Internal energy

In thermodynamics, the internal energy is the total energy contained by a thermodynamic system. It is the energy necessary 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 and potential energy. The kinetic energy is due to the motion of the system's particles (translations, rotations, vibrations), and the potential energy is associated with the static constituents of matter, static electric energy of atoms within molecules or crystals, the static energy of chemical bonds. The internal energy of a system can be changed by heating the system or by doing work on it; the first law of thermodynamics 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 with changes of the internal energy.

The internal energy is a state function of a system, 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 unit of energy is the joule. Sometimes physicists define a corresponding intensive thermodynamic property called specific internal energy, which is internal energy per a unit of mass of the system in question. As such, the SI unit of specific internal energy is J/kg. If intensive internal energy is expressed per amount of substance, then it is referred to as molar internal energy and the unit is J/mol.

From the standpoint of statistical mechanics, the internal energy is equal to the ensemble average of the total energy of the system. It is also called intrinsic energy.

Composition and interactions

Internal energy is the sum of all forms of energy intrinsic to a system. It is the energy needed to create the system. It may be divided into kinetic energy and potential energy 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, the chemical composition, i.e. the potential energy stored in bonds, and the physical force fields within the system, such as due to internal induced electric or magnetic dipole moment, as well as the energy of deformation of solids (stress-strain).

At any temperature greater than absolute zero, potential energy and kinetic energy are constantly exchanged into one another. 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.

Internal energy does not include the energy due to motion of a system as a whole. It further excludes any kinetic or potential energy a body may have because of its location in external gravitational, electrostatic, or electromagnetic force fields.

Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as a rough 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 and the other noble gases. Here the kinetic energy consists only of the translational energy of the individual atoms. Monoatomic particles are not considered to rotate or vibrate, and are not electronically excited to higher energies except at very high temperatures.

The kinetic energy portion of the internal energy gives rise to the temperature of the system. Statistical mechanics 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 to the human experience of hot and cold.

Expressions for the internal 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. 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.

Changes in 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).

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:

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

$\delta W=p\mathrm {d} V\,$.

$p$ 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

$\delta Q=T\mathrm {d} S\,$.

$T$ is temperature

$S$ is entropy

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

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

If we also include 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. U is an extensive function, so when considered as a function of the extensive variables variables S, V, and the particle numbers $N_{1},N_{2},\ldots$, we have:

$U(\alpha S,\alpha V,\alpha N_{1},\alpha N_{2},\ldots )=\alpha U(S,V,N_{1},N_{2},\ldots )\,$

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 potentials for the particles of type i in the system. These are defined as:

$\mu _{i}=\left({\frac {\partial U}{\partial N_{i}}}\right)_{S,V,N_{j\neq i}}$

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:

$\mathrm {d} U=T\mathrm {d} S+V\sigma _{ij}\mathrm {d} \varepsilon _{ij}$

where Einstein notation has been used for the tensors, in which there is a summation over all repeated indices in the product term. The Euler theorem yields for the internal energy:

$U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}$

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

$\sigma _{ij}=C_{ijkl}\varepsilon _{kl}$

The path integral Monte Carlo method is a numerical approach for determining the values of internal energy, based on quantum dynamical principles.

Changes in internal energy

The change in internal energy of a system may be written as

$\Delta U=Q+W_{exp}+W_{extra}\,$

where

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

Q is heat added to a system;

W_exp is the mechanical work (pressure-volume work) due to expansion of a system

W_extra is energy added by all other processes, such as an electrical current introduced into an electronic circuit.

This assumes that a positive energy denotes heat added or work done to the system, while a negative energy denotes work of the system on the environment. Typically this relationship is expressed in infinitesimal terms using the differentials of each term. Only the internal energy is an exact differential.

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.

Changes due to temperature and volume

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

$dU=C_{V}dT+\left[T\left({\frac {\partial p}{\partial T}}\right)_{V}-p\right]dV\,\,{\text{ (1)}}\,$

This is useful if the equation of state is known. In case of an ideal gas, $p=NkT/V$ 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.

Derivation of dU in terms of dT and dV

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

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

in the fundamental thermodynamic relation

$dU=TdS-pdV\,$

This gives:

$dU=T\left({\frac {\partial S}{\partial T}}\right)_{V}dT+\left[T\left({\frac {\partial S}{\partial V}}\right)_{T}-p\right]dV\,$

The term $T\left({\frac {\partial S}{\partial T}}\right)_{V}$ is the heat capacity at constant volume $C_{V}$.

