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Heat & Thermodynamics
Temperature, heat transfer, the laws of thermodynamics, engines and entropy. 62 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Thermodynamics studies heat, temperature and energy transfer. This area covers temperature scales and thermometers, thermal expansion, heat capacity and latent heat, the zeroth, first and second laws, and the efficiency of heat engines and refrigerators. Entropy and the idea of irreversibility explain why some processes happen in only one direction.
How the topics connect
Heat and thermodynamics explain how energy moves between objects and how it is converted into work. Temperature and heat are kept separate, the laws of thermodynamics set the limits on engines and refrigerators, and entropy gives the direction of natural change. Many later topics, from chemistry to climate, depend on these ideas.
Real-world uses
Calculate the maximum efficiency of a heat engine operating between two temperatures, and see why power plants lose so much energy as waste heat.
Understand how a refrigerator moves heat from a cold space to a warm one, and what work it requires.
Estimate heat loss through walls and windows, and the energy needed to heat a room or a building.
Relate specific heat capacity and latent heat to how fast water boils and how much energy an ice cube absorbs when melting.
Engineers use the second law and the Carnot limit to judge how efficient a real engine could ever be.
Architects choose insulation and glazing with conduction and radiation in mind, while climate scientists use radiative balance to study the Earth.
Temperature
A measure of the average kinetic energy of particles.
Temperature measures how hot something is: the average translational kinetic energy per particle, and the property that decides which way heat flows. Two bodies in contact exchange heat until their temperatures match.
Room temperature air has molecules averaging about 500 m/s.
Temperature isn't heat; a bathtub of warm water holds more heat than a boiling cup.
Temperature scales
Celsius, Kelvin and Fahrenheit.
K = °C + 273.15Celsius sets 0 and 100 at water's freezing and boiling points; Kelvin starts at absolute zero, K = °C + 273.15; Fahrenheit F = 1.8C + 32. Physics formulas need kelvin.
300 K = 26.85 °C = 80.3 °F.
A change of 1 K equals a change of 1 °C.
Thermometers
Devices using thermal expansion or electrical change.
Thermometers use a property that changes with temperature: liquid expansion, bimetal bending, electrical resistance (RTD, thermistor), thermocouple voltage or infrared emission.
Clinical thermometers use mercury-free alcohol or digital sensors.
Calibrate at two fixed points.
Zeroth law
Bodies in equilibrium with a third body are in equilibrium with each other.
If A is in thermal equilibrium with C and B is too, then A and B are in equilibrium with each other. This makes temperature a well-defined property and thermometers possible.
Two cups both matching a thermometer have the same temperature.
Equilibrium means no net heat flow.
Thermal expansion
Materials expand when heated.
ΔL = αLΔTMost materials expand when heated: ΔL = αLΔT, with α about 10⁻⁵ K⁻¹ for metals. Gaps are left in bridges and railway lines.
A 100 m steel rail (α = 12×10⁻⁶) heated by 30 K grows 3.6 cm.
Holes expand too, like the material around them.
Area and volume expansion
Coefficients are about 2α and 3α.
β = 2α, γ = 3αArea expands by β = 2α and volume by γ = 3α for isotropic solids. Liquids have a larger γ than solids.
A steel sheet heated 50 K: area +0.12%.
Use the right coefficient for area or volume.
Anomalous expansion of water
Water is densest at 4 °C.
From 0 to 4 °C water contracts on heating; density peaks at 4 °C. As a result ice floats and lakes freeze from the top, protecting aquatic life.
A pond at 4 °C at the bottom under ice.
Most liquids don't behave like this.
Heat
Energy in transit due to temperature difference.
Heat is energy transferred between bodies due to a temperature difference, measured in joules. A body has internal energy, not heat.
A hot pan loses heat to the air.
Don't say a body "contains heat".
Specific heat capacity
Heat needed per mass per degree.
Q = mcΔTThe energy needed to raise 1 kg by 1 K: Q = mcΔT. Water has a high c = 4186 J/kg·K, so it's a good coolant and moderates climate.
Heating 2 kg of water by 30 K: 251 kJ.
The substance and the units matter.
Molar heat capacity
Heat per mole per degree.
C = Q/nΔTThe heat for 1 mole per kelvin; C = Q/nΔT. For an ideal monatomic gas Cv = 3R/2 and Cp = 5R/2.
