Internal energy
Overview

In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, the internal energy is the total energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

contained by a thermodynamic system
Thermodynamic system
A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....

. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal energy has two major components, kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...

and potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

. The kinetic energy is due to the motion of the system's particles (translations
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...

, rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s, vibrations
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

), and the potential energy is associated with the static constituents of matter, static electric
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...

energy of atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s within molecules or crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

s, and the static energy of chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...

s.
Discussions
Encyclopedia
In thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...

, the internal energy is the total energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

contained by a thermodynamic system
Thermodynamic system
A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....

. It is the energy needed to create the system, but excludes the energy to displace the system's surroundings, any energy associated with a move as a whole, or due to external force fields. Internal energy has two major components, kinetic energy
Kinetic energy
The kinetic energy of an object is the energy which 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 acceleration, the body maintains this kinetic energy unless its speed changes...

and potential energy
Potential energy
In physics, potential energy is the energy stored in a body or in a system due to its position in a force field or due to its configuration. The SI unit of measure for energy and work is the Joule...

. The kinetic energy is due to the motion of the system's particles (translations
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...

, rotation
Rotation
A rotation is a circular movement of an object around a center of rotation. A three-dimensional object rotates always around an imaginary line called a rotation axis. If the axis is within the body, and passes through its center of mass the body is said to rotate upon itself, or spin. A rotation...

s, vibrations
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...

), and the potential energy is associated with the static constituents of matter, static electric
Electricity
Electricity is a general term encompassing a variety of phenomena resulting from the presence and flow of electric charge. These include many easily recognizable phenomena, such as lightning, static electricity, and the flow of electrical current in an electrical wire...

energy of atom
Atom
The atom is a basic unit of matter that consists of a dense central nucleus surrounded by a cloud of negatively charged electrons. The atomic nucleus contains a mix of positively charged protons and electrically neutral neutrons...

s within molecules or crystal
Crystal
A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

s, and the static energy of chemical bond
Chemical bond
A chemical bond is an attraction between atoms that allows the formation of chemical substances that contain two or more atoms. The bond is caused by the electromagnetic force attraction between opposite charges, either between electrons and nuclei, or as the result of a dipole attraction...

s. The internal energy of a system can be changed by heating the system or by doing work
Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...

on it; the first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...

states that the increase in internal energy is equal to the total heat added and work done. If the system is isolated, its internal energy cannot change.

For practical considerations in thermodynamics or engineering it is rarely necessary, nor convenient, 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. Thermodynamics is chiefly concerned only with changes of the internal energy.

The internal energy is a state function
State function
In thermodynamics, a state function, function of state, state quantity, or state variable is a property of a system that depends only on the current state of the system, not on the way in which the system acquired that state . A state function describes the equilibrium state of a system...

of a system
System
System is a set of interacting or interdependent components forming an integrated whole....

, because its value depends only on the current state of the system and not on the path taken or process undergone to arrive at this state. It is an extensive quantity. The SI
Si
Si, si, or SI may refer to :- Measurement, mathematics and science :* International System of Units , the modern international standard version of the metric system...

unit of energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

is the joule
Joule
The joule ; symbol J) is a derived unit of energy or work in the International System of Units. It is equal to the energy expended in applying a force of one newton through a distance of one metre , or in passing an electric current of one ampere through a resistance of one ohm for one second...

(J). Sometimes a corresponding intensive
Intensive and extensive properties
In the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...

thermodynamic property called specific internal energy is defined, which is internal energy per a unit of mass (kilogram
Kilogram
The kilogram or kilogramme , also known as the kilo, is the base unit of mass in the International System of Units and is defined as being equal to the mass of the International Prototype Kilogram , which is almost exactly equal to the mass of one liter of water...

