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The internal energy of a
thermodynamic system A thermodynamic system is a body of matter and/or radiation, confined in space by walls, with defined permeabilities, which separate it from its surroundings. The surroundings may include other thermodynamic systems, or physical systems that a ...
is the total
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of he ...
contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinetic energy. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state. It does not include the
kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its ac ...
of motion of the system as a whole, or any external energies from surrounding force fields. The internal energy of an
isolated system In physical science, an isolated system is either of the following: # a physical system so far removed from other systems that it does not interact with them. # a thermodynamic system enclosed by rigid immovable walls through which neither ...
is constant, which is expressed as the law of conservation of energy, a foundation of the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...
. The internal energy is an extensive property. The internal energy cannot be measured directly and knowledge of all its components is rarely interesting, such as the static rest mass energy of its constituent matter. Thermodynamics is chiefly concerned only with ''changes'' in the internal energy, not with its absolute value. Instead, it is customary to define a reference state, and measure any changes in a
thermodynamic process Classical thermodynamics considers three main kinds of thermodynamic process: (1) changes in a system, (2) cycles in a system, and (3) flow processes. (1)A Thermodynamic process is a process in which the thermodynamic state of a system is change ...
from this state. The processes that change the internal energy are transfers of matter, or of energy as
heat In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
, or by
thermodynamic work In thermodynamics, work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. An exchange of energy is facilitated by a mechanism through which the system can spontaneously exer ...
. Born, M. (1949), Appendix 8
pp. 146–149
These processes are measured by changes in the system's properties, such as temperature,
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
, volume, and molar constitution. When transfer of matter is prevented by impermeable containing walls, the system is said to be closed. If the containing walls pass neither matter nor energy, the system is said to be isolated and its internal energy cannot change. The internal energy depends only on the state of the system and not on the particular choice from many possible processes by which energy may pass to or from the system. It is a thermodynamic potential. Microscopically, the internal energy can be analyzed in terms of the kinetic energy of microscopic motion of the system's particles from
translations Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
,
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s, and vibrations, and of the potential energy associated with microscopic forces, including chemical bonds. The unit of
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of he ...
in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes pleonastically as the SI system, is the modern form of the metric system and the world's most widely used system of measurement. ...
(SI) is the
joule The joule ( , ; symbol: J) is the unit of energy in the International System of Units (SI). It is equal to the amount of work done when a force of 1 newton displaces a mass through a distance of 1 metre in the direction of the force appli ...
(J). The internal energy relative to the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
with unit J/kg is the ''specific internal energy''. The corresponding quantity relative to the
amount of substance In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions ...
with unit J/ mol is the ''molar internal energy''.

# Cardinal functions

The internal energy of a system depends on its entropy S, its volume V and its number of massive particles: . It expresses the thermodynamics of a system in the ''energy representation''. As a function of state, its arguments are exclusively extensive variables of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, , of the same list of extensive variables of state, except that the entropy, , is replaced in the list by the internal energy, . It expresses the ''entropy representation''.Tschoegl, N.W. (2000), p. 17. Callen, H.B. (1960/1985), Chapter 5. 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 Massieu functions. 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. 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.

