Internal energy

 ( Physical Quantities ) 

The internal energy of a thermodynamic system is the energy contained within it, measured as the quantity of energy necessary to bring the system from its Standard state internal state to its present internal state of interest, accounting for the gains and losses of energy due to changes in its internal state, including such quantities as magnetization.Crawford, F. H. (1963), pp. 106–107.Haase, R. (1971), pp. 24–28. It excludes the kinetic energy of motion of the system as a whole and the potential energy of position of the system as a whole, with respect to its surroundings and external force fields. It includes the thermal energy, i.e., the constituent particles' kinetic energies of motion relative to the motion of the system as a whole. The internal energy of an isolated system cannot change, as expressed in the law of conservation of energy, a foundation of the first law of thermodynamics.

The internal energy cannot be measured absolutely. Thermodynamics concerns changes in the internal energy, not its absolute value. The processes that change the internal energy are transfers, into or out of the system, of matter, or of energy, as heat, or by thermodynamic work.Max Born (1949), Appendix 8, pp. 146–149. These processes are measured by changes in the system's properties, such as temperature, entropy, volume, electric polarization, and molar constitution. The internal energy depends only on the internal state of the system and not on the particular choice from many possible processes by which energy may pass into or out of the system. It is a State function, a thermodynamic potential, and an extensive property.

Thermodynamics defines internal energy macroscopically, for the body as a whole. In statistical mechanics, the internal energy of a body can be analyzed microscopically in terms of the kinetic energies of microscopic motion of the system's particles from translations, , and oscillation, and of the potential energies associated with microscopic forces, including chemical bonds.

The unit of energy in the International System of Units (SI) is the joule (J). The internal energy relative to the mass with unit J/kg is the specific internal energy. The corresponding quantity relative to the amount of substance with unit J/mol is the molar internal energy.

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Cardinal functions The internal energy of a system depends on its entropy S, its volume V and its number of massive particles: . It expresses the thermodynamics of a system in the energy representation. As a State function, its arguments are exclusively extensive variables of state. Alongside the internal energy, the other cardinal function of state of a thermodynamic system is its entropy, as a function, , of the same list of extensive variables of state, except that the entropy, , is replaced in the list by the internal energy, . It expresses the entropy representation.Tschoegl, N.W. (2000), p. 17.Herbert Callen (1960/1985), Chapter 5.Münster, A. (1970), p. 6.

Each cardinal function is a monotonic function of each of its natural or canonical variables. Each provides its characteristic or fundamental equation, for example , that by itself contains all thermodynamic information about the system. The fundamental equations for the two cardinal functions can in principle be interconverted by solving, for example, for , to get .

In contrast, Legendre transforms are necessary to derive fundamental equations for other thermodynamic potentials and . The entropy as a function only of extensive state variables is the one and only cardinal function of state for the generation of Massieu functions. It is not itself customarily designated a 'Massieu function', though rationally it might be thought of as such, corresponding to the term 'thermodynamic potential', which includes the internal energy.Münster, A. (1970), Chapter 3.Bailyn, M. (1994), pp. 206–209.

For real and practical systems, explicit expressions of the fundamental equations are almost always unavailable, but the functional relations exist in principle. Formal, in principle, manipulations of them are valuable for the understanding of thermodynamics.

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Description and definition The internal energy $U$ of a given state of the system is determined relative to that of a standard state of the system, by adding up the macroscopic transfers of energy that accompany a change of state from the reference state to the given state:

$\backslash Delta\; U\; =\; \backslash sum\_i\; E\_i,$

where $\backslash Delta\; U$ denotes the difference between the internal energy of the given state and that of the reference state, and the $E\_i$ are the various energies transferred to the system in the steps from the reference state to the given state. It is the energy needed to create the given state of the system from the reference state. From a non-relativistic microscopic point of view, it may be divided into microscopic potential energy, $U\_\backslash text\{micro,pot\}$, and microscopic kinetic energy, $U\_\backslash text\{micro,kin\}$, components:

$U\; =\; U\_\backslash text\{micro,pot\}\; +\; U\_\backslash text\{micro,kin\}.$

The microscopic kinetic energy of a system arises as the sum of the motions of all the system's particles with respect to the center-of-mass frame, whether it be the motion of atoms, molecules, atomic nuclei, electrons, or other particles. The microscopic potential energy algebraic summative components are those of the Chemical energy and nuclear particle bonds, and the physical force fields within the system, such as due to internal induced electric or magnetism dipole moment, as well as the energy of deformation of solids (stress-strain). Usually, the split into microscopic kinetic and potential energies is outside the scope of macroscopic thermodynamics.

