The internal energy of a

pp. 146–149

These processes are measured by changes in the system's properties, such as temperature,

– 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

''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, .

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 are ...

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 ...

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 acce ...

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 am ...

. 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. Born, M. (1949), Appendix 8pp. 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
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...

. 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, 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 ...

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, io ...

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 andMassieu 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: : $\backslash Delta\; U\; =\; \backslash sum\_i\; E\_i,$ where $\backslash Delta\; U$ 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, $U\_\backslash text$, and microscopic kinetic energy, $U\_\backslash text$, components: : $U\; =\; U\_\backslash text\; +\; U\_\backslash text.$ 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 andnuclear
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
*Nuclear ...

particle bonds, and the physical force fields within the system, such as due to internal induced 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 particl ...

dipole moment
Moment or Moments may refer to:
* Present time
Music
* The Moments, American R&B vocal group Albums
* ''Moment'' (Dark Tranquillity album), 2020
* ''Moment'' (Speed album), 1998
* ''Moments'' (Darude album)
* ''Moments'' (Christine Guldbrand ...

, as well as the energy of deformation of solids ( stress- strain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.
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 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, io ...

it contains.
At any temperature greater than absolute zero
Absolute zero is the lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as zero kelvin. The fundamental particles of nature have minimum vibra ...

, 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
Zero-point energy (ZPE) is the lowest possible energy that a quantum mechanical system may have. Unlike in classical mechanics, quantum systems constantly fluctuate in their lowest energy state as described by the Heisenberg uncertainty ...

. 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. 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 '' ar ...

value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence:
: $U\; =\; \backslash sum\_^N\; p\_i\; \backslash ,E\_i.$
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 $\backslash Delta\; U$. For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer $Q$ and due to thermodynamic work $W$ 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 $\backslash Delta\; U$ for a process may be written $$\backslash Delta\; U\; =\; Q\; -\; W\; \backslash quad\; \backslash text.$$ 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 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, $\backslash Delta\; U\_\backslash mathrm$ 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: $$\backslash Delta\; U\; =\; Q\; -\; W\; +\; \backslash Delta\; U\_\backslash text\; \backslash quad\; \backslash text.$$ 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, 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 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 theirmean 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 ...

between collisions is much larger than their diameter. Such systems approximate monatomic 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 gases. For an ideal gas the kinetic energy consists only of the translational
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 ...

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 o ...

s.
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 $T$
: $U\; =\; C\_V\; n\; T,$
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
: $U(S,V,n)\; =\; \backslash mathrm\; \backslash cdot\; e^\backslash frac\; V^\backslash frac\; n^\backslash frac,$
where $\backslash mathrm$ is an arbitrary positive constant and where $R$ is the universal gas constant. 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 ''d ...

of the three variables (that is, it is ''extensive'' in these variables), and that it is weakly convex. Knowing temperature and pressure to be the derivatives
$T\; =\; \backslash frac,$ $P\; =\; -\backslash frac,$ 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 ininfinitesimal
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 re ...

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 is
: $\backslash mathrm\; U\; =\; \backslash delta\; Q\; -\; \backslash delta\; W,$
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 am ...

. It may be expressed in terms of other thermodynamic parameters. Each term is composed of an intensive variable
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

(a generalized force) and its conjugate infinitesimal extensive variable
Physical properties of materials and systems can often be categorized as being either intensive or extensive, according to how the property changes when the size (or extent) of the system changes. According to IUPAC, an intensive quantity is one ...

(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 $\backslash mathrmV$. The pressure is the intensive generalized force, while the volume change is the extensive generalized displacement:
: $\backslash delta\; W\; =\; P\; \backslash ,\; \backslash mathrmV.$
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
: $\backslash delta\; Q\; =\; T\; \backslash mathrmS,$
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 o ...

, 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
: $\backslash mathrmU\; =\; T\; \backslash ,\; \backslash mathrmS\; -\; P\; \backslash ,\; \backslash mathrmV.$
Changes due to temperature and volume

The expression relating changes in internal energy to changes in temperature and volume is This is useful if theequation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or intern ...

