heat does not spontaneously flow from a colder to a hotter body; equivalently, the entropy of an isolated system never decreases (ΔS ≥ 0).
4. Third law — as the temperature approaches absolute zero, the entropy of a perfect crystal approaches a constant (zero, conventionally). Absolute zero is unreachable in a finite number of steps.

Thermodynamic processes

Process	Constant quantity	Work done
Isothermal	Temperature T	W = nRT ln(V₂/V₁)
Isobaric	Pressure P	W = P(V₂ − V₁)
Isochoric	Volume V	W = 0
Adiabatic	No heat exchange (Q = 0)	W = (P₁V₁ − P₂V₂)/(γ − 1)

In an adiabatic process for an ideal gas, PVγ = constant and TVγ⁻¹ = constant, where γ = Cₚ/Cᵥ is the adiabatic index (γ ≈ 1.4 for diatomic gases).

Heat engines and refrigerators

A heat engine operates in a cycle, absorbing heat Q_H from a hot reservoir at temperature T_H, doing work W, and rejecting heat Q_C to a cold reservoir at T_C. Its thermal efficiency is:

η = W/Q_H = 1 − Q_C/Q_H

The Carnot engine (Sadi Carnot, 1824) is the ideal reversible engine; its efficiency depends only on the reservoir temperatures:

η_Carnot = 1 − T_C/T_H

No heat engine operating between two reservoirs can be more efficient than a Carnot engine — this is Carnot's theorem.

Entropy and the arrow of time

Entropy S is a state function whose change in a reversible process is dS = dQ_rev/T. Clausius's inequality, dS ≥ dQ/T, implies that the entropy of an isolated system never decreases. Statistically, Boltzmann showed that

S = k_B ln Ω

where Ω is the number of microstates consistent with the macroscopic state and k_B = 1.381 × 10⁻²³ J/K is the Boltzmann constant. This connects thermodynamics to statistical mechanics and gives a microscopic origin to the second law: nature evolves toward configurations of higher multiplicity.

Thermodynamic potentials

Beyond internal energy U, four potentials are useful depending on which variables are held fixed:

• Enthalpy H = U + PV — useful at constant pressure (chemistry).
• Helmholtz free energy F = U − TS — minimum at equilibrium at constant T, V.
• Gibbs free energy G = H − TS — minimum at equilibrium at constant T, P; central to chemical equilibrium and phase transitions.
• A process is spontaneous at constant T and P if ΔG < 0.

Kinetic theory of gases

The kinetic theory derives ideal-gas behaviour from molecular motion. Key results:

• Average translational KE per molecule: ⟨½mv²⟩ = (3/2)k_B T.
• Root-mean-square speed: v_rms = √(3RT/M).
• Maxwell–Boltzmann distribution describes the spread of molecular speeds.
• Equipartition theorem: each quadratic degree of freedom contributes ½k_B T to the average energy.

Real-world applications

Thermodynamics governs power plants, refrigeration, climate science, biology, and even black-hole physics (Hawking's discovery that black holes have temperature and entropy). The Carnot limit sets an unbreakable ceiling on the efficiency of any thermal device — a reminder that the laws of thermodynamics are as fundamental as those of mechanics.