Internal energy does not include the energy due to motion or location of a system as a whole. That is to say, it excludes any kinetic or potential energy the body may have because of its motion or location in external , electrostatics, or electromagnetics fields. It does, however, include the contribution of such a field to the energy due to the coupling of the internal degrees of freedom of the object with 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.I. Klotz, R. Rosenberg, Chemical Thermodynamics - Basic Concepts and Methods, 7th ed., Wiley (2008), p.39 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 it contains.

At any temperature greater than absolute zero, microscopic potential energy and kinetic energy are constantly converted into one another, but the sum remains constant in an isolated system (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 of given composition has attained its minimum attainable entropy.

The microscopic 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 microscopic kinetic energy to the macroscopically observed empirical property that is expressed as temperature of the system. While temperature is an intensive measure, this energy expresses the concept as an extensive property of the system, often referred to as the thermal energy,Leland, T. W. Jr., Mansoori, G. A., pp. 15, 16. Thermal energy – Hyperphysics. The scaling property between temperature and thermal energy is the entropy change of the system.

Statistical mechanics considers any system to be statistically distributed across an ensemble of $N$ microstates. In a system that is in thermodynamic contact equilibrium with a heat reservoir, each microstate has an energy $E\_i$ and is associated with a probability $p\_i$. The internal energy is the mean value of the system's total energy, i.e., the sum of all microstate energies, each weighted by its probability of occurrence:

$U\; =\; \backslash sum\_\{i=1\}^N\; p\_i\; \backslash ,E\_i.$

This is the statistical expression of the law of conservation of energy.

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Internal energy changes Thermodynamics is chiefly concerned with the changes in internal energy $\backslash Delta\; U$.

For a closed system, with matter transfer excluded, the changes in internal energy are due to heat transfer $Q$ and due to thermodynamic work $W$ done by the system on its surroundings.This article uses the sign convention of the mechanical work as often defined in engineering, which is different from the convention used in physics and chemistry; in engineering, work performed by the system against the environment, e.g., a system expansion, is taken to be positive, while in physics and chemistry, it is taken to be negative. Accordingly, the internal energy change $\backslash Delta\; U$ for a process may be written $$\backslash Delta\; U\; =\; Q\; -\; W\; \backslash quad\; \backslash text\{(closed\; system,\; no\; transfer\; of\; matter)\}.$$

When a closed system receives energy as heat, this energy increases the internal energy. It is distributed between microscopic kinetic and microscopic potential energies. In general, thermodynamics does not trace this distribution. In an ideal gas all of the extra energy results in a temperature increase, as it is stored solely as microscopic kinetic energy; such heating is said to be Sensible heat.

A second kind of mechanism of change in the internal energy of a closed system changed is in its doing of work on its surroundings. Such work may be simply mechanical, as when the system expands to drive a piston, or, for example, when the system changes its electric polarization so as to drive a change in the electric field in the surroundings.

If the system is not closed, the third mechanism that can increase the internal energy is transfer of matter into the system. This increase, $\backslash Delta\; U\_\backslash mathrm\{matter\}$ cannot be split into heat and work components. If the system is so set up physically that heat transfer and work that it does are by pathways separate from and independent of matter transfer, then the transfers of energy add to change the internal energy: $$\backslash Delta\; U\; =\; Q\; -\; W\; +\; \backslash Delta\; U\_\backslash text\{matter\}\; \backslash quad\; \backslash text\{(matter\; transfer\; pathway\; separate\; from\; heat\; and\; work\; transfer\; pathways)\}.$$

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 does not change is called latent energy or latent heat, in contrast to sensible heat, which is associated with temperature change.

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Internal energy of the ideal gas Thermodynamics often uses the concept of the ideal gas for teaching purposes, and as an approximation for working systems. The ideal gas consists of particles considered as point objects that interact only by elastic collisions and fill a volume such that their mean free path between collisions is much larger than their diameter. Such systems approximate monatomic gases such as helium and other . For an ideal gas the kinetic energy consists only of the translational energy of the individual atoms. Monatomic particles do not possess rotational or vibrational degrees of freedom, and are not energy level to higher energies except at very high .

