PhysicsDifficulty 2-5

Thermodynamics

Thermodynamics is the physics of heat, temperature and energy transfer. Temperature measures the average kinetic energy of particles; heat is energy that flows because of a temperature difference, always spontaneously from hot to cold. Its laws state that internal energy changes by heat added minus work done (ΔE = Q − W), that the entropy of a closed system never decreases, and that no heat engine can beat the efficiency of a reversible (Carnot) engine.

Engines, Efficiency and Entropy

A heat engine takes heat from a hot reservoir, does work W and rejects the rest to a cold reservoir; its efficiency is the work done divided by the heat taken in. Carnot's principle: no engine between two fixed temperatures beats a reversible one. For the ideal-gas Carnot cycle the ratio of heat rejected to heat taken in equals the ratio of the cold to the hot reservoir temperature.

e = \dfrac{W}{Q_h} = 1 - \dfrac{Q_c}{Q_h} \;\le\; 1 - \dfrac{T_c}{T_h}

Engine efficiency and the Carnot upper bound (temperatures in kelvins).

Between 600 K and 300 K the bound is 50 %. Entropy makes the second law quantitative: for reversible heat transfer dS = dQ/T, so 1200 J added reversibly at 300 K raises entropy by 4.0 J/K. For any process the total entropy of a closed system never decreases.

\Delta S = \int \dfrac{dQ_{\text{rev}}}{T}, \qquad \Delta S_{\text{universe}} \ge 0

Entropy change along a reversible path, and the entropy statement of the second law.

Info: The first law cannot rule out a 100 %-efficient engine — energy would still be conserved. It is the second law that forbids it.

Full explanation — the complete reference version every reading depth is based on

Temperature and heat are different things

Temperature tells you how hot something is; at the particle level it is proportional to the average kinetic energy of the particles' random motion. Heat is energy on the move: the energy that flows from one object to another because their temperatures differ. A full bath and a cup of the same warm water are at the same temperature, yet the bath holds far more internal energy because it contains far more particles.

Zeroth law: two objects each in thermal equilibrium with a third are in equilibrium with each other — the reason thermometers work.
The SI unit of temperature is the kelvin (K). 0 K is absolute zero; water freezes at 273.15 K and boils at 373.15 K at standard pressure.
T(K) = T(°C) + 273.15, and a change of 1 K equals a change of 1 °C.
The kelvin is defined by fixing the Boltzmann constant at exactly 1.380649 × 10⁻²³ J K⁻¹.

Heating things up

Q = mc\,\Delta T

Heat Q needed to change the temperature of mass m by ΔT; c is the specific heat (water: about 4186 J/(kg·K)).

Worked example: warming water and iron

Worked example (our calculation): warming 1.0 kg of water by 10 K needs Q = 1.0 × 4186 × 10 ≈ 41,900 J, about 42 kJ. Iron's specific heat is about 452 J/(kg·K), so the same heat would warm 1.0 kg of iron by roughly nine times as much.

Q = mcΔT stops applying during a change of state. While ice melts at 0 °C its temperature stays constant even though it keeps absorbing heat — about 333 kJ for every kilogram — because that energy goes into breaking apart the attractive forces between the molecules rather than speeding them up.

Conduction: heat passes through stationary matter by contact — from a stove burner through the base of a pan.
Convection: heat is carried by the bulk movement of a fluid — warm air circulating in a room, or weather systems.
Radiation: heat is carried by electromagnetic radiation such as infrared and light — the Sun warming Earth across empty space.

The first law: energy is conserved

\Delta E_{\text{int}} = Q - W

Change in internal energy = heat added to the system − work done by the system.

If 500 J of heat flows into a gas while the gas does 200 J of work pushing a piston out, its internal energy rises by 500 − 200 = 300 J. The first law is the law of conservation of energy applied to heat and work, so an isolated system's internal energy stays constant.

The second law: the direction of change

Clausius: heat never flows spontaneously from a colder object to a hotter one. A fridge can move heat 'uphill' only with work from outside.
Kelvin: no engine can turn heat from a single source entirely into work with no other effect — some heat must always be released to a colder place.
Entropy: for heat Q added reversibly at temperature T, ΔS = Q/T. The entropy of a closed system, or of the universe, never decreases.
Statistically, entropy measures disorder: a shuffled deck of cards has higher entropy than a sorted one.

e_{\text{Carnot}} = 1 - \dfrac{T_c}{T_h}

The maximum possible efficiency of a heat engine working between a hot reservoir and a cold reservoir, with both temperatures in kelvins.

Worked example: an ideal engine between 500 K and 300 K can convert at most 1 − 300/500 = 0.40, or 40 %, of the heat it takes in into work. Real engines do worse; none can do better.

Third law: absolute zero cannot be reached in any finite number of cooling steps. Laboratories have pushed ever closer — a temperature of about 1 × 10⁻¹⁰ K was reported at the Helsinki University of Technology in 2008 — but never to 0 K itself.

Where thermodynamics connects

Thermodynamics extends Energy conservation to heat. Electricity heats resistors (P = I²R); radiation links it to Light; and the motion of particles links it back to Motion and Momentum through the kinetic theory of gases.

