In the development and implementation of Equations of State, the concept of residual properties is valuable, as it allows one to separate the behaviour of a fluid that stems from non-ideality from that stemming from the properties of an ideal gas. For example, the isochoric heat capacity is given by

${\displaystyle C_{V}=\left({\frac {\partial U}{\partial T}}\right)_{V,n}=T\left({\frac {\partial S}{\partial T}}\right)_{V,n}=T\left[\left({\frac {\partial S^{id}}{\partial T}}\right)_{V,n}+\left({\frac {\partial S^{res}}{\partial T}}\right)_{V,n}\right]=C_{V}^{id}+C_{V}^{res}}$ 

Where the ideal gas heat capacity, ${\displaystyle C_{V}^{id}}$ , can be measured experimentally, by measuring the heat capacity at very low pressure. After measurement it is typically represented using a polynomial fit such as the Shomate equation. The residual heat capacity is given by

${\displaystyle T\left({\frac {\partial S^{res}}{\partial T}}\right)_{V,n}=-T\left({\frac {\partial ^{2}A^{res}}{\partial T^{2}}}\right)_{V,n}}$ ,

and the accuracy of a given equation of state in predicting or correlating the heat capacity can be assessed by regarding only the residual contribution, as the ideal contribution is independent of the equation of state.

In Equilibrium Calculations edit

In fluid phase equilibria (i.e. liquid-vapour or liquid-liquid equilibria), the notion of the fugacity coefficient is crucial, as it can be shown that the equilibrium condition for a system consisting of phases ${\displaystyle \alpha }$ , ${\displaystyle \beta }$ , ${\displaystyle \gamma }$ , ... the condition for chemical equilibrium is

${\displaystyle x_{i}^{\alpha }\Phi _{i}^{\alpha }=x_{i}^{\beta }\Phi _{i}^{\beta }=x_{i}^{\gamma }\Phi _{i}^{\gamma }=\quad ...}$ 

for all species ${\displaystyle i}$ , where ${\displaystyle x_{i}^{j}}$  denotes the mole fraction of species ${\displaystyle i}$  in phase ${\displaystyle j}$ , and ${\displaystyle \Phi _{i}^{j}}$  is the fugacity coefficient of species ${\displaystyle i}$  in phase ${\displaystyle j}$ . The fugacity coefficient, being defined by

${\displaystyle \mu _{i}=\mu _{i}^{\circ }+RT\ln {\frac {\Phi _{i}x_{i}p}{p^{\circ }}}}$ 

is directly related to the residual chemical potential, as

${\displaystyle \mu _{i}=\mu _{i}^{id}+\mu _{i}^{res}=\mu _{i}^{\circ }+RT\ln {\frac {x_{i}p}{p^{\circ }}}+\mu _{i}^{res}\implies \mu _{i}^{res}=RT\ln \Phi _{i}}$ ,

thus, because ${\displaystyle \mu _{i}^{res}=\left({\frac {\partial A^{res}}{\partial n_{i}}}\right)_{T,V}}$ , we can see that an accurate description of the residual Helmholtz energy, rather than the total Helmholtz energy, is the key to accurately computing the equilibrium state of a system.

Residual Entropy Scaling edit

The residual entropy of a fluid has some special significance. In 1976, Yasha Rosenfeld published a landmark paper, showing that the transport coefficients of pure liquids, when expressed as functions of the residual entropy, can be treated as monovariate functions, rather than as functions of two variables (i.e. temperature and pressure, or temperature and density).^[1] This discovery lead to the concept of residual entropy scaling, which has spurred a large amount of research, up until the modern day, in which various approaches for modelling transport coefficients as functions of the residual entropy have been explored.^[2] Residual entropy scaling is still very much an area of active research.

While any real state variable ${\displaystyle X}$ , in a real state (${\displaystyle T,V,p,n}$ ), is independent of whether one evaluates ${\displaystyle X(T,p,n)}$  or ${\displaystyle X(T,V,n)}$ , one should be aware that the residual property is in general dependent on the variable set, i.e.

${\displaystyle X^{res}(T,p,n)\neq X^{res}(T,V,n)}$ 

This arises from the fact that the real state ${\displaystyle (T,V,p,n)}$  is in general not a valid ideal gas state, such that the ideal part of the property will be different depending on variable set. Take for example the chemical potential of a pure fluid: In a state ${\displaystyle (T,V,p,n)}$  that does not satisfy the ideal gas law, but may be a real state for some real fluid. The ideal gas chemical potential computed as a function of temperature, pressure and mole number is

${\displaystyle \mu ^{id}(T,p,n)=\mu ^{\circ }+RT\ln {\frac {p}{p^{\circ }}}}$ ,

while computing it as a function of concentration (${\displaystyle c=n/V}$ ), we have

${\displaystyle \mu ^{id}(T,V,n)=\mu ^{\circ }+RT\ln {\frac {c}{c^{\circ }}}}$ ,

such that

${\displaystyle \mu ^{id}(T,p,n)-\mu ^{id}(T,V,n)=RT\ln {\frac {p}{p^{\circ }}}-RT\ln {\frac {c}{c^{\circ }}}=RT\ln {\frac {pV}{nRT}}=RT\ln Z}$ ,

where we have used ${\displaystyle p^{\circ }=c^{\circ }RT}$ , and ${\displaystyle Z}$  denotes the compressibility factor. This leads to the result

${\displaystyle \mu _{i}(T,p,n)-\mu _{i}(T,V,n)=0\implies \mu _{i}^{res}(T,V,n)-\mu _{i}^{res}(T,p,n)=RT\ln Z}$ .

In practice, the most significant residual property is the residual Helmholtz energy. The reason for this is that other residual properties can be computed from the residual Helmholtz energy as various derivatives (see: Maxwell relations). We note that

${\displaystyle \left({\frac {\partial A}{\partial V}}\right)_{T,n}=\left({\frac {\partial A^{id}}{\partial V}}\right)_{T,n}+\left({\frac {\partial A^{res}}{\partial V}}\right)_{T,n}\iff \left({\frac {\partial A^{res}}{\partial V}}\right)_{T,n}=\left({\frac {\partial A}{\partial V}}\right)_{T,n}-\left({\frac {\partial A^{id}}{\partial V}}\right)_{T,n}=-p(T,V,n)-(-p^{id}(T,V,n))}$ 

such that

${\displaystyle A(T,V=\infty ,n)=A^{id}(T,V=\infty ,n)\iff A^{res}(T,V=\infty ,n)=0}$ .

Thus, for any Equation of State that is explicit in pressure, such as the van der Waals Equation of State, we may compute

${\displaystyle A^{res}(T,V,n)=\int _{\infty }^{V}{\frac {nRT}{V'}}-p(T,V',n)dV'}$ .

However, in modern approaches to developing Equations of State, such as SAFT, it is found that it can be simpler to develop the equation of state by directly developing an equation for ${\displaystyle A^{res}}$ , rather than developing an equation that is explicit in pressure.

• Departure function

• J. M. Smith, H.C.Van Ness, M. M. Abbot Introduction to Chemical Engineering Thermodynamics 2000, McGraw-Hill 6TH edition ISBN 0-07-240296-2
• Robert Perry, Don W. Green Perry's Chemical Engineers' Handbook 2007 McGraw-Hill 8TH edition ISBN 0-07-142294-3