February 15, 2012  Tagged with:

This is just a summary of many of the definitions and identities I find useful when working with the basic thermodynamics of a system or process.  It is not meant to be a comprehensive list.

# Mathematical Identities

Using the reasoning behind the notation used for derivatives of thermodynamic potentials, there are some useful identities given the following prototypical system

Eq. 1) $\displaystyle X=X\left(Y,Z\right)$

This equation represents a generic thermodynamic potential of an arbitrary system or process as a function of two implicitly extensive thermodynamic potentials.  Adopting the notation

Eq. 2) $\displaystyle\underline{X}=\frac{X}{N}$

To denote the relationship between intensive variables, extensive variables, and the size of the system allows for the trivial conversion between system size dependent and system size agnostic thermodynamic relationships.  The total differential for the extensive form of the thermodynamic potential for the prototypical system is given by

Eq. 3) $\displaystyle dX=\left(\frac{\partial X}{\partial Y}\right)_{Z,N}dY+\left(\frac{\partial X}{\partial Z}\right)_{Y,N}dZ+\left(\frac{\partial X}{\partial N}\right)_{Y,Z}dN$

Whereas the total differential for the intensive form of the thermodynamic potential is given as

Eq. 4) $\displaystyle d\underline{X}=\left(\frac{\partial\underline{X}}{\partial\underline{Y}}\right)_{\underline{Z}}d\underline{Y}+\left(\frac{\partial\underline{X}}{\partial\underline{Z}}\right)_{\underline{Y}}d\underline{Z}$

The latter form rearranges to yield the identity

Eq. 5) $\displaystyle\left(\frac{\partial\underline{X}}{\partial\underline{Y}}\right)_{\underline{Z}}\left(\frac{\partial\underline{Z}}{\partial\underline{X}}\right)_{\underline{Y}}\left(\frac{\partial\underline{Y}}{\partial\underline{Z}}\right)_{\underline{X}}=-1$

Which is obtained, in part, by using the inversion property

Eq. 6) $\displaystyle\left(\frac{\partial\underline{X}}{\partial\underline{Y}}\right)_{\underline{Z}}=\left(\frac{\partial\underline{Y}}{\partial\underline{X}}\right)^{-1}_{\underline{Z}}$

Furthermore, the chain rule also applies given like conditional constraints

Eq. 7) $\displaystyle\left(\frac{\partial\underline{X}}{\partial\underline{K}}\right)_{\underline{Z}}=\left(\frac{\partial\underline{X}}{\partial\underline{Y}}\right)_{\underline{Z}}\left(\frac{\partial\underline{Y}}{\partial\underline{K}}\right)_{\underline{Z}}$

# Definitions of Some Common Thermodynamic Potentials

Enthalpy

Eq. 8 ) $\displaystyle H=U+PV$

Helmholtz Energy

Eq. 9) $\displaystyle A=U-TS$

Gibbs Energy

Eq. 10) $\displaystyle G=U+PV-TS=H-TS$

# Total Differentials of Common Thermodynamic Potentials

The following are the total differentials of the common thermodynamic potentials for both open and closed systems respectively.

Internal Energy

Eq. 11) $\displaystyle dU=TdS-PdV+\underline{G}dN=\left(\frac{\partial U}{\partial S}\right)_{V,N}dS+\left(\frac{\partial U}{\partial V}\right)_{S,N}dV+\left(\frac{\partial U}{\partial N}\right)_{S,V}dN$

Eq. 12) $\displaystyle d\underline{U}=Td\underline{S}-Pd\underline{V}=\left(\frac{\partial\underline{U}}{\partial\underline{S}}\right)_{\underline{V}}d\underline{S}+\left(\frac{\partial\underline{U}}{\partial\underline{V}}\right)_{\underline{S}}d\underline{V}$

Entropy

Eq. 13) $\displaystyle dS=\frac{1}{T}dU+\frac{P}{T}dV-\frac{\underline{G}}{T}dN=\left(\frac{\partial S}{\partial U}\right)_{V,N}dU+\left(\frac{\partial S}{\partial V}\right)_{U,N}dV+\left(\frac{\partial S}{\partial N}\right)_{U,V}dN$

