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The internal energy of a
thermodynamic system A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, whic ...
is the
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

contained within it. It is the energy necessary to create or prepare the system in any given internal state. It does not include the
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
of motion of the system as a whole, nor the
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

of the system as a whole due to external force fields, including the energy of displacement of the surroundings of the system. It keeps account of the gains and losses of energy of the system that are due to changes in its internal state. The internal energy is measured as a difference from a reference zero defined by a standard state. The difference is determined by
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 change in a system is defined by a passage from an initial to a final state of thermodyna ...
es that carry the system between the reference state and the current state of interest. The internal energy is an
extensive property Extensive may refer to: * Extensive property * Extensive function * Extensional See also * Extension (disambiguation) {{Dab ...
, and cannot be measured directly. The thermodynamic processes that define the internal energy are transfers of
chemical substance A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which ...
s or of energy as
heat In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these ...

, and
thermodynamic work In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is govern ...
. (1949), Appendix 8
pp. 146–149
These processes are measured by ''changes'' in the system's extensive variables, such as entropy, volume, and
chemical composition {{Unreferenced, date=December 2017 Chemical composition refers to identity and number of the chemical element Image:Simple Periodic Table Chart-blocks.svg, 400px, Periodic table, The periodic table of the chemical elements In chemistry, an el ...
. It is often not necessary to consider all of the system's intrinsic energies, for example, the static rest mass energy of its constituent matter. When
mass transfer Mass transfer is the net movement of mass from one location, usually meaning stream, phase, fraction or component, to another. Mass transfer occurs in many processes, such as absorption Absorption may refer to: Chemistry and biology *Absorptio ...
is prevented by impermeable containing walls, the system is said to be closed and the
first law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
defines the change in internal energy as the difference between the energy added to the system as heat and the thermodynamic work done by the system on its surroundings. If the containing walls pass neither substance nor energy, the system is said to be isolated and its internal energy cannot change. The internal energy describes the entire thermodynamic information of a system, and is an equivalent representation to the entropy, both cardinal
state function In the thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form ...
s of only extensive state variables. Thus, its value depends only on the current 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 A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermodyna ...
. 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 ''transla ...
,
rotation A rotation is a circular movement of an object around a center (or point) of rotation. The plane (geometry), geometric plane along which the rotation occurs is called the ''rotation plane'', and the imaginary line extending from the center an ...

s, and
vibrations Vibration is a mechanical phenomenon whereby oscillation Oscillation is the repetitive variation, typically in time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparentl ...

, and of the potential energy associated with microscopic forces, including
chemical bonds A chemical bond is a lasting attraction between atom An atom is the smallest unit of ordinary matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyd ...
. The unit of
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

in the
International System of Units The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms_and_initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wi ...
(SI) is the
joule The joule ( ; symbol: J) is a derived unit of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates ...

(J). Also defined is a corresponding intensive energy density, called ''specific internal energy'', which is either relative to the mass of the system, with the unit J/kg, or relative to the
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
with unit J/ mol (''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 In the Thermodynamics#Equilibrium_thermodynamics, thermodynamics of equilibrium, a state function, function of state, or point function is a function defined for a system relating several state variables or state quantities that depends only on the ...
, 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. (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 function In thermodynamics, Massieu function (sometimes called Massieu–Gibbs function, Massieu potential, or Gibbs function, or characteristic (state) function in its original terminology), symbol \Psi (Psi), is defined by the following relation: ...
s. 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, $U_\mathrm$, and microscopic kinetic energy, $U_\mathrm$, components: :$U = U_ + U_$ 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 A chemical substance is a form of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which ...
and nuclear particle bonds, and the physical force fields within the system, such as due to internal electric or
magnetic Magnetism is a class of physical attributes that are mediated by magnetic field A magnetic field is a vector field In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For in ...

dipole In electromagnetism Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is c ...

