Material properties (thermodynamics)

Thermodynamics

The classical Carnot heat engine

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|  State

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System properties

Note: Conjugate variables in italics

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Material properties

* Property databases

| Specific heat capacity |  c = {\displaystyle c=} | |  T {\displaystyle T} |  ∂ S {\displaystyle \partial S}

|

N {\displaystyle N} |  ∂ T {\displaystyle \partial T}
Compressibility |  β = − {\displaystyle \beta =-} | |  1 {\displaystyle 1} |  ∂ V {\displaystyle \partial V}

|

V {\displaystyle V} |  ∂ p {\displaystyle \partial p}
Thermal expansion |  α = {\displaystyle \alpha =} | |  1 {\displaystyle 1} |  ∂ V {\displaystyle \partial V}

|

V {\displaystyle V} |  ∂ T {\displaystyle \partial T}

Equations

* Carnot's theorem
* Clausius theorem
* Fundamental relation
* Ideal gas law

* Maxwell relations
* Onsager reciprocal relations
* Bridgman's equations
* Table of thermodynamic equations

Potentials

* Free energy
* Free entropy

* Internal energy
U ( S , V ) {\displaystyle U(S,V)}
* Enthalpy
H ( S , p ) = U + p V {\displaystyle H(S,p)=U+pV}
* Helmholtz free energy
A ( T , V ) = U − T S {\displaystyle A(T,V)=U-TS}
* Gibbs free energy
G ( T , p ) = H − T S {\displaystyle G(T,p)=H-TS}

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The thermodynamic properties of materials are intensive thermodynamic parameters which are specific to a given material. Each is directly related to a second order differential of a thermodynamic potential. Examples for a simple 1-component system are:

* Compressibility (or its inverse, the bulk modulus)

* Isothermal compressibility

κ T = − 1 V ( ∂ V ∂ P ) T = − 1 V ∂ 2 G ∂ P 2 {\displaystyle \kappa _{T}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{T}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P^{2}}}}

* Adiabatic compressibility

κ S = − 1 V ( ∂ V ∂ P ) S = − 1 V ∂ 2 H ∂ P 2 {\displaystyle \kappa _{S}=-{\frac {1}{V}}\left({\frac {\partial V}{\partial P}}\right)_{S}\quad =-{\frac {1}{V}}\,{\frac {\partial ^{2}H}{\partial P^{2}}}}

* Specific heat (Note - the extensive analog is the heat capacity)

* Specific heat at constant pressure

c P = T N ( ∂ S ∂ T ) P = − T N ∂ 2 G ∂ T 2 {\displaystyle c_{P}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{P}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}G}{\partial T^{2}}}}

* Specific heat at constant volume

c V = T N ( ∂ S ∂ T ) V = − T N ∂ 2 A ∂ T 2 {\displaystyle c_{V}={\frac {T}{N}}\left({\frac {\partial S}{\partial T}}\right)_{V}\quad =-{\frac {T}{N}}\,{\frac {\partial ^{2}A}{\partial T^{2}}}}

* Coefficient of thermal expansion

α = 1 V ( ∂ V ∂ T ) P = 1 V ∂ 2 G ∂ P ∂ T {\displaystyle \alpha ={\frac {1}{V}}\left({\frac {\partial V}{\partial T}}\right)_{P}\quad ={\frac {1}{V}}\,{\frac {\partial ^{2}G}{\partial P\partial T}}}

where P is pressure, V is volume, T is temperature, S is entropy, and N is the number of particles.

For a single component system, only three second derivatives are needed in order to derive all others, and so only three material properties are needed to derive all others. For a single component system, the "standard" three parameters are the isothermal compressibility  κ T {\displaystyle \kappa _{T}} , the specific heat at constant pressure  c P {\displaystyle c_{P}} , and the coefficient of thermal expansion  α {\displaystyle \alpha } .

For example, the following equations are true:

c P = c V + T V α 2 N κ T {\displaystyle c_{P}=c_{V}+{\frac {TV\alpha ^{2}}{N\kappa _{T}}}}

κ T = κ S + T V α 2 N c P {\displaystyle \kappa _{T}=\kappa _{S}+{\frac {TV\alpha ^{2}}{Nc_{P}}}}

The three "standard" properties are in fact the three possible second derivatives of the Gibbs free energy with respect to temperature and pressure. Moreover, considering derivatives such as  ∂ 3 G ∂ P ∂ T 2 {\displaystyle {\frac {\partial ^{3}G}{\partial P\partial T^{2}}}} and the related Schwartz relations, shows that the properties triplet is not independent. In fact, one property function can be given as an expression of the two others, up to a reference state value.[1]

The second principle of thermodynamics has implications on the sign of some thermodynamic properties such isothermal compressibility.[1][2]

## See also

[edit]

* Physics portal

* List of materials properties (thermal properties)
* Heat capacity ratio
* Statistical mechanics
* Thermodynamic equations
* Thermodynamic databases for pure substances
* Heat transfer coefficient
* Latent heat
* Specific heat of melting (Enthalpy of fusion)
* Specific heat of vaporization (Enthalpy of vaporization)
* Thermal mass

## External links

[edit]

* The Dortmund Data Bank is a factual data bank for thermodynamic and thermophysical data.

## References

[edit]

1. ^ a b S. Benjelloun, "Thermodynamic identities and thermodynamic consistency of Equation of States", Link to Archiv e-print Link to Hal e-print
2. ^ Israel, R. (1979). Convexity in the Theory of Lattice Gases. Princeton, New Jersey: Princeton University Press. doi:10.2307/j.ctt13x1c8g

* Callen, Herbert B. (1985). Thermodynamics and an Introduction to Thermostatistics (2nd ed.). New York: John Wiley & Sons. ISBN 0-471-86256-8.

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