The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its Standard state internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. It excludes the kinetic energy of motion and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields, but it includes the thermal energy (i.e. internal kinetic energy). The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.
The internal energy cannot be measured absolutely. Thermodynamics concerns changes in the internal energy, not its absolute value. The processes that change the internal energy are transfers, into or out of the system, of matter, or of energy, as heat, or by thermodynamic work.Max Born (1949), Appendix 8, pp. 146–149. These processes are measured by changes in the system's properties, such as temperature, entropy, volume, electric polarization, and molar constitution. The internal energy depends only on the internal state of the system and not on the particular choice from many possible processes by which energy may pass into or out of the system. It is a State function, a thermodynamic potential, and an extensive property.
Thermodynamics defines internal energy macroscopically, for the body as a whole. In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, , and oscillation, and of the potential energies associated with microscopic forces, including chemical bonds.
The unit of energy in the International System of Units (SI) is the joule (J). The internal energy relative to the mass with unit J/kg is the specific internal energy. The corresponding quantity relative to the amount of substance with unit J/mol is the molar internal energy.
Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example , that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, for , to get .
In contrast, Legendre transforms are necessary to derive fundamental equations for other thermodynamic potentials and . The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation of 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.Münster, A. (1970), Chapter 3.Bailyn, M. (1994), pp. 206–209.
For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics.
The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the centerofmass 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 energy and nuclear particle bonds, and the physical force fields within the system, such as due to internal induced electric or magnetism dipole moment, as well as the energy of deformation of solids (stressstrain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.
Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external , electrostatics, or electromagnetics fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the object with the field. In such a case, the field is included in the thermodynamic description of the object in the form of an additional external parameter.
For practical considerations in thermodynamics or engineering, it is rarely necessary, convenient, nor even possible, to consider all energies belonging to the total intrinsic energy of a sample system, such as the energy given by the equivalence of mass. Typically, descriptions only include components relevant to the system under study. Indeed, in most systems under consideration, especially through thermodynamics, it is impossible to calculate the total internal energy.I. Klotz, R. Rosenberg, Chemical Thermodynamics  Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 Therefore, a convenient null reference point may be chosen for the 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.
At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (cf. table). In the classical picture of thermodynamics, kinetic energy vanishes at zero temperature and the internal energy is purely potential energy. However, quantum mechanics has demonstrated that even at zero temperature particles maintain a residual energy of motion, the zero point energy. A system at absolute zero is merely in its quantummechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable entropy.
The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics relates the pseudorandom kinetic energy of individual particles to the mean kinetic energy of the entire ensemble of particles comprising a system. Furthermore, it relates the mean microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. While temperature is an intensive measure, this energy expresses the concept as an extensive property of the system, often referred to as the thermal energy,Leland, T. W. Jr., Mansoori, G. A., pp. 15, 16. Thermal energy – Hyperphysics. The scaling property between temperature and thermal energy is the entropy change of the system.
Statistical mechanics considers any system to be statistically distributed across an ensemble of $N$ microstates. In a system that is in thermodynamic contact equilibrium with a heat reservoir, each microstate has an energy $E\_i$ and is associated with a probability $p\_i$. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence:
For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer $Q$ and due to thermodynamic work $W$ done by the system on its surroundings.This article uses the sign convention of the mechanical work as often defined in engineering, which is different from the convention used in physics and chemistry; in engineering, work performed by the system against the environment, e.g., a system expansion, is taken to be positive, while in physics and chemistry, it is taken to be negative. Accordingly, the internal energy change $\backslash Delta\; U$ for a process may be written $$\backslash Delta\; U\; =\; Q\; \; W\; \backslash quad\; \backslash text\{(closed\; system,\; no\; transfer\; of\; matter)\}.$$
When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be Sensible heat.
A second kind of mechanism of change in the internal energy of a closed system changed is in its doing of work on its surroundings. Such work may be simply mechanical, as when the system expands to drive a piston, or, for example, when the system changes its electric polarization so as to drive a change in the electric field in the surroundings.