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 (mechanics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
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 Gravity (), or gravitation, is a natural phenomenon Types of natural phenomena include: Weather, fog, thunder, tornadoes; biological processes, decomposition, germination seedlings, three days after germination. Germination is t ...

al,
electrostatic Electrostatics is a branch of physics that studies electric charges at Rest (physics), rest (static electricity). Since classical antiquity, classical times, it has been known that some materials, such as amber, attract lightweight particles af ...
, or
electromagnetic Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electric charge, electrically charged particles. The electromagnetic force is carried by electromagneti ...
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grassl ...
s. 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 Physical properties A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of ...
: it depends on the size of the system, or on the
amount of substance In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that make up matter to the composed of s, s and s: their composition, structure, properties, behavior and the changes they undergo during a ...
it contains. At any temperature greater than
absolute zero Absolute zero is the lowest limit of the thermodynamic temperature Thermodynamic temperature is the measure of ''absolute temperature'' and is one of the principal parameters of thermodynamics. A thermodynamic temperature reading of zero deno ...
, 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , i ...
(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 Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly Quantum fluctuation, fluctuate in their lowest energy state as described by the Heisenb ...
. 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 thermodynamic ...

. The microscopic kinetic energy portion of the internal energy gives rise to the temperature of the system.
Statistical mechanics In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
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 Image:BlankMap-World-v6 small states.png, upright=1.4, Map of the smallest states in the world by land area. Note many of these are not considered microstates A microstate or ministate is a sovereign state having a very small population or very ...
. 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. For a data set, the ''arithmetic mean'', also known as arithmetic average, is a central value of a finite set of numbers: specifically, the sum of the values divided by ...
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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
.

## 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is govern ...
$W$ done ''by'' the system on its surroundings.This article uses the sign convention of the mechanical work as usually defined in physics, which is different from the convention used in chemistry. In chemistry, work performed by the system against the environment, e.g., a system expansion, is negative, while in physics this is taken to be positive. Accordingly, the internal energy change $\Delta U$ for a process may be written $\Delta U = Q - W \qquad \text.$ 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) Work or labor is intentional activity people perform to support themselves, others, or the needs and wants of a wider community. Alternatively, work can be viewed as the human activity that cont ...
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: $\Delta U = Q - W + \Delta U_ ~~ \text.$ 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 A thermodynamic system is a body of matter In classical physics and general chemistry, matter is any su ...
, 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 Gas is one of the four fundamental states of matter In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion ...
for teaching purposes, and as an approximation for working systems. The ideal gas is a gas of particles considered as point objects that interact only by elastic collisions and fill a volume such that their
mean free path In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

between collisions is much larger than their diameter. Such systems approximate the
monatomic In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...
gases,
helium Helium (from el, ἥλιος, helios Helios; Homeric Greek: ), Latinized as Helius; Hyperion and Phaethon are also the names of his father and son respectively. often given the epithets Hyperion ("the one above") and Phaethon ("the shining" ...

and the other
noble gas The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical element In chemistry Chemistry is the study of the properties and behavior of . It is a that covers the that m ...
es. Here the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not rotate or vibrate, and are not electronically excited to higher energies except at very high
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

s. Therefore, internal energy changes in an ideal gas may be described solely by changes in its kinetic energy. Kinetic energy is simply the internal energy of the perfect gas and depends entirely on its
pressure Pressure (symbol: ''p'' or ''P'') is the force In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

,
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...

and
thermodynamic temperature Thermodynamic temperature is a quantity defined in thermodynamics as distinct from Kinetic theory of gases, kinetic theory or statistical mechanics. A thermodynamic temperature reading of zero is of particular importance for the third law of therm ...

. 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. The internal energy may be written as a function of the three extensive properties $S$, $V$, $n$ (entropy, volume, mass) 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 The Boltzmann constant ( or ) is the proportionality fac ...
. 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 (mathematics), scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneit ...
of the three variables (that is, it is ''extensive'' in these variables), and that it is weakly
convex Convex means curving outwards like a sphere, and is the opposite of concave. Convex or convexity may refer to: Science and technology * Convex lens A lens is a transmissive optics, optical device which focuses or disperses a light beam by me ...

. 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the na ...

