Internal energy

Statistical mechanics
$S = k_B \, \ln\Omega$
Statistical thermodynamics Kinetic theory
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In thermodynamics, the internal energy of a thermodynamic system, or a body with well-defined boundaries, denoted by U, or sometimes E, is the total of the kinetic energy due to the motion of molecules (translational, rotational, vibrational) and the potential energy associated with the vibrational and electric energy of atoms within molecules or crystals. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Thermodynamics, statistical thermodynamics is the study of the microscopic behaviors of Thermodynamic systems using Probability theory. Kinetic theory (or kinetic theory of gases) attempts to explain Macroscopic properties of Gases such as pressure temperature or volume by considering In Physics, thermodynamics (from the Greek θερμη therme meaning " Heat " and δυναμις dynamis meaning " In Thermodynamics, a thermodynamic system, originally called a working substance, is defined as that part of the universe that is under consideration In Physics, a physical body (sometimes called simply a body or even an object) is a collection of Masses taken to be one In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it The kinetic energy of an object is the extra Energy which it possesses due to its motion In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by In Physics, translation is movement that changes the position of an object as opposed to Rotation. A rotation is a movement of an object in a circular motion A two- Dimensional object rotates around a center (or point) of rotation Oscillation is the repetitive variation typically in Time, of some measure about a central value (often a point of Equilibrium) or between two or more different states Potential energy can be thought of as Energy stored within a physical system History See also Atomic theory, Atomism The concept that matter is composed of discrete units and cannot be divided into arbitrarily tiny In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating It includes the energy in all the chemical bonds, and the energy of the free, conduction electrons in metals. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός A chemical bond is the physical process responsible for the attractive interactions between Atoms and Molecules and which confers stability to diatomic and polyatomic Electrical conduction is the movement of electrically charged particles through a Transmission medium ( Electrical conductor) The electron is a fundamental Subatomic particle that was identified and assigned the negative charge in 1897 by J The M acro E xpansion T emplate A ttribute L anguage complements TAL, providing macros which allow the reuse of code across

One can also calculate the internal energy of electromagnetic or blackbody radiation. Electromagnetic radiation takes the form of self-propagating Waves in a Vacuum or in Matter. In Physics, a black body is an object that absorbs all light that falls on it It is a state function of a system, and is an extensive quantity. In Thermodynamics, a state function, state quantity, or a function of state, is a property of a system that depends only on the current System (from Latin systēma, in turn from Greek systēma is a set of interacting or interdependent Entities, real or abstract In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the The SI unit of energy is the joule although other historical, conventional units are still in use, such as the (small and large) calorie for heat. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity This article is about the unit of energy For its use in Nutrition and Food labelling regulations, see the article on Food energy. In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature

1 Overview
2 Composition
3 The first law of thermodynamics
4 Expressions for the internal energy
5 References
6 Notes
7 See also

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 = mc². Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object It excludes any potential energy a body may have because of its location in external gravitational or electrostatic field, although the potential energy it has in a field due to an induced electric or magnetic dipole moment does count, as does the energy of deformation of solids (stress-strain). In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it Gravitation is a natural Phenomenon by which objects with Mass attract one another Electrostatics is the branch of Science that deals with the Phenomena arising from what seems to be stationary Electric charges Since Classical In Physics, a field is a Physical quantity associated to each point of Spacetime. Electrostatic induction is a redistribution of Electrical charge in an object caused by the influence of nearby charges In Physics, magnetism is one of the Phenomena by which Materials exert attractive or repulsive Forces on other Materials. In physics there are two kinds of dipoles ( Hellènic: di(s- = two- and pòla = pivot hinge An electric dipole is a In Physics, the moment of force (often just moment, though there are other quantities of that name such as Moment of inertia) is a Pseudovector In Materials science, deformation is a change in the shape or size of an object due to an applied force. Stress is a measure of the average amount of Force exerted per unit Area.