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 A is given by:

$dA=-SdT-pdV\,$

The symmetry of second derivatives of A with respect to T and V yields the Maxwell relation:

$\left({\frac {\partial S}{\partial V}}\right)_{T}=\left({\frac {\partial p}{\partial T}}\right)_{V}\,$

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:

$dU=\left(C_{p}-\alpha pV\right)dT+\left(\beta _{T}p-\alpha T\right)Vdp\,$

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

$C_{p}=C_{V}+VT{\frac {\alpha ^{2}}{\beta _{T}}}\,$

Derivation of dU in terms of dT and dP

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

$\alpha \equiv {\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{p}\,$

and the isothermal compressibility

$\beta _{T}\equiv -{\frac {1}{V}}\left({\frac {\partial V}{\partial p}}\right)_{T}\,$

by writing:

$dV=\left({\frac {\partial V}{\partial p}}\right)_{T}dp+\left({\frac {\partial V}{\partial T}}\right)_{p}dT=V\left(\alpha dT-\beta _{T}dp\right)\,\,{\text{ (2)}}\,$

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

$\left({\frac {\partial p}{\partial T}}\right)_{V}=-{\frac {\left({\frac {\partial V}{\partial T}}\right)_{p}}{\left({\frac {\partial V}{\partial p}}\right)_{T}}}={\frac {\alpha }{\beta _{T}}}\,\,{\text{ (3)}}\,$

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

Changes due to volume at constant pressure

Main article: Internal pressure

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

$\pi _{T}=\left({\frac {\partial U}{\partial V}}\right)_{T}$

Equipartition theorem

The principle of equipartition of energy in classical statistical mechanics states that each molecular quadratic degree of freedom receives 1/2 kT of energy, a result which had to be modified when quantum mechanics explained certain anomalies, such as discrepancies in the observed specific heats of crystals when the expected thermal energy per degree of freedom is less than the energy necessary to move that degree of freedom up one quantum energy level.

The equipartition theorem yields simple expressions for the thermal energy. In case of an ideal gas, the thermal energy is exactly given by the kinetic energy of the constituent particles. The internal energy per particle is equivalent to the average translational kinetic energy of each particle. Ignoring quantum effects, this is given by equipartition of energy.

According to the equipartition theorem, the thermal energy of a molecule in a thermal bath is

$U_{thermal}=f\cdot {\tfrac {1}{2}}kT$

where f is the number of degrees of freedom, T is the temperature, and k is Boltzmann's constant. For example, for a monatomic ideal gas, each particle has three degrees of freedom, and thus

$U_{thermal,monatomic}={\tfrac {3}{2}}kT.$

When the spacing between the energy levels of a particular degree of freedom becomes of the order of k T or less, the energy in that degree of freedom becomes less than given by the equipartition theorem and it vanishes exponentially as k T becomes much less than the energy difference. The system is then frozen in the ground state.

For a gas at room temperature at normal densities, the vibrational degrees of freedom are usually frozen, while the rotational and vibrational degrees of freedom can be treated classically. Quantum effects for the translational degrees of freedom become important when the specific volume per particle is of the same order or smaller than $\Lambda ^{3}$ where $\Lambda$ is the thermal de Broglie wavelength. In this regime quantum statistical effects become important. Depending on whether the molecules are Fermions or Bosons, the gas will become a degenerate Fermi gas or a Bose-Einstein condensate, respectively. E.g., at room temperature, the electrons in a metal form a degenerate Fermi gas, the internal energy per electron is of the order of $kT_{f}$ with $T_{f}$ the Fermi temperature which can be of the order of 80,000 K. So, in this case, the equipartition theorem underestimates the internal energy by many orders of magnitude.

History

James Joule 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 4200 joules of energy were needed to raise the temperature of a kilogram of water by one degree Celsius.

References

• Alberty, R. A. (2001). "Use of Legendre transforms in chemical thermodynamics" (PDF). Pure Appl. Chem. 73 (8): 1349–1380. doi:10.1351/pac200173081349.
• Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). Thermodynamics (2nd ed.). New York, NY USA: McGraw-Hill Book Co. ISBN 0-07-113809-9.{{cite book}}: CS1 maint: multiple names: authors list (link)
• 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. {{cite book}}: Invalid |ref=harv (help)

See also

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