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Internal energy

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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. It includes the energy in all the chemical bonds, and the energy of the free, conduction electrons in metals.

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.

Overview

Internal energy does not include the translational or rotational kinetic energy of a body as a whole. It also does not include the relativistic mass-energy equivalent E = mc. It excludes any potential energy a body may have because of its location in an 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).

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.

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

The First Law of Thermodynamics

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

Δ U = Q + W + W ′ {\displaystyle \Delta U=Q+W+W'\,}

where

ΔU is the value of the internal energy after a process minus its value before.
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:

d U = δ Q + δ W + δ W ′ {\displaystyle dU=\delta Q+\delta W+\delta W'\,}

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

From a microscopic point of view, the internal energy may be found in many different forms. For a gas it may consist almost entirely of the kinetic energy of the gas molecules. It may also consist of the potential energy of these molecules in a gravitational, electric, 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.

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.

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:

δ W = − P d V {\displaystyle \delta W=-PdV\,}

The heat energy term can be related to the temperature T and the entropy S:

δ Q = T d S {\displaystyle \delta Q=TdS\,}

Although the internal energy is not exactly measureable, it may be expressed in terms of other similarly unmeasureable quantities. Using the above two equations to construct one possible expression for the internal energy gives:

d U = T d S − P d V {\displaystyle dU=TdS-PdV\,}

The internal energy function may be written as U ( S , V ) {\displaystyle U(S,V)} in which case it follows that, since U, S, and V are extensive

U ( α S , α V ) = α U ( S , V ) {\displaystyle U(\alpha S,\alpha V)=\alpha U(S,V)\,}

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

U = T S − P V {\displaystyle U=TS-PV\,}

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

U = T S − P V + ∑ i μ i N i {\displaystyle U=TS-PV+\sum _{i}\mu _{i}N_{i}\,}

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

d U = T d S + V σ i j d ε i j {\displaystyle dU=TdS+V\sigma _{ij}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. For a linearly elastic material, the stress can be related to the strain by:

σ i j = C i j k l ε k l {\displaystyle \sigma _{ij}=C_{ijkl}\varepsilon _{kl}}

and the Euler theorem yields for the internal energy Template:Ref harvard:

U = T S + 1 2 σ i j ε i j {\displaystyle U=TS+{\frac {1}{2}}\sigma _{ij}\varepsilon _{ij}}

References

  • Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). Thermodynamics (2nd Edition ed.). New York, NY USA: McGraw-Hill Book Co. ISBN 0071138099. {{cite book}}: |edition= has extra text (help)CS1 maint: multiple names: authors list (link)
  • Landau, L. D. (1986). Theory of Elasticity (Course of Theoretical Physics Volume 7). (Translated from Russian by J.B. Sykes and W.H. Reid) (Third ed. ed.). Boston, MA: Butterworth Heinemann. ISBN 075062633X. {{cite book}}: |edition= has extra text (help); Unknown parameter |coauthors= ignored (|author= suggested) (help); Unknown parameter |languange= ignored (help)

See also

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