- Internal energy
In

thermodynamics , the**internal energy**of athermodynamic system , or a body with well-defined boundaries, denoted by "U", or sometimes "E", is the total of thekinetic energy due to the motion ofmolecule s (translational,rotation al, vibrational) and thepotential energy associated with the vibrational and electric energy ofatom s within molecules orcrystal s. It includes theenergy in all thechemical bond s, and the energy of the free, conductionelectron s inmetal s.One can also calculate the internal energy of electromagnetic or blackbody radiation. It is a

state function of asystem , and is anextensive quantity . TheSI unit ofenergy is thejoule although other historical, conventional units are still in use, such as the (small and large)calorie forheat .**Overview**"Internal" energy does not include the translational or rotational kinetic energy of a body "as a whole". It also does not include the relativistic

mass -energy equivalent "E" = "mc"^{2}. It excludes any potential energy a body may have because of its location in externalgravitation al or electrostatic field, although the potential energy it has in a field due to an induced electric or magneticdipole moment does count, as does the energy ofdeformation of solids (stress-strain).The principle of equipartition of energy in classical

statistical mechanics states that each molecular quadratic degree of freedom receives 1/2 "kT" of energy, [*cite book | last = Reif | first = Frederick | title = Statistical Physics*] a result which was modified when

publisher = McGraw-Hill Book Company | year = 1965 | location = New York

pages = 246-250quantum mechanics explained certain anomalies; e.g., in the observed specific heats of crystals (when "h"ν > "kT"). Formonoatomic helium and othernoble gas es, the internal energy consists only of the translational kinetic energy of the individual atoms. Monoatomic particles, of course, do not (sensibly) rotate or vibrate, and are not electronically excited to higher energies except at very hightemperature s.From the standpoint of

statistical mechanics , the internal energy is equal to theensemble average of the total energy of the system.**Composition**– the sum of all microscopic forms of energy of a system. It is related to the molecular structure and the degree of molecular activity and may be viewed as the sum of kinetic and potential energies of the molecules; it is composed of the following types of energies: [Internal energy *cite book | last = Cengel | first = Yungus, A. | coauthors = Boles, Michael | title = Thermodynamics - An Engineering Approach, 4th ed. | pages = 17-18 | publisher = McGraw-Hill | year = 2002 | id = ISBN 0-07-238332-1*]**Sensible energy**and**latent energy**may be further combined into.thermal energy **The first law of thermodynamics**The internal energy is essentially defined by the

first law of thermodynamics which states that energy is conserved::$Delta\; U\; =\; Q\; +\; W\; +\; W\text{'}\; ,$

where

:Δ"U" is the change in internal energy of a system during a process.

:"Q" is

heat "added to" a system (measured injoule s inSI ); that is, apositive value for "Q" represents heat flow "into" a system while anegative value denotes heat flow "out of" a system.:"W" is the

mechanical work "done on" a system (measured in joules in SI): "W' " is energy added by all other processes

The first law may be stated equivalently in

infinitesimal terms as::$mathrm\{d\}U\; =\; delta\; Q\; +\; delta\; W\; +\; delta\; W\text{'},$

where the terms now represent infinitesimal amounts of the respective quantities. The "d" before the internal energy function indicates that it is an exact differential. In other words it is a state function or a value which can be assigned to the system. On the other hand, the δ's before the other terms indicate that they describe increments of energy which are not state functions but rather they are processes by which the internal energy is changed. (See the discussion in the first law article.)

From a microscopic point of view, the internal energy may be found in many different forms. For a gas it may consist almost entirely of the

kinetic energy of the gas molecules. It may also consist of the potential energy of these molecules in a gravitational, electric, ormagnetic field . For any material, solid, liquid or gaseous, it may also consist of the potential energy of attraction or repulsion between the individual molecules of the material.**Expressions for the internal energy**The internal energy may be expressed in terms of other thermodynamic parameters. Each term is composed of an

intensive variable (a generalized force) and its conjugate infinitesimalextensive variable (a generalized displacement).For example, for a non-viscous fluid, the mechanical work done on the system may be related to the

pressure "p" andvolume "V". The pressure is the intensive generalized force, while the volume is the extensive generalized displacement:Taking the default direction of work, $W$, to be from the working fluid to the surroundings,:$mathrm\{d\}\; W\; =\; p\; mathrm\{d\}V,$.::$p$ is the

pressure ::$V$ is thevolume Taking the default direction of heat transfer, $q$, to be into the working fluid and assuming a

reversible process , we have:$delta\; q\; =\; T\; mathrm\{d\}S,$.::$T$ is

temperature ::$S$ isentropy The above two equations in the

first law of thermodynamics imply for aclosed system ::$mathrm\{d\}U\; =\; delta\; q\; -\; d\; W\; =\; Tmathrm\{d\}S-pmathrm\{d\}V,$

If we also incude the dependence on the numbers of particles in the system, the internal energy function may be written as $U(S,V,N\_\{1\},\; N\_\{2\},ldots)$ where the $N\_\{j\}$ are the numbers of particles of type j in the system. The fact that U is an extensive function when considered as a function of the variables S, V, $N\_\{1\},\; N\_\{2\},ldots$, we have: : $U(alpha\; S,alpha\; V,alpha\; N\_\{1\},alpha\; N\_\{2\},ldots\; )=alpha\; U(S,V,N\_\{1\},N\_\{2\}),$

From Euler's homogeneous function theorem we may now write the internal energy as:

: $U=TS-pV\; +\; sum\_\{i\}mu\_\{i\}N\_\{i\},$

where the $mu\_\{i\}$ are the

chemical potential s for the particles of type i in the system. These are defined as::$mu\_i\; =\; left(\; frac\{partial\; U\}\{partial\; N\_i\}\; ight)\_\{S,V,\; N\_\{j\; e\; i$