Helium at constant volume: 12.5 J/mol·K.
Differs at constant pressure and constant volume for gases.
Latent heat
Energy absorbed or released in a phase change.
Q = mLDuring melting or boiling temperature stays constant while energy changes the phase: Q = mL. Water: L_f = 334 kJ/kg and L_v = 2260 kJ/kg.
Boiling 1 kg of water needs 2.26 MJ, over 5 times the energy to heat it from 0 to 100 °C.
The heat goes into breaking bonds, not raising temperature.
Calorimetry
Heat lost equals heat gained in an isolated mixture.
In an insulated mix, heat lost by hot parts equals heat gained by cold parts: Σmc ΔT = 0. It lets you find specific heats or final temperatures.
200 g of water at 80 °C mixed with 300 g at 20 °C: T_f = 44 °C.
Remember to include the calorimeter's heat capacity.
Phase diagrams
Pressure–temperature maps of solid, liquid and gas regions.
P–T diagrams show regions of solid, liquid and gas separated by lines. Crossing a line changes phase; slopes tell how melting and boiling points shift with pressure.
Water's melting line slopes backward, which helps ice skating.
Phase boundaries are equilibrium lines.
Triple and critical points
Special conditions for phase coexistence and fluid merging.
At the triple point three phases coexist (water: 273.16 K, 611 Pa). Above the critical point (water: 374 °C, 22 MPa) liquid and gas become one supercritical fluid.
Supercritical CO₂ decaffeinates coffee.
Defines the Kelvin scale.
Conduction
Heat flow through a material by particle collisions.
Q/t = kAΔT/LHeat flows through a material from hot to cold by particle collisions and free electrons: Q/t = kAΔT/L. Metals have high k, gases low.
A 5 cm thick brick wall (k = 0.8) of 10 m² and 20 K difference: 3.2 kW.
Doubling thickness halves the rate.
Thermal resistance
Resistances add like resistors in series.
R = L/kAR = L/kA, and for layers in series the resistances add. Insulation aims for large R.
Double glazing traps a low-k air gap.
Heat flow = ΔT/R_total.
Convection
Heat carried by moving fluid.
Heat transported by bulk motion of a fluid: warm fluid rises, cool sinks. It gives breezes and ocean currents.
A radiator heats a room by convection currents.
Not possible in a solid or vacuum.
Radiation
Heat transfer by EM waves, needing no medium.
All bodies emit electromagnetic radiation depending on temperature, needing no medium. Dark, rough surfaces absorb and emit better than shiny ones.
Sunlight warms Earth through vacuum.
A hot object radiates even when surrounded by hotter air.
Stefan–Boltzmann law
Radiated power grows as the fourth power of temperature.
P = εσAT⁴The power radiated is P = εσAT⁴, with σ = 5.67×10⁻⁸ W/m²K⁴. Doubling T gives 16 times the power.
Sun's surface at 5800 K emits 64 MW/m².
Net radiation = σεA(T⁴ − T_s⁴).
Newton's law of cooling
Cooling rate is proportional to temperature difference.
dT/dt = −k(T−T₀)The cooling rate is proportional to the temperature difference with the surroundings: T(t) = T_s + (T₀ − T_s)e^{−kt}.
Coffee at 90 °C in a 20 °C room halves its excess in minutes.
Valid for small differences.
Wien's displacement law
Peak wavelength of blackbody radiation shifts with temperature.
λ_max T = 2.9×10⁻³ m·KThe peak wavelength of blackbody radiation is λ_max = b/T, with b = 2.898×10⁻³ m·K. Hotter objects glow bluer.
Sun: 500 nm; human body (310 K): 9.3 μm infrared.
Gives temperature of stars.
Blackbody radiation
An ideal absorber and emitter.
blackbodyAn ideal blackbody absorbs all radiation and emits a characteristic spectrum that depends only on temperature. Explaining it led Planck to quantum theory.
Stars and furnaces approximate blackbodies.
Real materials have emissivity less than 1.
Ideal gas laws
Boyle, Charles and Gay-Lussac laws relate P, V and T.
At low density gases follow simple laws linking pressure, volume and temperature, summarised by the ideal gas equation.
Aerosol cans explode when heated.