The principle of equipartition of energy in classical statistical mechanics states that each molecular quadratic degree of freedom receives 1/2 kT of energy, ^[1] a result which was modified when quantum mechanics explained certain anomalies; e. In classical Statistical mechanics, the equipartition theorem is a general formula that relates the Temperature of a system with its average energies Classical mechanics is used for describing the motion of Macroscopic objects from Projectiles to parts of Machinery, as well as Astronomical objects Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics For information on degrees of freedom in other sciences see Degrees of freedom. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons g. , in the observed specific heats of crystals (when hν > kT). In scientific inquiry an experiment ( Latin: Ex- periri, "to try out" is a method of investigating particular types of research questions or Specific heat capacity, also known simply as specific heat, is the measure of the heat energy required to increase the Temperature of a unit quantity For monoatomic helium and other noble gases, the internal energy consists only of the translational kinetic energy of the individual atoms. In Physics and Chemistry, monatomic is a combination of the words "mono" and "atomic" and means "single Atom. Helium ( He) is a colorless odorless tasteless non-toxic Inert Monatomic Chemical History Noble gas is translated from the German noun de ''Edelgas'' first used in 1898 by Hugo Erdmann to indicate their extremely low level of reactivity In Physics, translation is movement that changes the position of an object as opposed to Rotation. Monoatomic particles, of course, do not (sensibly) rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures. A quantum mechanical system or particle that is bound, confined spacially can only take on certain discrete values of energy as opposed to classical particles which Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature

From the standpoint of statistical mechanics, the internal energy is equal to the ensemble average of the total energy of the system. Statistical mechanics is the application of Probability theory, which includes mathematical tools for dealing with large populations to the field of Mechanics In Statistical mechanics, the ensemble average is defined as the Mean of a quantity that is a function of the micro-state of a system (the ensemble of possible

Composition

Internal energy – 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 comprised of the following types of energies:^[2]

Type	Composition of Internal Energy (U)
Sensible energy	the portion of the internal energy of a system associated with kinetic energies (molecular translation, rotation, and vibration; electron translation and spin; and nuclear spin) of the molecules. Sensible heat is Potential energy in the form of thermal energy or Heat.
Latent energy	the internal energy associated with the phase of a system. In Thermochemistry, latent heat is the amount of Energy in the form of Heat released or absorbed by a substance during a change of phase A state of matter (or physical state, or form of matter) has physical properties which are qualitatively different from other states of matter
Chemical energy	the internal energy associated with the atomic bonds in a molecule. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός A chemical bond is the physical process responsible for the attractive interactions between Atoms and Molecules and which confers stability to diatomic and polyatomic
Nuclear energy	the very large amount of energy associated with the strong bonds within the nucleus of the atom itself. Nuclear Energy is released by the splitting (fission or merging together (fusion of the nuclei of Atom (s Nuclear Energy is released by the splitting (fission or merging together (fusion of the nuclei of Atom (s
Energy interactions	those types of energies not stored in the system (e. In Physics, a fundamental interaction or fundamental force is a mechanism by which particles interact with each other and which cannot be explained in terms g. heat transfer, mass transfer, and work), but which are recognized at the system boundary as they cross it, which represent gains or losses by a system during a process. In thermal physics, heat transfer is the passage of Thermal energy from a hot to a colder body Mass transfer is the phrase commonly used in engineering for physical processes that involve molecular and convective transport of Atoms and Molecules In Thermodynamics, a thermodynamic system, originally called a working substance, is defined as that part of the universe that is under consideration
Thermal energy	the sum of sensible and latent forms of internal energy. Thermal energy is the sum of the sensible energy and latent energy.

The first law of thermodynamics

The internal energy is essentially defined by the first law of thermodynamics which states that energy is conserved:

$\Delta U = Q + W + W' \,$

where

ΔU is the change in internal energy of a system during a process. In Thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the Conservation of energy.