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
:$\backslash mathrmU\; =C\_\; \backslash ,\; \backslash mathrmT\; +\backslash left;\; href="/html/ALL/l/\backslash left(\backslash frac\backslash right)\_\_\_-\_P\backslash right.html"\; ;"title="\backslash left(\backslash frac\backslash right)\_\; -\; P\backslash right">\backslash left(\backslash frac\backslash right)\_\; -\; P\backslash right$
The equation of state is the ideal gas law
:$P\; V\; =\; n\; R\; T.$
Solve for pressure:
:$P\; =\; \backslash frac.$
Substitute in to internal energy expression:
:$dU\; =C\_\backslash mathrmT\; +\backslash left;\; href="/html/ALL/l/\backslash left(\backslash frac\backslash right)\_\_\_-\_\backslash frac\backslash right.html"\; ;"title="\backslash left(\backslash frac\backslash right)\_\; -\; \backslash frac\backslash right">\backslash left(\backslash frac\backslash right)\_\; -\; \backslash frac\backslash right$
Take the derivative of pressure with respect to temperature:
:$\backslash left(\; \backslash frac\; \backslash right)\_\; =\; \backslash frac.$
Replace:
:$dU\; =\; C\_\; \backslash ,\; \backslash mathrmT\; +\; \backslash left;\; href="/html/ALL/l/\backslash frac\_\_-\_\backslash frac\_\backslash right.html"\; ;"title="\backslash frac\; -\; \backslash frac\; \backslash right">\backslash frac\; -\; \backslash frac\; \backslash right$
And simplify:
:$\backslash mathrmU\; =C\_\; \backslash ,\; \backslash mathrmT.$
To express $\backslash mathrmU$ in terms of $\backslash mathrmT$ and $\backslash mathrmV$, the term
:$\backslash mathrmS\; =\; \backslash left(\backslash frac\backslash right)\_\backslash mathrmT\; +\; \backslash left(\backslash frac\backslash right)\_\; \backslash mathrmV$
is substituted in the fundamental thermodynamic relation
:$\backslash mathrmU\; =\; T\; \backslash ,\; \backslash mathrmS\; -\; P\; \backslash ,\; \backslash mathrmV.$
This gives
:$dU\; =\; T\backslash left(\backslash frac\backslash right)\_\; \backslash ,\; dT\; +\backslash left;\; href="/html/ALL/l/\backslash left(\backslash frac\backslash right)\_\_-\_P\backslash right.html"\; ;"title="\backslash left(\backslash frac\backslash right)\_\; -\; P\backslash right">\backslash left(\backslash frac\backslash right)\_\; -\; P\backslash right$
The term $T\backslash left(\backslash frac\backslash right)\_$ 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 en ...

$A$ is given by
:$dA\; =\; -S\; \backslash ,\; dT\; -\; P\; \backslash ,\; dV.$
The symmetry of second derivatives
In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function
:f\left(x_1,\, x_2,\, \ldots,\, x_n\right)
of ''n ...

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 volu ...

:
:$\backslash left(\backslash frac\backslash right)\_\; =\; \backslash left(\backslash frac\backslash right)\_.$
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: :$dU\; =\; \backslash left(C\_-\backslash alpha\; P\; V\backslash right)\; \backslash ,\; dT\; +\backslash left(\backslash beta\_P-\backslash alpha\; T\backslash right)V\; \backslash ,\; dP,$ where it is assumed that the heat capacity at constant pressure isrelated
''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
:$C\_\; =\; C\_\; +\; V\; T\backslash frac.$
The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the coefficient of thermal expansion
:$\backslash alpha\; \backslash equiv\; \backslash frac\backslash left(\backslash frac\backslash right)\_$
and the isothermal compressibility
:$\backslash beta\_\; \backslash equiv\; -\backslash frac\backslash left(\backslash frac\backslash right)\_$
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

Theinternal 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:
:$\backslash pi\; \_T\; =\; \backslash left\; (\; \backslash frac\; \backslash right\; )\_T.$
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: :$U\; =\; U(S,V,N\_1,\backslash ldots,N\_n),$ 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 linearlyhomogeneous 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 ''d ...

of first degree:
: $U(\backslash alpha\; S,\backslash alpha\; V,\backslash alpha\; N\_,\backslash alpha\; N\_,\backslash ldots\; )\; =\; \backslash alpha\; U(S,V,N\_,N\_,\backslash ldots),$
where $\backslash alpha$ is a factor describing the growth of the system. The differential internal energy may be written as
:$\backslash mathrm\; U\; =\; \backslash frac\; \backslash mathrm\; S\; +\; \backslash frac\; \backslash mathrm\; V\; +\; \backslash sum\_i\backslash \; \backslash frac\; \backslash mathrm\; N\_i\backslash \; =\; T\; \backslash ,\backslash mathrm\; S\; -\; P\; \backslash ,\backslash mathrm\; V\; +\; \backslash sum\_i\backslash mu\_i\; \backslash mathrm\; N\_i,$
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 $V$,
: $T\; =\; \backslash frac,$
: $P\; =\; -\backslash frac,$
and where the coefficients $\backslash mu\_$ are the chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...

s 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:
:$\backslash mu\_i\; =\; \backslash left(\; \backslash frac\; \backslash right)\_.$
As conjugate variables to the composition $\backslash lbrace\; N\_\; \backslash rbrace$, 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, the differential $\backslash mathrm\; d\; U$ may be integrated and yields an expression for the internal energy:
:$U\; =\; T\; S\; -\; P\; V\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i.$
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 an ...

:
:$G\; =\; \backslash sum\_i\; \backslash mu\_i\; N\_i$
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 $\backslash lbrace\; N\_\; \backslash rbrace$.
Internal energy in an elastic medium

For anelastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togethe ...

medium the mechanical energy term of the internal energy is expressed in terms of the stress $\backslash sigma\_$ and strain $\backslash varepsilon\_$ 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 ...

for tensors, with summation over repeated indices, for unit volume, the infinitesimal statement is
: $\backslash mathrmU=T\backslash mathrmS+\backslash sigma\_\backslash mathrm\backslash varepsilon\_.$
Euler's theorem yields for the internal energy:
: $U=TS+\backslash frac\backslash sigma\_\backslash varepsilon\_.$
For a linearly elastic material, the stress is related to the strain by
: $\backslash sigma\_=C\_\; \backslash varepsilon\_,$
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 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

See also

*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 re ...

*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
* 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 an ...

*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 en ...

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)