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. It includes the energy in all the chemical bonds, and the energy of the free, conduction electrons in metals. This article needs to be cleaned up to conform to a higher standard of quality. ... In thermodynamics, the Helmholtz free energy is a thermodynamic potential which measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature. ... t In thermodynamics and molecular chemistry, the enthalpy or heat content (denoted as H or ΔH, or rarely as χ) is a quotient or description of thermodynamic potential of a system, which can be used to calculate the useful work obtainable from a closed thermodynamic system under constant pressure. ... In thermodynamics, the Gibbs free energy is a thermodynamic potential which measures the useful work obtainable from a closed thermodynamic system at a constant temperature and pressure. ... Thermodynamics (from the Greek θερμη, therme, meaning heat and δυναμις, dynamis, meaning power) is a branch of physics that studies the effects of changes in temperature, pressure, and volume on physical systems at the macroscopic scale by analyzing the collective motion of their particles using statistics. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... A physical body is an object which can be described by the theories of classical mechanics, or quantum mechanics, and experimented upon by physical instruments. ... 2-dimensional renderings (ie. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... In physics, a translation is the operation changing the positions of all objects according to the formula where is a constant vector. ... This article is about rotation as a movement of a physical body. ... Oscillation is the 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. ... Electricity (from New Latin Ä“lectricus, amberlike) is a general term for a variety of phenomena resulting from the presence and flow of electric charge. ... Properties For other meanings of Atom, see Atom (disambiguation). ... For other uses, see Crystal (disambiguation). ... A chemical bond is the physical process responsible for the attractive interactions between atoms and molecules, and that which confers stability to diatomic and polyatomic chemical compounds. ... Conduction is the movement of electrically charged particles through a transmission medium (electrical conductor). ... For other uses, see Electron (disambiguation). ... This article is about metallic materials. ...

The internal energy is a thermodynamic potential and for a closed thermodynamic system held at constant entropy, it will be minimized. In thermodynamics, thermodynamic potentials are parameters associated with a thermodynamic system and have the dimensions of energy. ... In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... The neutrality of this article is disputed. ...

One can also calculate the internal energy of electromagnetic or blackbody radiation. It is a state function of a system, an extensive quantity. 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. Electromagnetic waves can be imagined as a self-propagating transverse oscillating wave of electric and magnetic fields. ... As the temperature decreases, the peak of the black body radiation curve moves to lower intensities and longer wavelengths. ... In thermodynamics, a state function, or state quantity, is a property of a system that depends only on the current state of the system, not on the way in which the system got to that state. ... For other uses, see System (disambiguation). ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... Etymology: French calorie, from Latin calor (heat), from calere (to be warm). ... For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ...

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. 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). For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... For other uses, see Mass (disambiguation). ... Gravity redirects here. ... Electrostatics (also known as static electricity) is the branch of physics that deals with the phenomena arising from what seem to be stationary electric charges. ... The magnitude of an electric field surrounding two equally charged (repelling) particles. ... Electrostatic induction is a method by which an electrically charged object can be used to create an electrical charge in a second object, without contact between the two objects. ... For other senses of this word, see magnetism (disambiguation). ... The Earths magnetic field, which is approximately a dipole. ... It has been suggested that this article or section be merged with torque. ... In engineering mechanics, deformation is a change in shape due to an applied force. ... Stress is a measure of force per unit area within a body. ... This article is about the deformation of materials. ...

The principle of equipartition of energy in classical statistical mechanics states that each molecular degree of freedom receives 1/2 kT of energy, a result which was modified when quantum mechanics explained certain anomalies; e.g., in the observed specific heats of crystals (when hν > kT). For monatomic helium and other noble gases, the internal energy consists only of the translational kinetic energy of the individual atoms. Monatomic particles, of course, do not (sensibly) rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures. Figure 1. ... Classical mechanics (commonly confused with Newtonian mechanics, which is a subfield thereof) is used for describing the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. ... Statistical mechanics is the application of probability theory, which includes mathematical tools for dealing with large populations, to the field of mechanics, which is concerned with the motion of particles or objects when subjected to a force. ... Degrees of freedom is a general term used in explaining dependence on parameters, and implying the possibility of counting the number of those parameters. ... For a less technical and generally accessible introduction to the topic, see Introduction to quantum mechanics. ... In the scientific method, an experiment (Latin: ex- periri, of (or from) trying) is a set of observations performed in the context of solving a particular problem or question, to support or falsify a hypothesis or research concerning phenomena. ... 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 of a substance by a certain temperature interval. ... In physics and chemistry, monatomic is a combination of the words mono and atomic, and means single atom. ... For other uses, see Helium (disambiguation). ... This article is about the chemical series. ... In physics, a translation is the operation changing the positions of all objects according to the formula where is a constant vector. ... A quantum mechanical system can only be in certain states, so that only certain energy levels are possible. ... For other uses, see Temperature (disambiguation). ...

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, which is concerned with the motion of particles or objects when subjected to a force. ... In statistical mechanics, the ensemble average is defined as the weighted average of a molecular property of a system, over the set of states available to the system. ...

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:^[1]

Sensible heat is heat energy that is transported by a body that has a temperature higher than its surroundings via conduction, convection, or both. ... In thermochemistry, latent heat is the amount of energy in the form of heat released or absorbed by a substance during evaporation. ... This article is about a portion of a periodic process. ... In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... In chemistry, a chemical bond is the force which holds together atoms in molecules or crystals. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... This article concerns the energy stored in the nuclei of atoms; for the use of nuclear fission as a power source, see Nuclear power. ... A fundamental interaction is a mechanism by which particles interact with each other, and which cannot be explained by another more fundamental interaction. ... In thermal physics, heat transfer is the passage of thermal energy from a hot to a cold body. ... Mass transfer is the phrase commonly used in engineering for physical processes that involve molecular and convective transport of atoms and molecules within physical systems. ... Look up work in Wiktionary, the free dictionary. ... Thermodynamics (Greek: thermos = heat and dynamic = change) is the physics of energy, heat, work, entropy and the spontaneity of processes. ... In thermal physics, thermal energy is the energy portion of a system that increases with its temperature. ...