# Description and definition

The internal energy $U$ of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state: : $\Delta U = \sum_i E_i,$ where $\Delta U$ denotes the difference between the internal energy of the given state and that of the reference state, and the $E_i$ are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state. From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, $U_\text$, and microscopic kinetic energy, $U_\text$, components: : $U = U_\text + U_\text.$ 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 Nuclear may refer to: Physics Relating to the nucleus of the atom: * Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics * Nuclear space * Nucle ...
particle bonds, and the physical force fields within the system, such as due to internal
induced Induce may refer to: * Induced consumption * Induced innovation * Induced character * Induced coma * Induced menopause * Induced metric * Induced path * Induced topology * Induce (musician), American musician See also * Inducement (disa ...
electric or
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
dipole In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways: *An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this syste ...
moment, as well as the energy of
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Def ...
of solids ( stress- strain). 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
gravitation In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
al,
electrostatic Electrostatics is a branch of physics that studies electric charges at rest ( static electricity). Since classical times, it has been known that some materials, such as amber, attract lightweight particles after rubbing. The Greek word for ...
, or
electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions ...
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 In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions ...
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 In physical science, an isolated system is either of the following: # a physical system so far removed from other systems that it does not interact with them. # a thermodynamic system enclosed by rigid immovable walls through which neither ...
(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 quantum-mechanical ground state, the lowest energy state available. At absolute zero a system of given composition has attained its minimum attainable
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
. The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system.
Statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
relates the pseudo-random 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'',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 A microstate or ministate is a sovereign state having a very small population or very small land area, usually both. However, the meanings of "state" and "very small" are not well-defined in international law.Warrington, E. (1994). "Lilliputs ...
. 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 There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set. For a data set, the ''ari ...
value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence: : $U = \sum_^N p_i \,E_i.$ This is the statistical expression of the law of
conservation of energy In physics and chemistry, the law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be ''conserved'' over time. This law, first proposed and tested by Émilie du Châtelet, means tha ...
.

## Internal energy changes

Thermodynamics is chiefly concerned with the changes in internal energy $\Delta U$. For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer $Q$ and due to
thermodynamic work In thermodynamics, work is one of the principal processes by which a thermodynamic system can interact with its surroundings and exchange energy. An exchange of energy is facilitated by a mechanism through which the system can spontaneously exer ...
$W$ 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, where work performed by the system against the environment, e.g., a system expansion, is negative, while in engineering, this is taken to be positive. Accordingly, the internal energy change $\Delta U$ for a process may be written $\Delta U = Q - W \quad \text.$ 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''. A second kind of mechanism of change in the internal energy of a closed system changed is in its doing of
work Work may refer to: * Work (human activity), intentional activity people perform to support themselves, others, or the community ** Manual labour, physical work done by humans ** House work, housework, or homemaking ** Working animal, an animal ...
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, $\Delta U_\mathrm$ 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: $\Delta U = Q - W + \Delta U_\text \quad \text.$ 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 Latent heat (also known as latent energy or heat of transformation) is energy released or absorbed, by a body or a thermodynamic system, during a constant-temperature process — usually a first-order phase transition. Latent heat can be underst ...
, in contrast to sensible heat, which is associated with temperature change.

# Internal energy of the ideal gas

Thermodynamics often uses the concept of the
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is ...
for teaching purposes, and as an approximation for working systems. The ideal gas consists of particles considered as point objects that interact only by elastic collisions and fill a volume such that their
mean free path In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
between collisions is much larger than their diameter. Such systems approximate
monatomic In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
gases such as
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic ta ...
and other
noble gas The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low c ...
es. For an ideal gas the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not possess rotational or vibrational degrees of freedom, and are not electronically excited to higher energies except at very high
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...
s. Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): $U = U\left(n,T\right)$. 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 $T$ : $U = C_V n T,$ where $C_V$ is the molar heat capacity (at constant volume) 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 : $U\left(S,V,n\right) = \mathrm \cdot e^\frac V^\frac n^\frac,$ where $\mathrm$ is an arbitrary positive constant and where $R$ is the
universal gas constant The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
. It is easily seen that $U$ is a linearly
homogeneous function In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
of the three variables (that is, it is ''extensive'' in these variables), and that it is weakly
convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...
. Knowing temperature and pressure to be the derivatives $T = \frac,$ $P = -\frac,$ the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
$PV = nRT$ immediately follows.

