5. Thermodynamic Relations
5.1 Introduction

For any substance, regardless of whether it is pure or a mixture, all thermodynamic properties of interest can be calculated from thermal and volumetric measurements. Whenever there is a change of phase (e.g., fusion or vaporization) additional thermal and volumetric measurements are required to characterize that change.

Thermodynamic relations are useful to:

evaluate the thermodynamic properties of importance (such as \(G, A, H, U\)) from easily measurable experimental quantities such as \(P, V, T, C_P\), and \(C_V\).

prepare a table of data for various thermodynamic properties from generalized correlations for \(Z\) and from experimental data.


From the definition of Gibbs free energy (\(G\)) \[G = H  TS\] and from the definition of Helmholtz free energy (\(A\)) \[A = U  TS\]

Fundamental property relations: \[\begin{align*} dU &= TdS  PdV \\ dH &= TdS + VdP \\ dA &= PdV  SdT \\ dG &= VdP  SdT\end{align*}\]

If \(F=F(x,y)\) then \[dF = M dx + N dy\] Exactness Criteria: \[\left(\frac{\partial M}{\partial y}\right)_x = \left(\frac{\partial N}{\partial x} \right)_y\]

If \(F=F(x,y,z)\) then \[dF = M dx + N dy + P dz\] Exactness Criteria: \[\begin{align*} \left(\frac{\partial M}{\partial y} \right)_{x,z} &= \left(\frac{\partial N}{\partial x} \right)_{y,z} \\ \left(\frac{\partial M}{\partial z} \right)_{x,y} &= \left(\frac{\partial P}{\partial x} \right)_{y,z} \\ \left(\frac{\partial N}{\partial z} \right)_{x,y} &= \left(\frac{\partial P}{\partial y} \right)_{x,z} \end{align*}\]