The first law of thermodynamics

The internal energy is essentially defined by the first law of thermodynamics which states that energy is conserved: The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ...

where

ΔU is the change in internal energy of a system during a process.

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.

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 equivalently in infinitesimal terms as: For other uses, see Heat (disambiguation) In physics, heat, symbolized by Q, is energy transferred from one body or system to another due to a difference in temperature. ... The joule (IPA: or ) (symbol: J) is the SI unit of energy. ... Look up si, Si, SI in Wiktionary, the free dictionary. ... In common usage positive is sometimes used in affirmation, as a synonym for yes or to express certainty. Look up Positive on Wiktionary, the free dictionary In mathematics, a number is called positive if it is bigger than zero. ... Negative has meaning in several contexts: Look up negative in Wiktionary, the free dictionary. ... In physics, mechanical work is the amount of energy transferred by a force. ... Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. ...

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.) The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ...

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, or magnetic 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. The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ... Gravity is a force of attraction that acts between bodies that have mass. ... 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. ... Magnetic field lines shown by iron filings In physics, the space surrounding moving electric charges, changing electric fields and magnetic dipoles contains a magnetic field. ...

Expressions for the internal energy

Strictly speaking, the internal energy cannot be precisely measured. This is because only changes in the internal energy can be measured, and the total internal energy of a given system is the difference between the internal energy of the system and the internal energy of the same system at absolute zero temperature. Since absolute zero cannot be attained, the total internal energy cannot be precisely measured. The same is true of other thermodynamic parameters such as entropy and the chemical potential. For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a...

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 physics and chemistry, an intensive quantity (also referred to as an intensive variable) is a physical quantity whose value does not depend on the amount of the substance for which it is measured. ... Thermodynamic potentials Maxwell relations Bridgmans equations Exact differential (edit) In thermodynamics, the internal energy of a system is expressed in terms of pairs of conjugate variables such as pressure/volume or temperature/entropy. ... In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ...

For example, for a non-viscous fluid, the mechanical work done on the system may be related to the pressure p and volume V. The pressure is the intensive generalized force, while the volume is the extensive generalized displacement: This article is about pressure in the physical sciences. ... For other uses, see Volume (disambiguation). ...

Taking the default direction of work, W, to be from the working fluid to the surroundings,

.

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 This article is about pressure in the physical sciences. ... For other uses, see Volume (disambiguation). ... In thermodynamics, a reversible process (or reversible cycle if the process is cyclic) is a process that can be reversed by means of infinitesimal changes in some property of the system (Sears and Salinger, 1986). ...

.

T is temperature

S is entropy

Although the internal energy is not exactly measurable, it may be expressed in terms of other similarly unmeasurable quantities. Using the above two equations in the first law of thermodynamics to construct one possible expression for the internal energy of a closed system gives: For other uses, see Temperature (disambiguation). ... For other uses, see: information entropy (in information theory) and entropy (disambiguation). ... The first law of thermodynamics, a generalized expression of the law of the conservation of energy, states: // Description Essentially, the First Law of Thermodynamics declares that energy is conserved for a closed system, with heat and work being the forms of energy transfer. ... In thermodynamics, a closed system, as contrasted with an isolated system, can exchange heat and work, but not matter, with its surroundings. ...

The internal energy function may be written as U(S,V) in which case it follows that, since U, S, and V are extensive In physics and chemistry, an extensive quantity (also referred to as an extensive variable) is a physical quantity whose value is proportional to the size of the system it describes. ...

From Euler's homogeneous function theorem we may now write the internal energy as: In mathematics, a homogeneous function is a function with multiplicative scaling behaviour: if the argument is multiplied by some factor, then the result is multiplied by some power of this factor. ...

If the (non-viscous) fluid gains energy from the addition of particles, we add the chemical energy term:

.

μ_i is the chemical potential of chemical species i. It is an intensive variable.

N_i is the particle number of chemical species i. It is an extensive variable.

For an elastic substance the mechanical term must be replaced by the more general expression involving the stress σ_ij and strain . The infinitesimal statement is: In thermodynamics and chemistry, chemical potential, symbolized by μ, is a term introduced in 1876 by the American mathematical physicist Willard Gibbs, which he defined as follows: Gibbs noted also that for the purposes of this definition, any chemical element or combination of elements in given proportions may be considered a... Chemical species is a common, general name for atoms, molecules, molecular fragments and ions as entities being subjected to a chemical process or to a measurement. ... In physics and chemistry, an intensive quantity (also referred to as an intensive variable) is a physical quantity whose value does not depend on the amount of the substance for which it is measured. ... The particle number, N, is the number of so called elementary particles (or elementary constituents) in a thermodynamical system. ... Elasticity is a branch of physics which studies the properties of elastic materials. ... Stress is a measure of force per unit area within a body. ... Look up strain in Wiktionary, the free dictionary. ...

where Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. For a linearly elastic material, the stress can be related to the strain by: This article or section does not adequately cite its references or sources. ...

and the Euler theorem yields for the internal energy (Landau & Lifshitz 1986):

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). 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. 

Evgeny Mikhailovich Lifshitz (Russian: ; February 21, 1915 – October 29, 1985) was a notable Soviet physicist. ...

See also

• Calorimetry
• Thermodynamic equations
• Thermodynamic potentials
• Gibbs free energy