For an elastic substance the mechanical term must be replaced by the more general expression involving the stress $sigma\_\{ij\}$ and strain $varepsilon\_\{ij\}$. The infinitesimal statement is:

: $mathrm\{d\}U=Tmathrm\{d\}S+Vsigma\_\{ij\}mathrm\{d\}varepsilon\_\{ij\}$

where

Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. The Euler theorem yields for the internal energy ref_harvard|LL|Landau & Lifshitz 1986|:: $U=TS+frac\{1\}\{2\}sigma\_\{ij\}varepsilon\_\{ij\}$

For a linearly elastic material, the stress can be related to the strain by:

: $sigma\_\{ij\}=C\_\{ijkl\}\; varepsilon\_\{kl\}$

**Change in internal energy due to change in temperature and volume or pressure**The expressions given above for the internal energy involves the entropy. In practice one often wants to know the change in internal energy of a substance as a function of the change in temperature and volume, or as a function of the change in temperature and pressure.

To express dU in terms of dT and dV, we substitute

:$dS\; =\; left(frac\{partial\; S\}\{partial\; T\}\; ight)\_\{V\}dT\; +\; left(frac\{partial\; S\}\{partial\; V\}\; ight)\_\{T\}dV\; ,$

in the

fundamental thermodynamic relation :$dU\; =\; T\; dS\; -\; P\; dV,$

This gives:

:$dU\; =\; Tleft(frac\{partial\; S\}\{partial\; T\}\; ight)\_\{V\}dT\; +left\; [Tleft(frac\{partial\; S\}\{partial\; V\}\; ight)\_\{T\}\; -\; P\; ight]\; dV,$

The term $Tleft(frac\{partial\; S\}\{partial\; T\}\; ight)\_\{V\}$ is the heat capacity at constant volume $C\_\{V\}$.

The partial derivative of S w.r.t. 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 F is given by::$dF\; =\; -S\; dT\; -\; P\; dV,$

The

symmetry of second derivatives of F w.r.t. T and V yields theMaxwell relation ::$left(frac\{partial\; S\}\{partial\; V\}\; ight)\_\{T\}\; =\; left(frac\{partial\; P\}\{partial\; T\}\; ight)\_\{V\}\; ,$

This gives the expression:

:$dU\; =C\_\{V\}dT\; +left\; [Tleft(frac\{partial\; P\}\{partial\; T\}\; ight)\_\{V\}\; -\; P\; ight]\; dV,,\; ext\{\; (1)\},$

This is useful if the equation of state is known. In case of an ideal gas, $P\; =\; N\; k\; T/V$ which implies 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.

When dealing with fluids or solids, an expression in terms of the temperature and pressure is usually more useful. The partial derivative of the pressure w.r.t. temperature at constant volume can be expressed in terms of the

coefficient of thermal expansion :$alpha\; equiv\; frac\{1\}\{V\}left(frac\{partial\; V\}\{partial\; T\}\; ight)\_\{P\},$

and the isothermal

compressibility :$eta\_\{T\}\; equiv\; -frac\{1\}\{V\}left(frac\{partial\; V\}\{partial\; P\}\; ight)\_\{T\},$

by writing:

:$dV\; =\; left(frac\{partial\; V\}\{partial\; P\}\; ight)\_\{T\}dP\; +\; left(frac\{partial\; V\}\{partial\; T\}\; ight)\_\{P\}\; dT\; =\; Vleft(alpha\; dT-eta\_\{T\}dP\; ight),,\; ext\{\; (2)\}\; ,$

and equating dV to zero and solving for the ratio dP/dT. This gives:

:$left(frac\{partial\; P\}\{partial\; T\}\; ight)\_\{V\}=\; -frac\{left(frac\{partial\; V\}\{partial\; T\}\; ight)\_\{P\{left(frac\{partial\; V\}\{partial\; P\}\; ight)\_\{T=\; frac\{alpha\}\{eta\_\{T,,\; ext\{\; (3)\},$

Substituting (2) and (3) in (1) gives:

:$dU\; =\; left(C\_\{P\}-alpha\; P\; V\; ight)dT\; +left(eta\_\{T\}P-alpha\; T\; ight)VdP,$

where we have used that the heat capacity at constant pressure is related to the heat capacity at constant volume according to:

:$C\_\{P\}\; =\; C\_\{V\}\; +\; V\; Tfrac\{alpha^\{2\{eta\_\{T,$

as shown here.

**References*** cite journal

author=Alberty, R. A.

url = http://www.iupac.org/publications/pac/2001/pdf/7308x1349.pdf

title = Use of Legendre transforms in chemical thermodynamics

journal=Pure Appl. Chem.

year=2001 | volume=Vol. 73 | issue=8 | pages=1349–1380

doi = 10.1351/pac200173081349

format=PDF* cite book

author=Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo

title=Thermodynamics

publisher= McGraw-Hill Book Co.

location = New York, NY USA

edition=2nd Edition

year=1961

id =ISBN 0-07-113809-9*cite book

last = Landau

first = L. D.

authorlink = Lev Landau

coauthors = Lifshitz, E. M.

languange = English

others = (Translated from Russian by J.B. Sykes and W.H. Reid)

year = 1986

title = Theory of Elasticity (Course of Theoretical Physics Volume 7)

edition = Third ed.

publisher = Butterworth Heinemann

location = Boston, MA

id = ISBN 0-7506-2633-X**Notes****ee also***

Calorimetry

*Thermodynamic equations

*Thermodynamic potentials

*Gibbs free energy

*Wikimedia Foundation.
2010.*

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