) of the system in question. As such, the SI unit of specific internal energy is J/kg. If intensive internal energy is expressed relative to units of amount of substance
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

(mol
Mole (unit)
The mole is a unit of measurement used in chemistry to express amounts of a chemical substance, defined as an amount of a substance that contains as many elementary entities as there are atoms in 12 grams of pure carbon-12 , the isotope of carbon with atomic weight 12. This corresponds to a value...

), then it is referred to as molar internal energy and the unit is J/mol.

From the standpoint of statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

, the internal energy is equal to the ensemble average
Ensemble average
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 , according to the distribution of the system on its micro-states in this ensemble....

of the total energy of the system. It is also called intrinsic energy.

## Description and definition

The internal energy (U) is the sum of all forms of energy (Ei) intrinsic to a thermodynamic system
Thermodynamic system
A thermodynamic system is a precisely defined macroscopic region of the universe, often called a physical system, that is studied using the principles of thermodynamics....

:
It is the energy needed to create the system. It may be divided into potential energy (Upot) and kinetic energy (Ukin) components:
The kinetic energy of a system arises as the sum of the motions of all the system's particles, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The potential energy includes all energies given by the mass of particles, by the chemical composition, i.e. the chemical energy
Chemical energy
Chemical energy is the potential of a chemical substance to undergo a transformation through a chemical reaction or, to transform other chemical substances...

stored in chemical bonds having the potential to undergo chemical reactions, the nuclear energy stored by the configuration of protons, neutrons, and other elementary particles in atomic nuclei, and the physical force fields within the system, such as due to internal induced
Electrostatic induction
Electrostatic induction is a redistribution of electrical charge in an object, caused by the influence of nearby charges. Induction was discovered by British scientist John Canton in 1753 and Swedish professor Johan Carl Wilcke in 1762. Electrostatic generators, such as the Wimshurst machine, the...

electric or magnetic
Magnetism
Magnetism is a property of materials that respond at an atomic or subatomic level to an applied magnetic field. Ferromagnetism is the strongest and most familiar type of magnetism. It is responsible for the behavior of permanent magnets, which produce their own persistent magnetic fields, as well...

dipole
Dipole
In physics, there are several kinds of dipoles:*An electric dipole is a separation of positive and negative charges. The simplest example of this is a pair of electric charges of equal magnitude but opposite sign, separated by some distance. A permanent electric dipole is called an electret.*A...

moment
Moment (physics)
In physics, the term moment can refer to many different concepts:*Moment of force is the tendency of a force to twist or rotate an object; see the article torque for details. This is an important, basic concept in engineering and physics. A moment is valued mathematically as the product of the...

, as well as the energy of deformation of solids (stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

-strain
Strain (materials science)
In continuum mechanics, the infinitesimal strain theory, sometimes called small deformation theory, small displacement theory, or small displacement-gradient theory, deals with infinitesimal deformations of a continuum body...

).

Internal energy does not include the energy due to motion of a system as a whole. It further excludes any kinetic or potential energy the body may have because of its location in external gravitation
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...

al, electrostatic
Electrostatics
Electrostatics is the branch of physics that deals with the phenomena and properties of stationary or slow-moving electric charges....

, or electomagnetic field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...

s. It does, however, include the contribution to the energy due to the coupling of the internal degrees of freedom of the object to such 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. 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
Amount of substance
Amount of substance is a standards-defined quantity that measures the size of an ensemble of elementary entities, such as atoms, molecules, electrons, and other particles. It is sometimes referred to as chemical amount. The International System of Units defines the amount of substance to be...

it contains.

At any temperature greater than absolute zero
Absolute zero
Absolute zero is the theoretical temperature at which entropy reaches its minimum value. The laws of thermodynamics state that absolute zero cannot be reached using only thermodynamic means....

, potential energy and kinetic energy constantly converted into one another, but the sum remains constant in an isolated system
Isolated system
In the natural sciences an isolated system, as contrasted with an open system, is a physical system without any external exchange. If it has any surroundings, it does not interact with them. It obeys in particular the first of the conservation laws: its total energy - mass stays constant...