Eq. 14) $\displaystyle d\underline{S}=\frac{1}{T}d\underline{U}+\frac{P}{T}d\underline{V}=\left(\frac{\partial\underline{S}}{\partial\underline{U}}\right)_{\underline{V}}d\underline{U}+\left(\frac{\partial\underline{S}}{\partial\underline{V}}\right)_{\underline{U}}d\underline{V}$

Enthalpy

Eq. 15) $\displaystyle dH=TdS+VdP+\underline{G}dN=\left(\frac{\partial H}{\partial S}\right)_{P,N}dS+\left(\frac{\partial H}{\partial P}\right)_{S,N}dP+\left(\frac{\partial H}{\partial N}\right)_{S,P}dN$

Eq. 16) $\displaystyle d\underline{H}=Td\underline{S}+VdP=\left(\frac{\partial\underline{H}}{\partial\underline{S}}\right)_{P}d\underline{S}+\left(\frac{\partial\underline{H}}{\partial P}\right)_{\underline{S}}dP$

Helmholtz Energy

Eq. 17) $\displaystyle dA=-PdV-SdT+\underline{G}dN=\left(\frac{\partial A}{\partial V}\right)_{T,N}dV+\left(\frac{\partial A}{\partial T}\right)_{V,N}dT+\left(\frac{\partial A}{\partial N}\right)_{V,T}dN$

Eq. 18) $\displaystyle d\underline{A}=-Pd\underline{V}-\underline{S}dT=\left(\frac{\partial\underline{A}}{\partial\underline{V}}\right)_{T}d\underline{V}+\left(\frac{\partial\underline{A}}{\partial T}\right)_{\underline{V}}dT$

Gibbs Energy

Eq. 19) $\displaystyle dG=VdP-SdT+\underline{G}dN=\left(\frac{\partial G}{\partial P}\right)_{T,N}dP+\left(\frac{\partial G}{\partial T}\right)_{P,N}dT+\left(\frac{\partial G}{\partial N}\right)_{T,P}dN$

Eq. 20) $\displaystyle d\underline{G}=\underline{V}dP-\underline{S}dT=\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}dP+\left(\frac{\partial\underline{G}}{\partial T}\right)_{P}dT$

# The Maxwell Relations

Eq. 21) $\displaystyle\left(\frac{\partial T}{\partial\underline{V}}\right)_{\underline{S}}=-\left(\frac{\partial P}{\partial\underline{S}}\right)_{\underline{V}}$

Eq. 22) $\displaystyle\left(\frac{\partial T}{\partial P}\right)_{\underline{S}}=\left(\frac{\partial\underline{V}}{\partial\underline{S}}\right)_{P}$

Eq. 23) $\displaystyle\left(\frac{\partial P}{\partial T}\right)_{\underline{V}}=\left(\frac{\partial\underline{S}}{\partial\underline{V}}\right)_{T}$

Eq. 24) $\displaystyle\left(\frac{\partial\underline{V}}{\partial T}\right)_{P}=-\left(\frac{\partial\underline{S}}{\partial P}\right)_{T}$

# Definition of Heat Capacities

Isochoric Heat Capacity

Eq. 25) $\displaystyle C_V=\left(\frac{\partial\underline{U}}{\partial T}\right)_{\underline{V}}=T\left(\frac{\partial\underline{S}}{\partial T}\right)_{\underline{V}}$

Isobaric Heat Capacity

Eq. 26) $\displaystyle\underline{C_P}=\left(\frac{\partial\underline{H}}{\partial T}\right)_{P}=T\left(\frac{\partial\underline{S}}{\partial T}\right)_{P}$

The Relationship Between Heat Capacities

Eq. 27) $\displaystyle\underline{C_P}=C_V+T\left(\frac{\partial P}{\partial T}\right)_{\underline{V}}\left(\frac{\partial\underline{V}}{\partial T}\right)_{p}$