$PV = nRT$ 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, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not exist in the standard real number system, but do exist in many other number systems, such a ...
terms using the differentials of each term, though only the internal energy is an
exact differential In multivariate calculus, a differential is said to be exact or perfect, as contrasted with an inexact differential, if it is of the form ''dQ'', for some differentiable function In calculus Calculus, originally called infinitesimal calculu ...
. For a closed system, with transfers only as heat and work, the change in the internal energy is :$d U = \delta Q - \delta W$ expressing the
first law of thermodynamics The first law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter tran ...
. 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 In physics, a force is an influence that can change the motion (physics), motion of an Physical object, object. A force can cause an object with mass to change its velocity (e.g. moving fr ...

$P$ and
volume Volume is a scalar quantity expressing the amount Quantity or amount is a property that can exist as a multitude Multitude is a term for a group of people who cannot be classed under any other distinct category, except for their shared fact ...
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 :$\delta Q = T \mathrmS\,.$ *$T$ denotes the
temperature Temperature ( ) is a physical quantity that expresses hot and cold. It is the manifestation of thermal energy Thermal radiation in visible light can be seen on this hot metalwork. Thermal energy refers to several distinct physical concept ...

*$S$ 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 thermodynamic ...

and 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 In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through S ...
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: :$dU =C_ \, dT.$ To express $\mathrmU$ in terms of $\mathrmT$ and $\mathrmV$, the term :$dS = \left\left(\frac\right\right)_dT + \left\left(\frac\right\right)_ dV \,$ is substituted in the
fundamental thermodynamic relation In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
:$dU = T \, dS - P \, dV.\,$ 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 Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order ...
$A$ is given by: :$dA = -S \, dT - P \, dV.$ The
symmetry of second derivativesIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
of $A$ with respect to $T$ and $V$ yields the
Maxwell relation file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volum ...
: :$\left\left(\frac\right\right)_ = \left\left(\frac\right\right)_.$ 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 Comedy-drama, or dramedy, is a genre of dramatic works that combines elements of comedy and Drama (film and television), drama. History The advent of radio drama, film, cinema and in particular, televisi ...
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 A shape or figure is the form of an object or its external boundary, outline, or external surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask ...
:$\alpha \equiv \frac\left\left(\frac\right\right)_$ and the isothermal
compressibility In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quan ...
:$\beta_ \equiv -\frac\left\left(\frac\right\right)_$ 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 of a system changes when it expands or contracts at constant . It has the same dimensions as , the of which is the . Internal pressure is usually given the symbol \pi_T. It is defined as a of internal ...
is defined as a
partial derivative In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 (mathematics), scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneit ...
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 potential In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, radiation, and physical properties of matter. The behavior of these quantities is governed ...

s for the components of type $i$ in the system. The chemical potentials are defined as the partial derivatives of the 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 Physical properties A physical property is any property Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of ...
, 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, 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 Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
: :$G = \sum_i \mu_i N_i\,$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
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 Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...

, 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 A physicist is a scientist A scientist is a person who conducts scientific research The scientific method is an Empirical evidence, empirical m ...

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 Calorimetry is the science or act of measuring changes in '' state variables'' of a body for the purpose of deriving the heat transfer Heat transfer is a discipline of thermal engineering that concerns the generation, use, conversion, and ...
*
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 p ...

*
Exergy In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
*
Thermodynamic equations Thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of ...
*
Thermodynamic potentials A thermodynamic potential (or more accurately, a thermodynamic potential energy)ISO/IEC 80000-5, Quantities an units, Part 5 - Thermodynamics, item 5-20.4 Helmholtz energy, Helmholtz functionISO/IEC 80000-5, Quantities an units, Part 5 - Thermodyna ...

*
Gibbs free energy In thermodynamics Thermodynamics is a branch of physics that deals with heat, Work (thermodynamics), work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these qua ...
*
Helmholtz Free Energy In thermodynamics Thermodynamics is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order ...

# 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, . * (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, . * , (1923/1927). ''Treatise on Thermodynamics'', translated by A. Ogg, third English edition, , 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)