# Internal energy of a closed thermodynamic system

The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings. This relationship may be expressed in
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
terms using the differentials of each term, though only the internal energy is an
exact differential In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&n ...
. For a closed system, with transfers only as heat and work, the change in the internal energy is : $\mathrm U = \delta Q - \delta W,$ expressing the
first law of thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant amo ...
. It may be expressed in terms of other thermodynamic parameters. Each term is composed of an intensive variable (a generalized force) and its
conjugate Conjugation or conjugate may refer to: Linguistics *Grammatical conjugation, the modification of a verb from its basic form * Emotive conjugation or Russell's conjugation, the use of loaded language Mathematics * Complex conjugation, the change ...
infinitesimal extensive variable (a generalized displacement). For example, the mechanical work done by the system may be related to the
pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
$P$ and
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). ...
change $\mathrmV$. The pressure is the intensive generalized force, while the volume change is the extensive generalized displacement: : $\delta W = P \, \mathrmV.$ This defines the direction of work, $W$, to be energy transfer from the working system to the surroundings, indicated by a positive term. Taking the direction of heat transfer $Q$ to be into the working fluid and assuming a reversible process, the heat is : $\delta Q = T \mathrmS,$ where $T$ denotes the
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied on ...
, and $S$ denotes the
entropy Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
. The change in internal energy becomes : $\mathrmU = T \, \mathrmS - P \, \mathrmV.$

## Changes due to temperature and volume

The expression relating changes in internal energy to changes in temperature and volume is This is useful if the equation of state is known. In case of an ideal gas, we can derive that $dU = C_V \, dT$, 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 :$P V = n R T.$ Solve for pressure: :$P = \frac.$ Substitute in to internal energy expression: : Take the derivative of pressure with respect to temperature: :$\left\left( \frac \right\right)_ = \frac.$ Replace: : And simplify: :$\mathrmU =C_ \, \mathrmT.$ To express $\mathrmU$ in terms of $\mathrmT$ and $\mathrmV$, the term :$\mathrmS = \left\left(\frac\right\right)_\mathrmT + \left\left(\frac\right\right)_ \mathrmV$ is substituted in the fundamental thermodynamic relation :$\mathrmU = T \, \mathrmS - P \, \mathrmV.$ This gives : The term $T\left\left(\frac\right\right)_$ is the heat capacity at constant volume $C_.$ 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 In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ene ...
$A$ is given by :$dA = -S \, dT - P \, dV.$ The symmetry of second derivatives of $A$ with respect to $T$ and $V$ yields the Maxwell relation: :$\left\left(\frac\right\right)_ = \left\left(\frac\right\right)_.$ This gives the expression above.

## Changes due to temperature and pressure

When considering fluids or solids, an expression in terms of the temperature and pressure is usually more useful: :$dU = \left\left(C_-\alpha P V\right\right) \, dT +\left\left(\beta_P-\alpha T\right\right)V \, dP,$ where it is assumed that the heat capacity at constant pressure is
related ''Related'' is an American comedy-drama television series that aired on The WB from October 5, 2005, to March 20, 2006. It revolves around the lives of four close-knit sisters of Italian descent, raised in Brooklyn and living in Manhattan. The ...
to the heat capacity at constant volume according to :$C_ = C_ + V T\frac.$ The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the
coefficient of thermal expansion Thermal expansion is the tendency of matter to change its shape, area, volume, and density in response to a change in temperature, usually not including phase transitions. Temperature is a monotonic function of the average molecular kineti ...
:$\alpha \equiv \frac\left\left(\frac\right\right)_$ and the isothermal
compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a ...
:$\beta_ \equiv -\frac\left\left(\frac\right\right)_$ 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.

## Changes due to volume at constant temperature

The
internal pressure Internal pressure is a measure of how the internal energy of a system changes when it expands or contracts at constant temperature. It has the same dimensions as pressure, the SI unit of which is the pascal. Internal pressure is usually given the ...
is defined as a
partial derivative In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Pa ...
of the internal energy with respect to the volume at constant temperature: :$\pi _T = \left \left( \frac \right \right)_T.$