# Definition of Other Important Thermodynamic Potentials

Coefficient of Thermal Expansion

Eq. 28) $\displaystyle\alpha=\frac{1}{\underline{V}}\left(\frac{\partial\underline{V}}{\partial T}\right)_{P}$

Isothermal Compressibility

Eq. 29) $\displaystyle\kappa_T=-\frac{1}{\underline{V}}\left(\frac{\partial\underline{V}}{\partial P}\right)_{T}$

Joule-Thomson Coefficient

Eq. 30) $\displaystyle\mu=\left(\frac{\partial T}{\partial P}\right)_{\underline{H}}=-\cfrac{\left[\underline{V}-T\left(\frac{\partial\underline{V}}{\partial T}\right)_{P}\right]}{\underline{C_P}}$

# Useful Thermodynamic Identities

Temperature

Eq. 31) $\displaystyle\left(\frac{\partial\underline{H}}{\partial\underline{S}}\right)_{P}=\left(\frac{\partial\underline{U}}{\partial\underline{S}}\right)_{\underline{V}}=T$

Specific Volume

Eq. 32) $\displaystyle\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}=\left(\frac{\partial\underline{H}}{\partial P}\right)_{\underline{S}}=\underline{V}$

Pressure

Eq. 33) $\displaystyle\left(\frac{\partial\underline{U}}{\partial\underline{V}}\right)_{\underline{S}}=\left(\frac{\partial\underline{A}}{\partial\underline{V}}\right)_{T}=-P$

Specific Entropy

Eq. 34) $\displaystyle\left(\frac{\partial\underline{A}}{\partial T}\right)_{\underline{V}}=\left(\frac{\partial\underline{G}}{\partial T}\right)_{P}=-\underline{S}$

# Total Differentials in terms of Measurable Variables

Internal Energy

Eq. 35) $\displaystyle d\underline{U}=\underline{C_V}dT+\left[T\left(\frac{\partial P}{\partial T}\right)_{\underline{V}}-P\right]d\underline{V}$

Enthalpy

Eq. 36) $\displaystyle d\underline{H}=\underline{C_P}dT+\left[\underline{V}-T\left(\frac{\partial\underline{V}}{\partial T}\right)_{P}\right]dP$

# Common Thermodynamic Potentials in Terms of Gibbs Energy

Entropy

Eq. 37) $\displaystyle\underline{S}=-\left(\frac{\partial\underline{G}}{\partial T}\right)_{P}$

Specific Volume

Eq. 38) $\displaystyle\underline{V}=\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}$

Enthalpy

Eq. 39) $\displaystyle\underline{H}=\underline{G}-T\left(\frac{\partial\underline{G}}{\partial T}\right)_{P}$

Internal Energy

Eq. 40) $\displaystyle\underline{U}=\underline{G}-T\left(\frac{\partial\underline{G}}{\partial T}\right)_{P}-P\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}$

Helmholtz Energy

Eq. 41) $\displaystyle\underline{A}=\underline{G}-P\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}$

Isobaric Heat Capacity

Eq. 42) $\displaystyle\underline{C_P}=-T\left(\frac{\partial^2\underline{G}}{\partial T^2}\right)_{P}$

Isochoric Heat Capacity

Eq. 43) $\displaystyle\underline{C_V}=-T\left(\frac{\partial^2\underline{G}}{\partial T^2}\right)_{P}-T\left(\frac{\partial^2\underline{G}}{\partial T\partial P}\right)^2\left(\frac{\partial^2\underline{G}}{\partial P^2}\right)^{-1}_{T}$

Isothermal Compressibility

Eq. 44) $\displaystyle\kappa_T=-\cfrac{\left(\cfrac{\partial^2\underline{G}}{\partial P^2}\right)_{T}}{\left(\cfrac{\partial\underline{G}}{\partial P}\right)_{T}}$

Coefficient of Thermal Expansion

Eq. 45) $\displaystyle\alpha=\cfrac{\left(\frac{\partial^2\underline{G}}{\partial P\partial T}\right)}{\left(\frac{\partial\underline{G}}{\partial P}\right)_{T}}$

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