[3] The notion has been introduced to describe the systems characterized by temperature variations, temperature being added to the set of state parameters, the position variables known in mechanics (and their conjugated generalized force parameters), in a similar way to potential energy of the conservative fields of force, gravitational and electrostatic.

[4] These processes are measured by changes in the system's properties, such as temperature, entropy, volume, electric polarization, and molar constitution.

The corresponding quantity relative to the amount of substance with unit J/mol is the molar internal energy.

[6] The internal energy of a system depends on its entropy S, its volume V and its number of massive particles: U(S,V,{Nj}).

Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, S(U,V,{Nj}), of the same list of extensive variables of state, except that the entropy, S, is replaced in the list by the internal energy, U.

In contrast, Legendre transformations are necessary to derive fundamental equations for other thermodynamic potentials and Massieu functions.

It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.

[8][10][11] For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle.

, components: The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles.

The microscopic potential energy algebraic summative components are those of the chemical and nuclear particle bonds, and the physical force fields within the system, such as due to internal induced electric or magnetic dipole moment, as well as the energy of deformation of solids (stress-strain).

Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.

That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external gravitational, electrostatic, or electromagnetic fields.

In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter.

Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.

The internal energy is an extensive property: it depends on the size of the system, or on the amount of substance it contains.

Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system.

In a system that is in thermodynamic contact equilibrium with a heat reservoir, each microstate has an energy

Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as an approximation for working systems.

Monatomic particles do not possess rotational or vibrational degrees of freedom, and are not electronically excited to higher energies except at very high temperatures.

The internal energy of an ideal gas is proportional to its amount of substance (number of moles)

The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties

[16]: 33  For a closed system, with transfers only as heat and work, the change in the internal energy is expressing the first law of thermodynamics.

, to be energy transfer from the working system to the surroundings, indicated by a positive term.

, i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature.

When considering fluids or solids, an expression in terms of the temperature and pressure is usually more useful: where it is assumed that the heat capacity at constant pressure is related to the heat capacity at constant volume according to The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the coefficient of thermal expansion and the isothermal compressibility by writing and equating dV to zero and solving for the ratio dP/dT.

, the chemical potentials are intensive properties, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent.

For a single component system, the chemical potential equals the Gibbs energy per amount of substance, i.e. particles or moles according to the original definition of the unit for

In Einstein notation for tensors, with summation over repeated indices, for unit volume, the infinitesimal statement is Euler's theorem yields for the internal energy:[18] For a linearly elastic material, the stress is related to the strain by where the

Elastic deformations, such as sound, passing through a body, or other forms of macroscopic internal agitation or turbulent motion create states when the system is not in thermodynamic equilibrium.

Expressed in modern units, he found that c. 4186 joules of energy were needed to raise the temperature of one kilogram of water by one degree Celsius.