Notifications

Log in to see your notifications.

Heat

14 min read

Temperature, heat capacity and the kinetic theory

Temperature measures the average kinetic energy of the particles. The kelvin scale starts at absolute zero (−273 °C). Heat is energy transferred because of a temperature difference. The specific heat capacity c and the specific latent heats l relate heat to temperature change and change of state: Q = mcΔT and Q = ml.

Kinetic theory of an ideal gas: pressure comes from molecules hitting the walls. The mean translational kinetic energy per molecule is (3/2)kT, where k = 1.38 × 10⁻²³ J/K is the Boltzmann constant.

The first law of thermodynamics

First law of thermodynamics
The heat supplied to a system equals the increase in its internal energy plus the work done by the system: Q = ΔU + W.
Internal energy is the total kinetic and potential energy of the particles. For an ideal gas it depends only on temperature.
Pressure–volume diagramsA graph of pressure against volume for a gas. An isobaric process is a horizontal line at constant pressure. An isochoric process is a vertical line at constant volume. An isothermal process at constant temperature is a gentle curve, and an adiabatic process with no heat flow falls more steeply. The area under a curve is the work done by the gas.volume Vpressure Pisobaricisochoricisothermaladiabaticarea under the curve = work done
ProcessConstantResult
IsothermalTemperatureΔU = 0, so Q = W: all heat supplied is turned into work
IsobaricPressureGas expands and does work W = PΔV
Isochoric (isovolumetric)VolumeW = 0, so Q = ΔU
AdiabaticNo heat flow (Q = 0)ΔU = −W: a gas that expands does work and cools
Worked exampleUsing the first law

A gas absorbs 500 J of heat and expands, doing 200 J of work. By how much does its internal energy change?

  1. Use ΔU = Q − W.

Answer: The internal energy rises by 300 J.

Heat engines and efficiency

A heat engine takes heat Q_h from a hot reservoir, converts part of it to work W and rejects the rest Q_c to a cold reservoir. Its efficiency is W ÷ Q_h. No engine can be 100% efficient (the second law of thermodynamics). The most efficient possible engine between two temperatures is the Carnot engine.

Temperatures in kelvin. The bigger the temperature difference, the higher the possible efficiency.
Try itEfficiency of a heat engine

Choose the temperatures of the hot and cold reservoirs. Find the greatest possible (Carnot) efficiency.

Each cycle turns heat from burning fuel into work.

Heat transfer

Heat is transferred by conduction, convection and radiation. The rate of conduction through a slab is Q/t = kAΔT/d. A hot object radiates energy at a rate given by the Stefan–Boltzmann law: P = σAT⁴, with σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴. Doubling the absolute temperature increases the radiated power 16 times.

Check yourself In an adiabatic expansion of a gas…
Check yourself The greatest efficiency of an engine working between 600 K and 300 K is…
Check yourself At constant volume, heat supplied to a gas…

Key terms

Internal energy
The total kinetic and potential energy of the particles of a system.
Isothermal
At constant temperature.
Adiabatic
With no heat entering or leaving the system.
Efficiency
The fraction of the heat supplied that is converted to work.

Summary

  • Kinetic theory: PV = ⅓Nm⟨c²⟩; mean energy ½m⟨c²⟩ = 3kT/2.
  • First law: Q = ΔU + W.
  • Isothermal, isobaric, isochoric and adiabatic processes; work = area under the P–V curve.
  • Carnot efficiency = 1 − T_c ÷ T_h.
Properties of Matter Vibrations and Waves