Valid for real gases at low pressure and high temperature.
Boyle's law
At fixed temperature, pressure is inversely proportional to volume.
PV = constAt constant temperature, PV = constant. Compressing a gas to half the volume doubles the pressure.
1 L at 1 atm squeezed to 0.25 L: 4 atm.
Needs absolute pressure.
Charles's law
At fixed pressure, volume is proportional to temperature.
V/T = constAt constant pressure, V/T = constant (T in kelvin). Volume goes to zero at −273 °C when extrapolated.
A balloon at 300 K heated to 330 K grows 10%.
Use kelvin, not °C.
Ideal gas equation
Equation of state for ideal gases.
PV = nRTPV = nRT = NkT, R = 8.314 J/mol·K. One mole at STP occupies 22.4 L.
n = 2 mol at 300 K in 10 L: P = 5 atm approximately.
Use SI units.
Avogadro's number
Number of particles in one mole.
N_A = 6.022×10²³N_A = 6.022×10²³ per mole: the number of particles in a mole. Equal volumes of gases at the same T and P contain equal numbers of molecules.
18 g of water has 6×10²³ molecules.
It's a count, so unitless apart from per mole.
Internal energy
Total microscopic energy of a system.
U is the sum of microscopic kinetic and potential energies. For an ideal gas it depends only on temperature: U = (f/2)nRT.
1 mol of a monatomic gas at 300 K: 3.7 kJ.
U is a state function.
First law of thermodynamics
Heat added equals change in internal energy plus work done.
ΔU = Q − WΔU = Q − W: heat added minus work done by the system equals the change in internal energy. It's conservation of energy.
A gas absorbs 500 J and does 200 J of work: ΔU = +300 J.
Be careful with the sign convention for work.
Isothermal process
Constant temperature; ΔU=0.
W = nRT ln(V₂/V₁)At constant temperature, ΔU = 0 for an ideal gas, so Q = W = nRT ln(V₂/V₁). On a PV diagram it's a hyperbola.
1 mol at 300 K doubling in volume does 1.73 kJ.
Needs slow contact with a heat reservoir.
Adiabatic process
No heat exchange.
PV^γ = constNo heat is exchanged: Q = 0, PV^γ = constant and TV^{γ−1} = constant. A gas cools when it expands and warms when compressed.
A bicycle pump heats up while pumping.
Faster than heat can flow.
Isobaric process
Constant pressure.
W = PΔVConstant pressure: W = PΔV and Q = nCpΔT. A horizontal line on a PV diagram.
Heating gas in a piston with a freely moving lid.
Most heat goes to expansion work for gases.
Isochoric process
Constant volume, no work.
Q = nC_vΔTConstant volume: no work is done, so Q = ΔU = nCvΔT. A vertical line on the PV diagram.
Heating gas in a sealed rigid can.
Pressure rises with temperature.
Cyclic process
Returns to initial state, net ΔU=0.
The system returns to its initial state: ΔU = 0, so net heat equals net work. The area enclosed in the PV diagram is the net work.
Engines and refrigerators run in cycles.
Clockwise cycles produce work.
Heat capacities C_p and C_v
Cp exceeds Cv by R for ideal gases.
C_p − C_v = RFor an ideal gas C_p − C_v = R, since at constant pressure extra heat goes into work. Cv depends on degrees of freedom.
Air: Cv = 20.8, Cp = 29.1 J/mol·K.
Cp is always greater than Cv.
Ratio of specific heats
Adiabatic index.
γ = C_p/C_vγ = Cp/Cv: 1.67 for monatomic, 1.4 for diatomic, 1.33 for polyatomic gases. It sets adiabatic curves and sound speed.
Air: γ = 1.4.
γ depends on molecular structure.
P–V diagrams
Work equals the area under the curve.
carnotA PV diagram shows the thermodynamic path; the work done by the gas is the area under the curve; direction matters.
Expansion along a higher path does more work than along a lower one.
Work depends on the path, not just the endpoints.
Second law of thermodynamics
Heat does not flow spontaneously from cold to hot.
Heat doesn't flow spontaneously from a colder to a hotter body, and no cyclic engine can turn all heat into work. Entropy of an isolated system never decreases.
An ice cube melts in warm water but never regrows spontaneously.
Doesn't forbid local entropy decrease with external work.
