joule

SI coherent derived unit with special name and symbol

Name

Symbol

Derived quantity

Expressed in terms of SI base units

joule

energy, work,
amount of heat

kg m² s⁻²

Definition

The joule is the SI coherent derived unit of energy.

One joule is equal to the energy transferred to (or work done on) an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre.

The joule is named after the English physicist James Prescott Joule (1818 – 1889).

Mechanical energy

Energy, E, is transferred, or work is done, when a force, F, acts on an object in the direction of its motion through a distance, s.

Using SI coherent units,

$E = F \ s$

where:

energy, E, is measured in joules, symbol J,
force, F, is measured in newtons, symbol N,
distance, s, is measured in metres, symbol m.

$1 \ \text{J} = 1 \ \text{N m}$

Thus, one joule is equal to the energy transferred to an object when a force of one newton acts on that object in the direction of its motion through a distance of one metre.

Kinetic energy

The kinetic energy of an object is directly proportional to the product of its mass and the square of its velocity:

$E_k \propto mv^2$

Using SI coherent units, the proportionality constant is ¹⁄₂. Thus:

$E_k = \frac{1}{2} m v^2$

where:

E_k is the kinetic energy of the object in joules, symbol J,
m is the mass of the object in kilograms, symbol kg,
v is the velocity of the object measured in metres per second, symbol m s^-1.

1 J is equal to the energy required to accelerate a 1 kg mass at 1 m s^-2 through a distance of 1 m.

Gravitational potential energy

The gravitational potential energy of an object is the potential energy that it has when it is in a gravitational field. Gravitational potential energy, U, is dependent on the masses of two objects, m₁ and m₂, their distance apart, R, and the gravitational constant, G.

Using SI coherent units,

$U = -G \ \dfrac{m_1 m_2}{R}$

where:

U is the gravitational potential energy in joules, symbol J,
G is the gravitational constant, equal to 6.674 30(15) × 10⁻¹¹ m³ kg^-1 s^-2,
m₁ and m₂ are the masses of the two objects in kilograms, symbol kg,
R is the distance between the two objects in metres, symbol m.

For practical situations, close to the Earth’s surface, the gravitational field is considered to be constant. Gravitational potential energy is proportional to the mass of the object, the gravitational field strength, and the object’s height above the surface:

$U \propto mgh$

Using SI coherent units, the proportionality constant is 1. Thus:

$U = mgh$

where:

U is the potential gravitational energy in joules, symbol J,
m is the mass of the object in kilograms, symbol kg,
g is the acceleration due to gravity in metres per second squared, symbol m s^-2,
h is the height of the object in metres, symbol m.

Electric potential energy

The electric potential energy, U_E, of a point charge in the presence of another point charge in vacuum is dependent on the magnitudes of the two charges, q₁ and q₂, their distance apart, R, and the Coulomb constant, k_e.

Using SI coherent units,

$U_E = -k_e \ \dfrac{q_1 q_2}{R}$

where:

U_E is the electrostatic potential energy in joules, symbol J,
k_e is the Coulomb constant, equal to approximately 8.987 551 792 × 10⁹ N m² C^-2,
q₁ and q₂ are the magnitudes of the two charges in coulombs, symbol C,
R is the distance between the two charges in metres, symbol m.

Electrical energy

One joule is equal to the energy required to move an electric charge of one coulomb through an electrical potential difference of one volt, or one coulomb-volt (C V). This relationship can be used to define the volt.

Joule heating

One joule is equal to the energy dissipated as heat when an electric current of one ampere passes through a resistance of one ohm for one second.

Chemical energy

Chemical energy is the energy that is released or absorbed in a reaction between chemical substances. An endothermic reaction is one that absorbs energy. An exothermic reaction is one that releases energy.

When a chemical reaction occurs, the molecular bonds of the reactants are broken, and new bonds are formed to make the products.

Energy is always required to break a molecular bond, and energy is released when a bond is formed. This is known as bond energy. In a chemical reaction, the change in energy, or enthalpy, can be estimated from the difference in the bond energies of the reactants and products.