(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. A system at absolute zero is merely in its quantum-mechanical ground state, the lowest energy state available. At absolute zero a system has attained its minimum attainable entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

.

The kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...

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 kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. This energy is often referred to as the thermal energy of a system, relating this energy, like the temperature, to the human experience of hot and cold.

Statistical mechanics considers any system to be statistically distributed across an ensemble of N microstates. Each microstate has an energy Ei and is associated with a probability pi. The internal energy is the mean
Mean
In statistics, mean has two related meanings:* the arithmetic mean .* the expected value of a random variable, which is also called the population mean....

value of the system's total energy, i.e., the sum of all microstate energies, each weighted by their probability of occurrence:
This is the statistical expression of the first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...

.

### Internal energy changes

Thermodynamics is chiefly concerned only with the changes, ΔU, in internal energy:

The most important parameters in thermodynamics when considering the changes in total energy are the changes due to the flow of heat Q and due to mechanical work, i.e. changes in volume and pressure of the system. Accordingly, the internal energy change ΔU for a process may be written more specifically as
where Q is heat
Heat
In physics and thermodynamics, heat is energy transferred from one body, region, or thermodynamic system to another due to thermal contact or thermal radiation when the systems are at different temperatures. It is often described as one of the fundamental processes of energy transfer between...

added to a system and Wmech is the mechanical work
Mechanical work
In physics, work is a scalar quantity that can be described as the product of a force times the distance through which it acts, and it is called the work of the force. Only the component of a force in the direction of the movement of its point of application does work...

performed due to pressure or volume changes in the system. All other perturbations and energies added by other processes, such as an electric current introduced into an electronic circuit, is summarized as the term Wextra.

When a system is heated, it receives energy in form of heat. This energy increases the internal energy. However, it may be extremely difficult to determine how this extra energy is stored. In general, except in an ideal gas, it is redistributed between kinetic and potential energy. The net increase in kinetic energy is measurable by an increase in the temperature of the system. The equipartition theorem
Equipartition theorem
In classical statistical mechanics, the equipartition theorem is a general formula that relates the temperature of a system with its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition...

states that increase in thermal energy is distributed between the available degrees of freedom
Degrees of freedom (physics and chemistry)
A degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

of the fundamental oscillators in the system. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as kinetic energy. The heat introduced to a system while the temperature changes is often called sensible heat
Sensible heat
Sensible heat is the energy exchanged by a thermodynamic system that has as its sole effect a change of temperature.The term is used in contrast to a latent heat, which is the amount of energy exchanged that is hidden, meaning it cannot be observed as a change of temperature...

.

Another method to change the internal energy of a system is by performing work
Work (thermodynamics)
In thermodynamics, work performed by a system is the energy transferred to another system that is measured by the external generalized mechanical constraints on the system. As such, thermodynamic work is a generalization of the concept of mechanical work in mechanics. Thermodynamic work encompasses...

on the system, either in mechanical form by changing pressure or volume, or by other perturbations, such as directing an electrical current through the system. Finally, the internal energy increases when additional mass is transferred into the system.

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 did not change is called a latent energy, or latent heat
Latent heat
Latent heat is the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature. A typical example is a change of state of matter, meaning a phase transition such as the melting of ice or the boiling of water. The term was...

, in contrast to sensible heat. It increases only the potential energy of the system, but not its thermal energy component.

