Thermodynamics

What is Thermodynamics?

Thermodynamics is the branch of physics governing the relationships between heat, work, temperature, and energy. Its macroscopic laws describe which energy transformations are spontaneous and where hard limits exist — including the impossibility of perfect heat engines and perpetual motion machines. Applications span engines, refrigerators, chemical reactions, materials science, cosmology, and biological systems.

✦ Four Laws in One Sentence Each

Zeroth Law: if A is in thermal equilibrium with C and B is too, then A and B are in equilibrium — this defines temperature.

First Law: energy cannot be created or destroyed; ΔU = Q − W.

Second Law: entropy of an isolated system never decreases; spontaneous processes increase total entropy.

Third Law: entropy approaches a constant (usually zero) as temperature approaches absolute zero.

The Four Laws in Detail

⓪ Zeroth Law — Temperature Equilibrium

If two systems are each in thermal equilibrium with a third, they are in thermal equilibrium with each other. This law, added after the others, provides the conceptual foundation for the thermometer: temperature is a measurable, transitive property.

① First Law — Energy Conservation

The internal energy change ΔU of a closed system equals heat Q added to it, minus work W done by it: ΔU = Q − W . Energy may change form (heat ↔ work ↔ chemical energy) but the total is conserved. A perpetual motion machine of the first kind is impossible.

② Second Law — Entropy and Irreversibility

In any spontaneous process in an isolated system, entropy S either increases or stays constant (for reversible processes): ΔS ≥ 0 . Heat flows spontaneously from hot to cold, not the reverse. A perpetual motion machine of the second kind is impossible. The second law defines the "arrow of time."

③ Third Law — Absolute Zero

As temperature T → 0 K, the entropy of a perfect crystal approaches zero: S → 0 as T → 0 . It is impossible to reach absolute zero in a finite number of steps. This law underpins the absolute (Kelvin) temperature scale and explains why zero-point energy is non-zero in quantum mechanics.

Thermodynamic Processes

For an ideal gas undergoing adiabatic expansion, γ = C_p/C_v (heat capacity ratio, ~1.4 for diatomic gases).

Heat Engines and the Carnot Cycle

A heat engine converts heat into useful work by operating between a hot reservoir (T_H) and cold reservoir (T_C). The first law gives W = Q_H − Q_C. The thermal efficiency is:

Sadi Carnot (1824) proved that the most efficient possible heat engine operating between T_H and T_C is the Carnot engine , which uses two reversible isothermal and two adiabatic steps:

For a coal power station with T_H = 600 K (fire) and T_C = 300 K (atmosphere), η_Carnot = 50%. Real engines are less efficient due to friction, heat loss, and non-ideal working fluids. A 35–40% efficiency for modern steam turbines is typical.

η = W / Q_H = 1 − Q_C / Q_H

Entropy and the Second Law

Entropy is a state function measuring dispersal of energy or statistical disorder . For a reversible process:

For an irreversible process, dS > δQ/T. Total entropy of the universe always increases in spontaneous processes. This is why:

A broken egg doesn't spontaneously reassemble.

Heat flows from hot to cold, never the reverse.

Mixed gases don't spontaneously separate.

Living organisms maintain order locally by expelling entropy to their environment.

dS = δQ_rev / T (units: J/K)

Statistical Mechanics: Boltzmann's Contribution

Ludwig Boltzmann connected macroscopic entropy to microscopic statistics. Given Ω microstates corresponding to a macrostate:

This equation is engraved on Boltzmann's tombstone. It explains why entropy increases: there are astronomically more disordered microstates than ordered ones. For a system of just 100 gas molecules, if all 100 happen to be in the left half of a box, that is 1 microstate out of 2¹⁰⁰ ≈ 10³⁰ — statistically negligible. The Maxwell-Boltzmann distribution gives the probability of a particle having speed v:

This distribution predicts the spread of molecular speeds in a gas, successfully tested by molecular beam experiments.

S = k_B · ln Ω where k_B = 1.380649 × 10⁻²³ J/K (Boltzmann constant)

Free Energy: Gibbs and Helmholtz

Spontaneity under different conditions is captured by free energy functions:

The Gibbs free energy ΔG determines whether a chemical reaction proceeds spontaneously: ΔG = ΔH − TΔS. Reactions can be exothermic (ΔH < 0) but non-spontaneous if entropy decreases enough (TΔS is large and negative). Conversely, endothermic reactions can be spontaneous if entropy increases sufficiently — as in dissolving certain salts in water.

Helmholtz: A = U − TS (constant T, V; system work at const T) Gibbs: G = H − TS (constant T, P; most chemistry) where H = U + PV (enthalpy) Spontaneous process: ΔG < 0 (at constant T, P)

Phase Transitions

First-order transitions (melting, boiling) involve a latent heat L — energy that changes phase without changing temperature, explained by the disruption of intermolecular bonds:

Water's latent heat of vaporisation (2260 J/g at 100°C) is unusually high due to strong hydrogen bonding. Second-order (continuous) transitions (e.g., ferromagnetic ↔ paramagnetic at the Curie point, superconducting transitions) have no latent heat but exhibit discontinuities in heat capacity and are described by order parameters in modern statistical mechanics.