Molecular bond	Bond energy	Bond energy per mole
O=O	820 zJ	494 kJ/mol
H-H	717 zJ	432 kJ/mol
H-O	762 zJ	459 kJ/mol
C-H	686 zJ	413 kJ/mol
C-C	575 zJ	346 kJ/mol
C=C	1000 zJ	602 kJ/mol
C≡C	1387 zJ	835 kJ/mol

For example, when water is dissociated, the bonds of two water molecules are broken to form two molecules of hydrogen and one of oxygen:

$2\ \text{H}_2 \text{O} \rightarrow 2\ \text{H}_2 + \text{O}_2$

On the reactants side, 4 moles of O-H bonds are broken
= 4 × 459kJ = 1836 kJ energy absorbed
On the products side, 2 moles of H-H bonds, and 1 mole of O=O bonds are formed
= (2 × 432 kJ) + 494 kJ = 1358 kJ energy released

The total energy difference is 1836 kJ – 1358 kJ = 478 kJ, which indicates that the reaction is endothermic and that 478 kJ of enegy is needed to be supplied to carry out this reaction.

Food energy

Food energy is chemical energy derived from food through the metabolic process of cellular respiration. Carbohydrates, fats and proteins are all sources of food energy.

A minimum average amount of food energy is required to maintain human metabolism. The amount needed varies according to the amount of energy expended in physical activity. The recommended daily food energy intake in the UK is 8400 kJ. In Australia it is 8700 kJ.

A food energy intake of 8640 kJ/day corresponds to an energy intake rate of 100 J/s, or 100 W.

Mass-energy equivalence

A consequence of relativity theory is that any object that has a rest mass, a mass when stationary, also has an equivalent rest energy. The rest energy, E₀, of an object is directly proportional to its mass, m:

$E_0 \propto m$

Using SI coherent units, the proportionality constant is the speed of light in vacuum. Thus:

$E_0 = m c^2$

where:

rest energy, E₀, is measured in joules, symbol J,
mass, m, is measured in kilograms, symbol kg,
the speed of light, c, is in metres per second, symbol m s^-1.

The amount of energy, E, equivalent to 1 kg of matter can be calculated:

$E = 1 \ \text{kg} \times (299\ 792\ 458 \ \text{m s}^{-1})^2\\ \\ E = (299\ 792\ 458)^2 \ \text{kg m}^2{\text{ s}}^{-2}\\ \\ E = 8.987\ 551\ 787\ 368\ 176\ 4 \times 10^{16} \ \text{J}$

One kilogram of matter is equivalent to approximately 89.9 petajoules of energy.

Similarly, the mass, m, equivalent to one joule of energy, E, can be calculated:

$m = \dfrac{E}{c^2}\\ \\ m = \dfrac{1 \ \text{kg m}^2 \text{s}^{-2}} {(299\ 792\ 458\ \text{m s}^{-1})^2}\\ \\ \\ m = \dfrac{1}{(299\ 792\ 458)^2} \ \text{kg}\\ \\ \\ m = 1.112\ 650\ 056... \times 10^{-17} \ \text{kg}$

One joule of energy is equivalent to approximately 11.1 femtograms of mass.

Nuclear Binding Energy

The nucleus of an atom is made up of protons and neutrons, collectively called nucleons. The mass of a nucleus is always less than the sum of the individual masses of its nucleons. This difference is called the mass defect, and is equivalent to the nuclear binding energy which holds the nucleus together. The mass defect, Δm, is equal to the energy that would be released when forming the nucleus from it constituent nucleons. Nuclear binding energy can be calculated from the Einstein’s mass-energy equivalence relationship:

$E = \Delta m c^2$

For an alpha particle, consisting of 2 protons and 2 neutrons, Δm = 0.0505 yoctograms (0.0304 Da) which gives a binding energy of 4.53 × 10^-12 J (28.3 MeV). By comparison the binding energy of an electron in a hydrogen atom is 2.17 × 10^-18 J (13.6 eV).

Relation to power

Power is the rate at which energy is transferred. It is equal to the amount of energy transferred per unit time.

$P = \dfrac{E}{T}$

Inverting this relation gives an expression for energy in terms of power:

$E = P \ T$

Using SI coherent units,

power, P, is measured in watts, symbol W,
energy, E, is measured in joules, symbol J,
time, T, is measured in seconds, symbol s.

One joule is equal to the energy required to produce one watt of power for one second, or one watt second, symbol W s.

$1 \ \text{J} = 1 \ \text{W s}$