## Internal energy of the ideal gas

Thermodynamics often uses the concept of the ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...

for teaching purposes, and as an approximation for working systems. The ideal gas is a gas of particles considered as point objects of perfect spherical symmetry that interact only by elastic collisions and fill a volume such that their free mean path between collisions is much larger than their diameter. Such systems are approximated by the monoatomic gases, helium
Helium
Helium is the chemical element with atomic number 2 and an atomic weight of 4.002602, which is represented by the symbol He. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas that heads the noble gas group in the periodic table...

and the other noble gas
Noble gas
The noble gases are a group of chemical elements with very similar properties: under standard conditions, they are all odorless, colorless, monatomic gases, with very low chemical reactivity...

es. Here the kinetic energy consists only of the translational
Translation (physics)
In physics, translation is movement that changes the position of an object, as opposed to rotation. For example, according to Whittaker:...

energy of the individual atoms. Monoatomic particles are not considered to rotate or vibrate, and are not electronically excited
Energy level
A quantum mechanical system or particle that is bound -- that is, confined spatially—can only take on certain discrete values of energy. This contrasts with classical particles, which can have any energy. These discrete values are called energy levels...

to higher energies except at very high temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

s.

Therefore practical internal energy changes in an ideal gas may be described solely by changes in its kinetic energy.

## Internal energy of a closed thermodynamic system

This above summation of all components of change in internal energy assume that a positive energy denotes heat added to the system or work done on the system, while a negative energy denotes work of the system on the environment.

Typically this relationship is expressed in infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...

terms using the differentials of each term. Only the internal energy is an exact differential
Exact differential
A mathematical differential is said to be exact, as contrasted with an inexact differential, if it is of the form dQ, for some differentiable function Q....

. For a system undergoing only thermodynamics processes, i.e. a closed system that can exchange only heat and work, the change in the internal energy is
which constitutes the first law of thermodynamics
First law of thermodynamics
The first law of thermodynamics is an expression of the principle of conservation of work.The law states that energy can be transformed, i.e. changed from one form to another, but cannot be created nor destroyed...

. It may be expressed in terms of other thermodynamic parameters. Each term is composed of an intensive variable (a generalized force) and its conjugate
Conjugate variables (thermodynamics)
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 fact all thermodynamic potentials are expressed in terms of conjugate pairs....

infinitesimal extensive variable (a generalized displacement).

For example, for a non-viscous fluid, the mechanical work done on the system may be related to the pressure
Pressure
Pressure is the force per unit area applied in a direction perpendicular to the surface of an object. Gauge pressure is the pressure relative to the local atmospheric or ambient pressure.- Definition :...

p and volume V. The pressure is the intensive generalized force, while the volume is the extensive generalized displacement:.
This defines the direction of work, W, to be energy flow from the working system to the surroundings, indicated by a negative term. Taking the direction of heat transfer Q to be into the working fluid and assuming a reversible process
Reversible process (thermodynamics)
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 without loss or dissipation of energy. Due to these infinitesimal changes, the system is in thermodynamic...

, the heat is.
is temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...

is entropy
Entropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...

and the change in internal energy becomes

### Changes due to temperature and volume

The expression relating changes in internal energy to changes in temperature and volume is

This is useful if the equation of state is known.

In case of an ideal gas, we can derive that , for the change in internal energy.
The expression relating changes in internal energy to changes in temperature and volume is

The equation of state is the ideal gas law

Solve for pressure:

Substitute in to internal energy expression:

Take the derivative of pressure with respect to temperature:

Replace:

And simplify:

To express dU in terms of dT and dV, the term

is substituted in the fundamental thermodynamic relation

This gives:

The term is the heat capacity at constant volume

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
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

A is given by:

The symmetry of second derivatives
Symmetry of second derivatives
In mathematics, the symmetry of second derivatives refers to the possibility of interchanging the order of taking partial derivatives of a functionfof n variables...

of A with respect to T and V yields the Maxwell relation:

This gives the expression above.

### Changes due to temperature and pressure

When dealing with fluids or solids, an expression in terms of the temperature and pressure is usually more useful:

where it is assumed that the heat capacity at constant pressure is related to the heat capacity at constant volume according to:

The partial derivative of the pressure with respect to temperature at constant volume can be expressed in terms of the coefficient of thermal expansion

and the isothermal compressibility

by writing:

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

Substituting (2) and (3) in (1) gives the above expression.