Q = mL (m = mass, L = specific latent heat)

Frequently Asked Questions

The Carnot theorem, derived from the second law of thermodynamics, proves that any heat engine must reject some heat to a cold reservoir: η = 1 − T_C/T_H. To achieve 100% efficiency, you would need T_C = 0 K (absolute zero), which the third law of thermodynamics prohibits reaching in finitely many steps. Even a perfect, frictionless engine is limited by this fundamental constraint. Real engines are less efficient still due to irreversibilities like friction, turbulence, and heat conduction through finite temperature differences.

Entropy has two complementary descriptions: (1) Macroscopic (Clausius): entropy measures the dispersal or degradation of energy — high entropy means energy is spread over many degrees of freedom and less available to do useful work. (2) Microscopic (Boltzmann): S = k·lnΩ, entropy counts the number of microscopic configurations (microstates) consistent with a macroscopic observation. Systems evolve toward higher entropy because there are overwhelmingly more disordered microstates than ordered ones. Entropy is not "disorder" in a colloquial sense — it's a precise measure of statistical weight of a macrostate.

Yes — the second law applies to isolated systems. Open systems can locally decrease entropy by exporting it to their surroundings. Your body maintains its highly ordered structure by metabolising food (consuming free energy) and releasing heat and waste (entropy) to the environment. The total entropy of the system + surroundings still increases. This is why life doesn't violate the second law: living organisms are open systems, continuously exchanging energy and matter with their environment. The misconception that evolution violates thermodynamics confuses local and global entropy.

Temperature is an intensive property measuring the average kinetic energy of particles in a system — it doesn't depend on system size. Heat is a form of energy transfer (measured in joules) between objects at different temperatures; it flows from high to low temperature until thermal equilibrium is reached. A large glass of water at 30°C contains far more thermal energy than a teaspoon at 60°C, even though the teaspoon is hotter. Temperature tells you the direction heat flows; the amount of heat transferred depends on temperature difference, thermal conductivity, specific heat capacity, and contact time.

Absolute zero (0 K = −273.15°C) is the temperature at which all classical thermal motion ceases and quantum zero-point energy is the only remaining kinetic energy. The third law of thermodynamics states it cannot be reached in a finite number of thermodynamic steps — as a system approaches 0 K, each successive cooling step requires more effort and has diminishing returns. Experimentally, scientists have cooled systems to within billionths of a kelvin (nanokelvin) using laser cooling, magnetic evaporative cooling, and adiabatic demagnetisation, but absolute zero remains a theoretical asymptote.

A refrigerator is a heat engine run in reverse: it uses work W (electrical input) to move heat Q_C from a cold reservoir (food compartment, T_C) to a hot reservoir (room, T_H). The first law gives Q_H = Q_C + W. The coefficient of performance (COP) for a refrigerator is COP = Q_C/W = T_C/(T_H − T_C) for a Carnot refrigerator. A fridge in a 25°C room with its interior at 4°C has COP_Carnot = 277/(298−277) ≈ 13 — meaning up to 13 joules of heat could be moved per joule of electrical work. Real fridges achieve COP of 2–5 due to irreversibilities.

Spontaneity at constant temperature and pressure is determined by the change in Gibbs free energy ΔG = ΔH − TΔS. A process is spontaneous if ΔG < 0. Both enthalpy (heat released/absorbed) and entropy (change in disorder) matter: exothermic reactions (ΔH < 0) tend toward spontaneity; reactions that increase entropy (ΔS > 0) also favour spontaneity. When ΔH and ΔS have opposing effects, temperature determines which wins — some reactions become spontaneous only above or below a certain temperature (where ΔG changes sign). The condition ΔG = 0 defines equilibrium.

Enthalpy H = U + PV is a convenient state function for constant-pressure processes — which includes virtually all chemistry in open laboratory vessels. At constant pressure, the heat exchanged Q_p equals ΔH exactly, simplifying calculations. Changes in PV work (due to volume changes during reactions) are automatically included. Standard enthalpies of formation, combustion, and reaction are tabulated for this reason. Internal energy U is more fundamental but impractical for bench chemistry since it requires accounting for PV work separately whenever volume changes.

The Maxwell-Boltzmann distribution describes the statistical distribution of molecular speeds in an ideal gas at thermal equilibrium. The probability of a molecule having speed v is: f(v) ∝ v²·exp(−mv²/2k_BT). Key features: there is a most probable speed (peak of distribution), a higher mean speed, and an even higher root-mean-square speed. The distribution broadens and shifts to higher speeds as temperature increases. It successfully predicts measurable properties like viscosity, thermal conductivity, diffusion rates, and chemical reaction rates (those molecules in the high-speed tail have enough energy to cross activation energy barriers — Arrhenius equation).

The second law applied to the entire universe suggests it is evolving toward a state of maximum entropy — the "heat death" scenario: all thermal gradients eventually disappear, temperature becomes uniform, no more work can be extracted, and all macroscopic structure dissolves. The Big Bang created an extraordinarily low-entropy state; the universe has been increasing in entropy ever since (this is why time has a direction). Cosmologically, gravity complicates entropy: forming a black hole enormously increases entropy (Bekenstein-Hawking entropy). Stephen Hawking calculated that black holes have entropy proportional to their horizon area, not volume — a major clue toward quantum gravity theories.

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