Kelvin–Planck statement
No engine converts heat entirely into work.
It is impossible for a cyclic device to take heat from one reservoir and convert it completely to work. Some heat must be rejected to a colder reservoir.
A car engine always wastes heat through the exhaust.
Equivalent to Clausius' statement.
Clausius statement
No process only transfers heat from cold to hot.
It is impossible for a cyclic process to transfer heat from a colder body to a hotter body without work input. Refrigerators need electricity.
An air conditioner uses power to move heat outdoors.
Equivalent to the Kelvin–Planck form.
Heat engines
Convert heat to work in a cycle.
η = W/Q_hA heat engine absorbs Q_h from a hot reservoir, converts part to work W and rejects Q_c: η = W/Q_h = 1 − Q_c/Q_h.
A power plant takes 1000 MJ and outputs 350 MJ electricity: η = 35%.
Efficiency is limited by temperatures.
Carnot engine
The most efficient engine between two temperatures.
η = 1 − T_c/T_hThe most efficient reversible engine between T_h and T_c has η = 1 − T_c/T_h. No real engine can exceed it.
Between 600 K and 300 K: 50%.
Use kelvin; reaching 100% needs T_c = 0.
Carnot cycle
Two isothermal and two adiabatic steps.
carnotTwo isothermal and two adiabatic reversible steps: isothermal expansion at T_h, adiabatic expansion, isothermal compression at T_c, adiabatic compression.
Explore it in the Carnot lab.
Reversible processes are idealisations.
Refrigerators and heat pumps
Use work to move heat from cold to hot.
COP = Q_c/WThey use work to move heat from cold to hot. COP (fridge) = Q_c/W; for a heat pump COP = Q_h/W. Ideal values: T_c/(T_h − T_c) and T_h/(T_h − T_c).
A heat pump with COP 4 gives 4 kWh heat for 1 kWh electricity.
COP can exceed 1 because heat is moved, not created.
Otto cycle
Idealised petrol engine cycle.
η = 1 − r^{1−γ}Idealised petrol engine: adiabatic compression, constant-volume heating (spark), adiabatic expansion and constant-volume cooling. η = 1 − r^{1−γ} with compression ratio r.
r = 9, γ = 1.4: η = 58%.
Real engines achieve about 30%.
Diesel cycle
Idealised compression-ignition engine.
Compression-ignition engine: adiabatic compression, constant-pressure combustion, adiabatic expansion, constant-volume exhaust. Higher compression ratios give higher efficiency than petrol.
Diesel engines reach about 40%.
Air only compressed, fuel injected later.
Entropy
A measure of disorder or the number of microstates.
dS = δQ/TEntropy S measures disorder or the number of ways to arrange microscopic states; for reversible heat transfer dS = δQ/T. It allows us to state the second law quantitatively.
Melting 1 kg of ice at 273 K: ΔS = 334000/273 = 1224 J/K.
Entropy isn't "chaos", it's a measure of accessible states.
Entropy change
Entropy of the universe never decreases.
ΔS ≥ 0For heat flow Q from hot T_h to cold T_c: ΔS = Q/T_c − Q/T_h > 0. Total entropy increases in any real process.
1000 J from 400 K to 300 K: +0.83 J/K.
Entropy of a part can decrease if the surroundings gain more.
Statistical entropy
Boltzmann's formula links entropy to microstates.
S = k ln ΩS = k_B ln Ω, with Ω the number of microstates. More ways to arrange the particles means higher entropy.
Gas expanding to double the volume: ΔS = Nk ln 2.
Microstates must be counted correctly (indistinguishability).
Free energy
Helmholtz and Gibbs energies predict spontaneity.
G = H − TSGibbs free energy G = H − TS decides spontaneity at constant T and P: processes with ΔG < 0 happen. Helmholtz F = U − TS applies at constant volume.
Ice melts above 0 °C where TΔS beats ΔH.
Equilibrium has ΔG = 0.
Third law
Entropy approaches a constant as T approaches zero.
As temperature approaches absolute zero, the entropy of a perfect crystal approaches zero, and absolute zero cannot be reached in a finite number of steps.
Experiments have reached nanokelvin.
Heat capacities vanish at T → 0.
Maxwell relations
Equalities from the exactness of thermodynamic potentials.