### Changes due to volume at constant pressure

The internal pressure
Internal 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 1 pascal.Internal pressure is usually given the symbol \pi_T...

is defined as a partial derivative of the internal energy with respect to the volume at constant temperature:

## 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:

where the terms Nj are the numbers of constituents of type j in the system. The internal energy is an extensive function of the extensive variables variables S, V, and the set of components , the internal energy may be written as a linear homogeneous function
Homogeneous function
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 of this factor. More precisely, if is a function between two vector spaces over a field F, and k is an integer, then...

of first degree:

where α is a factor describing the growth of the system. The differential internal energy may be written as

where the coefficients are the chemical potential
Chemical potential
Chemical potential, symbolized by μ, is a measure first described by the American engineer, chemist and mathematical physicist Josiah Willard Gibbs. It is the potential that a substance has to produce in order to alter a system...

s for the components of type i in the system. The chemical potentials are defined as the partial derivatives of the energy with respect to the variations in composition:
As conjugate variables to the composition , the chemical potentials are intensive properties
Intensive and extensive properties
In the physical sciences, an intensive property , is a physical property of a system that does not depend on the system size or the amount of material in the system: it is scale invariant.By contrast, an extensive property In the physical sciences, an intensive property (also called a bulk...

, intrinsically characteristic of the system, and not dependent on its extent. Because of the extensive nature of U and its variables, the differential dU may be integrated and yields an expression for the internal energy:.

The sum over the composition of the system is the Gibbs energy:
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 .

## Internal energy in an elastic medium

For an elastic
Elasticity (physics)
In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

medium the mechanical energy term of the internal energy must be replaced by the more general expression involving the stress
Stress (physics)
In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

and strain . The infinitesimal statement is:

where Einstein notation
Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

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:

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

Where Cijkl is an element of the 4th-rank elastic constant tensor of the medium.

## Computational methods

The path integral Monte Carlo
Path integral Monte Carlo
Path integral Monte Carlo is a quantum Monte Carlo method in the path integral formulation of quantum mechanics.The equations often are applied assuming that quantum exchange does not matter...

method is a numerical approach for determining the values of the internal energy, based on quantum dynamical principles.

## History

James Joule
James Prescott Joule
James Prescott Joule FRS was an English physicist and brewer, born in Salford, Lancashire. Joule studied the nature of heat, and discovered its relationship to mechanical work . This led to the theory of conservation of energy, which led to the development of the first law of thermodynamics. The...

studied the relationship between heat, work, and temperature. He observed that if he did mechanical work on a fluid, such as water, by agitating the fluid, its temperature increased. He proposed that the mechanical work he was doing on the system was converted to thermal energy. Specifically, he found that 4185.5 joules of energy were needed to raise the temperature of a kilogram of water by one degree Celsius.

• Calorimetry
Calorimetry
Calorimetry is the science of measuring the heat of chemical reactions or physical changes. Calorimetry is performed with a calorimeter. The word calorimetry is derived from the Latin word calor, meaning heat...

• Enthalpy
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

• Gibbs free energy
Gibbs free energy
In thermodynamics, the Gibbs free energy is a thermodynamic potential that measures the "useful" or process-initiating work obtainable from a thermodynamic system at a constant temperature and pressure...

• Helmholtz free energy
Helmholtz free energy
In thermodynamics, the Helmholtz free energy is a thermodynamic potential that measures the “useful” work obtainable from a closed thermodynamic system at a constant temperature and volume...

• Thermodynamic equations
Thermodynamic equations
Thermodynamics is expressed by a mathematical framework of thermodynamic equations which relate various thermodynamic quantities and physical properties measured in a laboratory or production process...

• Thermodynamic potentials
Thermodynamic potentials
A thermodynamic potential is a scalar function used to represent the thermodynamic state of a system. The concept of thermodynamic potentials was introduced by Pierre Duhem in 1886. Josiah Willard Gibbs in his papers used the term fundamental functions. One main thermodynamic potential that has a...