Q is heat added to a system (measured in joules in SI); that is, a positive value for Q represents heat flow into a system while a negative value denotes heat flow out of a system. In Physics, heat, symbolized by Q, is Energy transferred from one body or system to another due to a difference in Temperature The joule (written in lower case ˈdʒuːl or /ˈdʒaʊl/ (symbol J) is the SI unit of Energy measuring heat, Electricity

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:

$\mathrm{d}U = \delta Q + \delta W + \delta W'\,$

where the terms now represent infinitesimal amounts of the respective quantities. In Physics, mechanical work is the amount of Energy transferred by a Force. Infinitesimals (from a 17th century Modern Latin coinage infinitesimus, originally referring to the " Infinite[[ th]]" member of a series have 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. In Thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the Conservation of energy. )

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. The kinetic energy of an object is the extra Energy which it possesses due to its motion It may also consist of the potential energy of these molecules in a gravitational, electric, or magnetic field. Gravitation is a natural Phenomenon by which objects with Mass attract one another In Physics, the space surrounding an Electric charge or in the presence of a time-varying Magnetic field has a property called an electric field (that can In Physics, a magnetic field is a Vector field that permeates space and which can exert a magnetic force on moving Electric charges 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 infinitesimal extensive variable (a generalized displacement). In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the In Thermodynamics, the Internal energy of a system is expressed in terms of pairs of conjugate variables such as temperature/entropy or pressure/volume In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the

For example, for a non-viscous fluid, the mechanical work done on the system may be related to the pressure p and volume V. Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically 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 the volume

Taking the default direction of heat transfer, $q$ , to be into the working fluid and assuming a reversible process, we have

$\delta q = T \mathrm{d}S\,$ . Pressure (symbol 'p' is the force per unit Area applied to an object in a direction perpendicular to the surface The volume of any solid plasma vacuum or theoretical object is how much three- Dimensional space it occupies often quantified numerically

T

is temperature

S

is entropy

The above two equations in the first law of thermodynamics imply for a closed system:

$\mathrm{d}U = \delta q - d W = T\mathrm{d}S-p\mathrm{d}V\,$

If we also incude the dependence on the numbers of paticles 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. Temperature is a physical property of a system that underlies the common notions of hot and cold something that is hotter generally has the greater temperature In Thermodynamics (a branch of Physics) entropy, symbolized by S, is a measure of the unavailability of a system ’s Energy In Thermodynamics, the first law of thermodynamics is an expression of the more universal physical law of the Conservation of energy. A Closed system is a System in the state of being isolated from the environment 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 $μ i$ are the chemical potentials for the particles of type i in the system. In the Physical sciences an intensive property (also called a bulk property) is a Physical property of a system that does not depend on the In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power In Thermodynamics and Chemistry, chemical potential, symbolized by μ, is a term introduced by the American engineer chemist and mathematical These are defined as:

$\mu_i = \left( \frac{\partial U}{\partial N_i} \right)_{S,V, N_{j \ne i}}$

For an elastic substance the mechanical term must be replaced by the more general expression involving the stress $σ i j$ and strain $\varepsilon_{ij}$ . A material is said to be elastic if it deforms under stress (e Stress is a measure of the average amount of Force exerted per unit Area. The infinitesimal statement is:

$\mathrm{d}U=T\mathrm{d}S+V\sigma_{ij}\mathrm{d}\varepsilon_{ij}$

where Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational The Euler theorem yields for the internal energy (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:

$\sigma_{ij}=C_{ijkl} \varepsilon_{kl}$

References

Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics". Pure Appl. Chem. Vol. 73 (8): 1349–1380.

Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). Thermodynamics, 2nd Edition, New York, NY USA: McGraw-Hill Book Co. . ISBN 0-07-113809-9.

Landau, L. D.; Lifshitz, E. M. (1986). Lev Davidovich Landau ( Russian language: Ле́в Дави́дович Ланда́у ( January 22, 1908 &ndash April 1, 1968 Evgeny Mikhailovich Lifshitz (Евгений Михайлович Лифшиц February 21 1915 &ndash October 29 1985) was a leading Soviet Theory of Elasticity (Course of Theoretical Physics Volume 7), (Translated from Russian by J. B. Sykes and W. H. Reid), Third ed. , Boston, MA: Butterworth Heinemann. ISBN 0-7506-2633-X.

Notes

^ Reif, Frederick (1965). Statistical Physics. New York: McGraw-Hill Book Company, 246-250.
^ Cengel, Yungus, A. ; Boles, Michael (2002). Thermodynamics - An Engineering Approach, 4th ed. . McGraw-Hill, 17-18. ISBN 0-07-238332-1.

Dictionary

-noun

(physics) a property, characteristic of the state of a thermodynamic system, the change of which is equal to the heat absorbed minus the work done by the system

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