Therefore, the internal energy of an ideal gas depends solely on its temperature (and the number of gas particles): $U\; =\; U(n,T)$. It is not dependent on other thermodynamic quantities such as pressure or density.

The internal energy of an ideal gas is proportional to its mass (number of moles) $n$ and to its temperature $T$

$U\; =\; C\_V\; n\; T,$

where $C\_V$ is the isochoric (at constant volume) molar heat capacity of the gas. $C\_V$ is constant for an ideal gas. The internal energy of any gas (ideal or not) may be written as a function of the three extensive properties $S$, $V$, $n$ (entropy, volume, mass). In case of the ideal gas it is in the following way

$U(S,V,n)\; =\; \backslash mathrm\{const\}\; \backslash cdot\; e^\backslash frac\{S\}\{C\_V\; n\}\; V^\backslash frac\{-R\}\{C\_V\}\; n^\backslash frac\{R+C\_V\}\{C\_V\},$

where $\backslash mathrm\; \{const\}$ is an arbitrary positive constant and where $R$ is the Gas constant. It is easily seen that $U$ is a linearly homogeneous function of the three variables (that is, it is extensive in these variables), and that it is weakly convex function. Knowing temperature and pressure to be the derivatives $T\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; S\},$ $P\; =\; -\backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\},$ the ideal gas law $PV\; =\; nRT$ immediately follows as below:

$T\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; S\}\; =\; \backslash frac\{U\}\{C\_V\; n\}$

$P\; =\; -\backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\}\; =\; U\; \backslash frac\{R\}\{C\_V\; V\}$

$\backslash frac\{P\}\{T\}\; =\; \backslash frac\{\backslash frac\{U\; R\}\{C\_V\; V\}\}\{\backslash frac\{U\}\{C\_V\; n\}\}\; =\; \backslash frac\{n\; R\}\{V\}$

$PV\; =\; nRT$

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Internal energy of a closed thermodynamic system The above summation of all components of change in internal energy assumes that a positive energy denotes heat added to the system or the negative of work done by the system on its surroundings.

This relationship may be expressed in infinitesimal terms using the differentials of each term, though only the internal energy is an exact differential.

Adkins, C. J. (Clement John) (1983). 9780521254458, Cambridge University Press. ISBN 9780521254458

For a closed system, with transfers only as heat and work, the change in the internal energy is

$\backslash mathrm\{d\}\; U\; =\; \backslash delta\; Q\; -\; \backslash delta\; W,$

expressing the first law of thermodynamics. It 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, the mechanical work done by the system may be related to the pressure $P$ and volume change $\backslash mathrm\{d\}V$. The pressure is the intensive generalized force, while the volume change is the extensive generalized displacement:

$\backslash delta\; W\; =\; P\; \backslash ,\; \backslash mathrm\{d\}V.$

This defines the direction of work, $W$, to be energy transfer from the working system to the surroundings, indicated by a positive term. Taking the direction of heat transfer $Q$ to be into the working fluid and assuming a reversible process, the heat is

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

where $T$ denotes the temperature, and $S$ denotes the entropy.

The change in internal energy becomes

$\backslash mathrm\{d\}U\; =\; T\; \backslash ,\; \backslash mathrm\{d\}S\; -\; P\; \backslash ,\; \backslash mathrm\{d\}V.$

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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 $dU\; =\; C\_V\; \backslash ,\; dT$, i.e. the internal energy of an ideal gas can be written as a function that depends only on the temperature.

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

$\backslash mathrm\{d\}U\; =C\_\{V\}\; \backslash ,\; \backslash mathrm\{d\}T\; +\backslash leftT\backslash left(\backslash frac\{\backslash partial\; \backslash mathrm\{d\}V.$

The equation of state is the ideal gas law

$P\; V\; =\; n\; R\; T.$

Solve for pressure:

$P\; =\; \backslash frac\{n\; R\; T\}\{V\}.$

Substitute in to internal energy expression:

$dU\; =C\_\{V\}\backslash mathrm\{d\}T\; +\backslash leftT\backslash left(\backslash frac\{\backslash partial\backslash mathrm\{d\}V.$

Take the derivative of pressure with respect to temperature:

$\backslash left(\; \backslash frac\{\backslash partial\; P\}\{\backslash partial\; T\}\; \backslash right)\_\{V\}\; =\; \backslash frac\{n\; R\}\{V\}.$

Replace:

$dU\; =\; C\_\{V\}\; \backslash ,\; \backslash mathrm\{d\}T\; +\; \backslash left\; \backslash mathrm\{d\}V.$

And simplify:

$\backslash mathrm\{d\}U\; =C\_\{V\}\; \backslash ,\; \backslash mathrm\{d\}T.$

To express $\backslash mathrm\{d\}U$ in terms of $\backslash mathrm\{d\}T$ and $\backslash mathrm\{d\}V$, the term

$\backslash mathrm\{d\}S\; =\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; T\}\backslash right)\_\{V\}\backslash mathrm\{d\}T\; +\; \backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; V\}\backslash right)\_\{T\}\; \backslash mathrm\{d\}V$

is substituted in the fundamental thermodynamic relation

$\backslash mathrm\{d\}U\; =\; T\; \backslash ,\; \backslash mathrm\{d\}S\; -\; P\; \backslash ,\; \backslash mathrm\{d\}V.$

This gives

$dU\; =\; T\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; T\}\backslash right)\_\{V\}\; \backslash ,\; dT\; +\backslash leftT\backslash left(\backslash frac\{\backslash partial\; dV.$

The term $T\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; T\}\backslash right)\_\{V\}$ 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

$dA\; =\; -S\; \backslash ,\; dT\; -\; P\; \backslash ,\; dV.$

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

$\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; V\}\backslash right)\_\{T\}\; =\; \backslash left(\backslash frac\{\backslash partial\; P\}\{\backslash partial\; T\}\backslash right)\_\{V\}.$

This gives the expression above.

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Changes due to temperature and pressure

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

$dU\; =\; \backslash left(C\_\{P\}-\backslash alpha\; P\; V\backslash right)\; \backslash ,\; dT\; +\backslash left(\backslash beta\_\{T\}P-\backslash alpha\; T\backslash right)V\; \backslash ,\; dP,$

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

$C\_\{P\}\; =\; C\_\{V\}\; +\; V\; T\backslash frac\{\backslash alpha^\{2\}\}\{\backslash beta\_\{T\}\}.$

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

$\backslash alpha\; \backslash equiv\; \backslash frac\{1\}\{V\}\backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}\backslash right)\_\{P\}$

and the isothermal compressibility

$\backslash beta\_\{T\}\; \backslash equiv\; -\backslash frac\{1\}\{V\}\backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; P\}\backslash right)\_\{T\}$

by writing

and equating d V to zero and solving for the ratio d P/d T. This gives

Substituting () and () in () gives the above expression.

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Changes due to volume at constant temperature

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

$\backslash pi\; \_T\; =\; \backslash left\; (\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\}\; \backslash right\; )\_T.$

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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:

$U\; =\; U(S,V,N\_1,\backslash ldots,N\_n),$

where $N\_j$ are the molar amounts of constituents of type $j$ in the system. The internal energy is an extensive function of the extensive variables $S$, $V$, and the amounts $N\_j$, the internal energy may be written as a linearly homogeneous function of first degree:

Landau (1980). 9780080230399 ISBN 9780080230399

$U(\backslash alpha\; S,\backslash alpha\; V,\backslash alpha\; N\_\{1\},\backslash alpha\; N\_\{2\},\backslash ldots\; )$

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

where $\backslash alpha$ is a factor describing the growth of the system. The differential internal energy may be written as

$\backslash mathrm\{d\}\; U\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; S\}\; \backslash mathrm\{d\}\; S\; +\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\}\; \backslash mathrm\{d\}\; V\; +\; \backslash sum\_i\backslash \; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; N\_i\}\; \backslash mathrm\{d\}\; N\_i\backslash \; =\; T\; \backslash ,\backslash mathrm\{d\}\; S\; -\; P\; \backslash ,\backslash mathrm\{d\}\; V\; +\; \backslash sum\_i\backslash mu\_i\; \backslash mathrm\{d\}\; N\_i,$

which shows (or defines) temperature $T$ to be the partial derivative of $U$ with respect to entropy $S$ and pressure $P$ to be the negative of the similar derivative with respect to volume $V$,