If the system is not closed, the third mechanism that can increase the internal energy is transfer of matter into the system. This increase, $\backslash Delta\; U\_\backslash mathrm\{matter\}$ cannot be split into heat and work components. If the system is so set up physically that heat transfer and work that it does are by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy: $$\backslash Delta\; U\; =\; Q\; \; W\; +\; \backslash Delta\; U\_\backslash text\{matter\}\; \backslash quad\; \backslash text\{(matter\; transfer\; pathway\; separate\; from\; heat\; and\; work\; transfer\; pathways)\}.$$
If a system undergoes certain phase transformations while being heated, such as melting and vaporization, it may be observed that the temperature of the system does not change until the entire sample has completed the transformation. The energy introduced into the system while the temperature does not change is called latent energy or latent heat, in contrast to sensible heat, which is associated with temperature change.
Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): $U\; =\; U(n,T)$. It is not dependent on other thermodynamic quantities such as pressure or density.
The internal energy of an ideal gas is proportional to its mass (number of moles) $n$ and to its temperature $T$
where $C\_V$ is the isochoric (at constant volume) molar heat capacity of the gas. $C\_V$ is constant for an ideal gas. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties $S$, $V$, $n$ (entropy, volume, mass). In case of the ideal gas it is in the following way
where $\backslash mathrm\; \{const\}$ is an arbitrary positive constant and where $R$ is the Gas constant. It is easily seen that $U$ is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex function. Knowing temperature and pressure to be the derivatives $T\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; S\},$ $P\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\},$ the ideal gas law $PV\; =\; nRT$ immediately follows as below:
This relationship may be expressed in infinitesimal terms using the differentials of each term, though only the internal energy is an exact differential.
For a closed system, with transfers only as heat and work, the change in the internal energy isFor example, the mechanical work done by the system may be related to the pressure $P$ and volume change $\backslash mathrm\{d\}V$. The pressure is the intensive generalized force, while the volume change is the extensive generalized displacement:
The change in internal energy becomes
This is useful if the equation of state is known.
In case of an ideal gas, we can derive that $dU\; =\; C\_V\; \backslash ,\; dT$, i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature.
The expression relating changes in internal energy to changes in temperature and volume is
The equation of state is the ideal gas law
Solve for pressure:
Substitute in to internal energy expression:
Take the derivative of pressure with respect to temperature:
Replace:
And simplify:
To express $\backslash mathrm\{d\}U$ in terms of $\backslash mathrm\{d\}T$ and $\backslash mathrm\{d\}V$, the term
is substituted in the fundamental thermodynamic relation
This gives
The term $T\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; T\}\backslash right)\_\{V\}$ is the heat capacity at constant volume $C\_\{V\}.$
The partial derivative of $S$ with respect to $V$ can be evaluated if the equation of state is known. From the fundamental thermodynamic relation, it follows that the differential of the Helmholtz free energy $A$ is given by
The symmetry of second derivatives of $A$ with respect to $T$ and $V$ yields the Maxwell relation:
This gives the expression above.
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 d V to zero and solving for the ratio d P/d T. This gives
Substituting () and () in () gives the above expression.
where $N\_j$ are the molar amounts of constituents of type $j$ in the system. The internal energy is an extensive function of the extensive variables $S$, $V$, and the amounts $N\_j$, the internal energy may be written as a linearly homogeneous function of first degree:
= \alpha U(S,V,N_{1},N_{2},\ldots),where $\backslash alpha$ is a factor describing the growth of the system. The differential internal energy may be written as
which shows (or defines) temperature $T$ to be the partial derivative of $U$ with respect to entropy $S$ and pressure $P$ to be the negative of the similar derivative with respect to volume $V$,
and where the coefficients $\backslash mu\_\{i\}$ are the chemical potentials for the components of type $i$ in the system. The chemical potentials are defined as the partial derivatives of the internal energy with respect to the variations in composition:
The sum over the composition of the system is the Gibbs free energy:
Euler's theorem yields for the internal energy:.
For a linearly elastic material, the stress is related to the strain by
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. While such energies of motion continue, they contribute to the total energy of the system; thermodynamic internal energy pertains only when such motions have ceased.