Equalities among partial derivatives of thermodynamic potentials, such as (∂T/∂V)_S = −(∂P/∂S)_V, allow us to relate hard-to-measure quantities to measurable ones.
Find entropy changes from P–V–T data.
They follow from the exactness of state functions.
Clausius–Clapeyron equation
Slope of the phase boundary.
dP/dT = L/TΔvThe slope of the coexistence curve dP/dT = L/(TΔv) relates latent heat to boiling-point changes. Higher altitudes lower the boiling point.
Water boils at 90 °C at 3000 m.
For ice, Δv is negative so the slope is negative.
Joule–Thomson effect
Temperature change when a gas expands through a valve.
A real gas expanding through a porous plug changes temperature at constant enthalpy; below the inversion temperature it cools. It's used for liquefying gases.
Refrigeration and liquid nitrogen production.
An ideal gas doesn't change temperature.
Heat death of the universe
Maximum entropy end state in some cosmologies.
If the universe expands forever, entropy increases until all energy is evenly spread and no work can be extracted. This ultimate equilibrium is the heat death.
Stars burn out in 10¹⁴ years.
It depends on cosmological assumptions.
Arrow of time
The second law gives time its direction.
The microscopic laws are time-symmetric, but the second law gives a direction to macroscopic processes because the early universe had very low entropy.
Broken eggs don't reassemble.
The explanation lies in initial conditions.
Humidity and dew point
Moisture content of air and condensation temperature.
Relative humidity is the ratio of water vapour pressure to saturated vapour pressure. The dew point is the temperature at which air becomes saturated and condenses.
At 25 °C and 60% RH, dew point ≈ 17 °C.
Cool surfaces below the dew point get wet.
Greenhouse effect
Atmospheric gases trap outgoing heat radiation.
Atmospheric gases (CO₂, CH₄, H₂O) transmit visible sunlight but absorb outgoing infrared, warming Earth from −18 °C to +15 °C. Increased concentrations strengthen it.
Venus's thick CO₂ atmosphere gives 460 °C.
The natural effect is essential for life.
Thermal equilibrium in the Sun-Earth system
Earth's temperature is set by radiative balance.
Earth's temperature is set when absorbed solar power equals emitted infrared: S(1 − A)πR² = 4πR²σT⁴, giving T ≈ 255 K without an atmosphere.
Albedo A ≈ 0.3.
Atmosphere adds about 33 K.
Thermal conductivity of materials
Metals conduct well; insulators poorly.
Metals (copper 400 W/m·K) conduct well; plastics and wood (0.1–0.2) and air (0.025) insulate. Metal feels colder than wood because it conducts heat from your hand faster.
Copper pans heat evenly.
Feeling of cold depends on conductivity, not temperature.
Insulation and R-value
Thicker low-k layers reduce heat loss.
R-value (thickness/conductivity) measures resistance to heat flow; layers add. Trapped air is the key to most insulators.
R-20 wall loses heat twice as slowly as R-10.
Gaps and thermal bridges reduce performance.
Heat transfer in everyday life
Thermos flasks, cooking and clothing.
Vacuum flasks reduce all three modes: vacuum stops conduction and convection and a silvered coating reflects radiation. Cooking uses conduction (pan), convection (oven) and radiation (grill).
Thermos keeps tea hot for hours.
Wearing layers traps air.
Frequently asked questions
What is the difference between heat and temperature?
Temperature measures the average kinetic energy of the particles in a substance. Heat is energy transferred because of a temperature difference. A bathtub holds more heat than a cup of boiling water, even though the water is cooler.
Why is no heat engine 100% efficient?
The second law of thermodynamics says some heat must always be rejected to a cold reservoir. The Carnot efficiency, 1 minus Tcold over Thot, gives the upper limit for any engine using those two temperatures.
What is entropy in simple terms?
Entropy is a measure of how many microscopic arrangements correspond to the same macroscopic state. Systems tend to move towards states with more arrangements, which is why heat flows from hot to cold and not the reverse.
Why can't a heat engine be 100 percent efficient?
The second law says some heat must always be rejected to a cold reservoir. Only an engine that rejected no heat could be perfectly efficient, which is not possible.
Is entropy the same as disorder?
It is often described that way, but more precisely it counts the number of microscopic arrangements compatible with the same overall state.
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