$T\; =\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; S\},$

$P\; =\; -\backslash frac\{\backslash partial\; U\}\{\backslash partial\; V\},$

and where the coefficients $\backslash mu\_\{i\}$ are the chemical potentials for the components of type $i$ in the system. The chemical potentials are defined as the partial derivatives of the internal energy with respect to the variations in composition:

$\backslash mu\_i\; =\; \backslash left(\; \backslash frac\{\backslash partial\; U\}\{\backslash partial\; N\_i\}\; \backslash right)\_\{S,V,\; N\_\{j\; \backslash ne\; i\}\}.$

As conjugate variables to the composition $\backslash lbrace\; N\_\{j\}\; \backslash rbrace$, the chemical potentials are intensive properties, intrinsically characteristic of the qualitative nature of the system, and not proportional to its extent. Under conditions of constant $T$ and $P$, because of the extensive nature of $U$ and its independent variables, using Euler's homogeneous function theorem, the differential $\backslash mathrm\; d\; U$ may be integrated and yields an expression for the internal energy:

$U\; =\; T\; S\; -\; P\; V\; +\; \backslash sum\_i\; \backslash mu\_i\; N\_i.$

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

$G\; =\; \backslash sum\_i\; \backslash mu\_i\; N\_i$

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 $\backslash lbrace\; N\_\{j\}\; \backslash rbrace$.

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Internal energy in an elastic medium For an elastic medium the mechanical energy term of the internal energy is expressed in terms of the stress $\backslash sigma\_\{ij\}$ and strain $\backslash varepsilon\_\{ij\}$ involved in elastic processes. In Einstein notation for tensors, with summation over repeated indices, for unit volume, the infinitesimal statement is

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

Euler's theorem yields for the internal energy:.

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

For a linearly elastic material, the stress is related to the strain by

$\backslash sigma\_\{ij\}=C\_\{ijkl\}\; \backslash varepsilon\_\{kl\},$

where the $C\_\{ijkl\}$ are the components of the 4th-rank elastic constant tensor of the medium.

Elastic deformations, such as sound, passing through a body, or other forms of macroscopic internal agitation or turbulent motion create states when the system is not in thermodynamic equilibrium. While such energies of motion continue, they contribute to the total energy of the system; thermodynamic internal energy pertains only when such motions have ceased.

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History James Joule studied the relationship between heat, work, and temperature. He observed that friction in a liquid, such as caused by its agitation with work by a paddle wheel, caused an increase in its temperature, which he described as producing a quantity of heat. Expressed in modern units, he found that c. 4186 joules of energy were needed to raise the temperature of one kilogram of water by one degree Celsius.

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Notes

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See also

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

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Bibliography of cited references

• Adkins, C. J. (1968/1975). Equilibrium Thermodynamics, second edition, McGraw-Hill, London, .
• Bailyn, M. (1994). A Survey of Thermodynamics, American Institute of Physics Press, New York, .
• Max Born (1949). Natural Philosophy of Cause and Chance, Oxford University Press, London.
• Callen, H. B. (1960/1985), Thermodynamics and an Introduction to Thermostatistics, (first edition 1960), second edition 1985, John Wiley & Sons, New York, .
• Crawford, F. H. (1963). Heat, Thermodynamics, and Statistical Physics, Rupert Hart-Davis, London, Harcourt, Brace & World, Inc.
• Haase, R. (1971). Survey of Fundamental Laws, chapter 1 of Thermodynamics, pages 1–97 of volume 1, ed. W. Jost, of Physical Chemistry. An Advanced Treatise, ed. H. Eyring, D. Henderson, W. Jost, Academic Press, New York, lcn 73–117081.
• .
• (1986). 9780750626330, Butterworth Heinemann. ISBN 9780750626330
• Münster, A. (1970), Classical Thermodynamics, translated by E. S. Halberstadt, Wiley–Interscience, London, .
• Max Planck, (1923/1927). Treatise on Thermodynamics, translated by A. Ogg, third English edition, Longman, London.
• Tschoegl, N. W. (2000). Fundamentals of Equilibrium and Steady-State Thermodynamics, Elsevier, Amsterdam, .

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Bibliography

• Lewis, Gilbert Newton; Randall, Merle: Revised by Pitzer, Kenneth S. & Brewer, Leo (1961). 9780071138093, McGraw-Hill Book Co.. ISBN 9780071138093

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