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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