# Internal energy of multi-component systems

In addition to including the entropy $S$ and volume $V$ terms in the internal energy, a system is often described also in terms of the number of particles or chemical species it contains: :$U = U\left(S,V,N_1,\ldots,N_n\right),$ 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 In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the '' ...
of first degree: : $U\left(\alpha S,\alpha V,\alpha N_,\alpha N_,\ldots \right) = \alpha U\left(S,V,N_,N_,\ldots\right),$ where $\alpha$ is a factor describing the growth of the system. The differential internal energy may be written as :$\mathrm U = \frac \mathrm S + \frac \mathrm V + \sum_i\ \frac \mathrm N_i\ = T \,\mathrm S - P \,\mathrm V + \sum_i\mu_i \mathrm N_i,$ 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 $V$, : $T = \frac,$ : $P = -\frac,$ and where the coefficients $\mu_$ 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: :$\mu_i = \left\left( \frac \right\right)_.$ As conjugate variables to the composition $\lbrace N_ \rbrace$, the chemical potentials are intensive properties, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent. Under conditions of constant $T$ and $P$, because of the extensive nature of $U$ and its independent variables, using
Euler's homogeneous function theorem In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the ''d ...
, the differential $\mathrm d U$ may be integrated and yields an expression for the internal energy: :$U = T S - P V + \sum_i \mu_i N_i.$ The sum over the composition of the system is the
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
: :$G = \sum_i \mu_i N_i$ that arises from changing the composition of the system at constant temperature and pressure. 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 $\lbrace N_ \rbrace$.

# Internal energy in an elastic medium

For an elastic medium the mechanical energy term of the internal energy is expressed in terms of the stress $\sigma_$ and strain $\varepsilon_$ involved in elastic processes. In
Einstein notation In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
for tensors, with summation over repeated indices, for unit volume, the infinitesimal statement is : $\mathrmU=T\mathrmS+\sigma_\mathrm\varepsilon_.$ Euler's theorem yields for the internal energy: : $U=TS+\frac\sigma_\varepsilon_.$ For a linearly elastic material, the stress is related to the strain by : $\sigma_=C_ \varepsilon_,$ where the $C_$ are the components of the 4th-rank elastic constant tensor of the medium. Elastic deformations, such as
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...
, 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.

# History

James Joule James Prescott Joule (; 24 December 1818 11 October 1889) was an English physicist, mathematician and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work (see energy). T ...
studied the relationship between heat, work, and temperature. He observed that friction in a liquid, such as caused by its agitation with work by a paddle wheel, caused an increase in its temperature, which he described as producing a ''quantity of heat''. 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.

# Notes

*
Calorimetry In chemistry and thermodynamics, calorimetry () is the science or act of measuring changes in '' state variables'' of a body for the purpose of deriving the heat transfer associated with changes of its state due, for example, to chemical rea ...
*
Enthalpy Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
*
Exergy In thermodynamics, the exergy of a system is the maximum useful work possible during a process that brings the system into equilibrium with a heat reservoir, reaching maximum entropy. When the surroundings are the reservoir, exergy is the pot ...
* Thermodynamic equations * Thermodynamic potentials *
Gibbs free energy In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and ...
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Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature (isothermal). The change in the Helmholtz ene ...

# References

## Bibliography of cited references

* Adkins, C. J. (1968/1975). ''Equilibrium Thermodynamics'', second edition, McGraw-Hill, London, . * Bailyn, M. (1994). ''A Survey of Thermodynamics'', American Institute of Physics Press, New York, . * Born, M. (1949)
''Natural Philosophy of Cause and Chance''
Oxford University Press, London. * Callen, H. B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, . * Crawford, F. H. (1963). ''Heat, Thermodynamics, and Statistical Physics'', Rupert Hart-Davis, London, Harcourt, Brace & World, Inc. * Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of ''Thermodynamics'', pages 1–97 of volume 1, ed. W. Jost, of ''Physical Chemistry. An Advanced Treatise'', ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081. * . * * Münster, A. (1970), Classical Thermodynamics, translated by E. S. Halberstadt, Wiley–Interscience, London, . * Planck, M., (1923/1927). ''Treatise on Thermodynamics'', translated by A. Ogg, third English edition, Longmans, Green and Co., London. * Tschoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam, .

# Bibliography

* * {{DEFAULTSORT:Internal Energy Physical quantities Thermodynamic properties State functions Statistical mechanics Energy (physics)