How we know

Thermodynamics grew from experiments on heat and work. Measuring the work needed to produce the same warming as a given amount of heat established the mechanical equivalent of heat (1 kcal = 4186 J), showing that heat is energy in transit. The Clausius statement of the second law summarises all the experiments ever done on spontaneous heat transfer: none has ever shown heat flowing by itself from cold to hot.

Common misconception: Misconception: 'cold' flows into things. There is no flow of cold — energy flows as heat from the hotter object to the colder one. An ice cube feels cold because heat flows out of your hand into it.

Common misconception: Misconception: heat and temperature are the same. Temperature is a property of an object; heat is energy transferred between objects. Two objects at the same temperature exchange no net heat however much internal energy each holds.

Info: Assumptions and limits: Q = mcΔT treats the specific heat as constant, which is only approximately true over a modest temperature range, and it fails during a change of state. The Carnot limit applies to reservoirs at fixed temperatures and needs kelvins. The laws of thermodynamics describe large collections of particles; 'temperature' and 'entropy' as used here are averages over enormous numbers of them.

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Sources and methodology

The zeroth law of thermodynamics states that if object A is in thermal equilibrium with object B, and B is in thermal equilibrium with C, then A is in thermal equilibrium with C. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.2: Temperature and Thermal Equilibrium — Textbook
Temperature is proportional to the average translational kinetic energy of the particles in a substance. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.2: Temperature and Thermal Equilibrium — Textbook
Heat is energy transferred spontaneously from one object to another because of a difference in temperature. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.5: Heat Transfer, Specific Heat, and Calorimetry — Textbook
On the Kelvin scale, 0 K is absolute zero and water freezes at 273.15 K and boils at 373.15 K (at standard pressure), so T(K) = T(°C) + 273.15 and a temperature change is the same size in kelvins and degrees Celsius. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.3: Thermometers and Temperature Scales — Textbook
The SI defines the kelvin by fixing the Boltzmann constant at exactly 1.380649 × 10⁻²³ J K⁻¹. (awaiting scientific review)
- SI base unit: kelvin (K) — Government or standards body
The heat needed to change the temperature of a mass m is Q = mcΔT, where c is the specific heat; for water c is about 4186 J/(kg·K) at 15 °C. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.5: Heat Transfer, Specific Heat, and Calorimetry — Textbook
The specific heat of iron and steel is about 452 J/(kg·K), roughly a ninth of water's 4186 J/(kg·K) (the ratio is our calculation). (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.5: Heat Transfer, Specific Heat, and Calorimetry — Textbook
During a phase change such as melting or boiling the temperature stays constant while heat is absorbed; melting ice at 0 °C absorbs about 333 kJ of heat per kilogram. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.6: Phase Changes — Textbook
Heat is transferred in only three ways: conduction through stationary matter by physical contact, convection by the bulk movement of a fluid, and radiation by the emission or absorption of electromagnetic radiation. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.7: Mechanisms of Heat Transfer — Textbook
The first law of thermodynamics, ΔEᵢₙₜ = Q − W, states that a system's internal energy changes by the heat added to it minus the work it does; it is a statement of energy conservation. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 3.4: First Law of Thermodynamics — Textbook
The Clausius statement of the second law of thermodynamics is that heat never flows spontaneously from a colder object to a hotter object. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.2: Reversible and Irreversible Processes — Textbook
The Kelvin statement of the second law of thermodynamics is that it is impossible to convert the heat from a single source into work without any other effect. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.5: Statements of the Second Law of Thermodynamics — Textbook
No engine working between two reservoirs at constant temperatures can be more efficient than a reversible engine; the ideal-gas Carnot engine's efficiency is e = 1 − T(cold)/T(hot), with both reservoir temperatures in kelvins. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.6: The Carnot Cycle — Textbook
For a reversible process at constant temperature T the entropy change is ΔS = Q/T, and the entropy of a closed system, or of the universe, never decreases. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.7: Entropy — Textbook
Statistically, entropy measures how disordered or randomised a system is: the more disordered, the higher its entropy. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.8: Entropy on a Microscopic Scale — Textbook
A temperature of about 1 × 10⁻¹⁰ K was reached in a low-temperature laboratory at the Helsinki University of Technology in 2008, cited in the source textbook as the lowest temperature achieved. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.8: Entropy on a Microscopic Scale — Textbook
The third law of thermodynamics states that absolute zero cannot be reached in any finite number of cooling steps. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.8: Entropy on a Microscopic Scale — Textbook
Joule's experiments established the mechanical equivalent of heat — the work needed to produce the same effects as heat transfer — as 1.000 kcal = 4186 J, showing that heat is a form of energy. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 1.5: Heat Transfer, Specific Heat, and Calorimetry — Textbook
In a textbook example, when two 4.00 kg blocks at 400 K and 300 K reach 350 K, the hotter block's entropy falls by 240 J/K and the cooler block's rises by 277 J/K, so the total increases by 37 J/K. (awaiting scientific review)
- University Physics Volume 2 (OpenStax), 4.8: Entropy on a Microscopic Scale — Textbook
The Boltzmann constant k is exactly 1.380649 × 10⁻²³ J K⁻¹ (CODATA 2022 exact value). (awaiting scientific review)
- CODATA value: Boltzmann constant k — Government or standards body

Claims marked “awaiting scientific review” cite the sources listed but have not yet been signed off by a scientific reviewer.

Content status: published 1 October 2026.

Scientific review: this version has not yet been signed off by a scientific reviewer.
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