In thermodynamics a residual property is defined as the difference between a real fluid property and an ideal gas property, both considered at the same density, temperature, and composition, typically expressed as
X(T,V,n)=Xid(T,V,n)+Xres(T,V,n)
where
X
Xid
Xres
Xid(T,V,n)=X\circ,(T,n)+\DeltaidX(T,V,n)
where
X\circ,
X
\DeltaidX
(T,V,n)
Residual properties should not be confused with excess properties, which are defined as the deviation of a thermodynamic property from some reference system, that is typically not an ideal gas system. Whereas excess properties and excess models (also known as activity coefficient models) typically concern themselves with strictly liquid-phase systems, such as smelts, polymer blends or electrolytes, residual properties are intimately linked to equations of state which are commonly used to model systems in which vapour-liquid equilibria are prevalent, or systems where both gases and liquids are of interest. For some applications, activity coefficient models and equations of state are combined in what are known as "
\gamma
\phi
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
CV=\left(
\partialU | |
\partialT |
\right)V,n=T\left(
\partialS | |
\partialT |
\right)V,n=T\left[\left(
\partialSid | |
\partialT |
\right)V,n+\left(
\partialSres | |
\partialT |
\right)V,n\right] =
id | |
C | |
V |
+
res | |
C | |
V |
Where the ideal gas heat capacity,
id | |
C | |
V |
T\left(
\partialSres | |
\partialT |
\right)V,=-T\left(
\partial2Ares | |
\partialT2 |
\right)V,
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 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
\alpha
\beta
\gamma
\alpha | |
x | |
i |
\alpha | |
\Phi | |
i |
=
\beta | |
x | |
i |
\beta | |
\Phi | |
i |
=
\gamma | |
x | |
i |
\gamma | |
\Phi | |
i |
= ...
for all species
i
j | |
x | |
i |
i
j
j | |
\Phi | |
i |
i
j
\mui=
\circ | |
\mu | |
i |
+RTln
\Phiixip | |
p\circ |
is directly related to the residual chemical potential, as
\mui=
id | |
\mu | |
i |
+
res | |
\mu | |
i |
=
\circ | |
\mu | |
i |
+RTln
xip | |
p\circ |
+
res | |
\mu | |
i |
\implies
res | |
\mu | |
i |
=RTln\Phii
thus, because
res | |
\mu | |
i |
=\left(
\partialAres | |
\partialni |
\right)T,
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
X
T,V,p,n
X(T,p,n)
X(T,V,n)
Xres(T,p,n) ≠ Xres(T,V,n)
This arises from the fact that the real state
(T,V,p,n)
(T,V,p,n)
\muid(T,p,n)=\mu\circ+RTln
p | |
p\circ |
while computing it as a function of concentration (
c=n/V
\muid(T,V,n)=\mu\circ+RTln
c | |
c\circ |
such that
\muid(T,p,n)-\muid(T,V,n)=RTln
p | |
p\circ |
-RTln
c | |
c\circ |
=RTln
pV | |
nRT |
=RTlnZ
where we have used
p\circ=c\circRT
Z
\mui(T,p,n)-\mui(T,V,n)=0\implies
res | |
\mu | |
i |
(T,V,n)-
res | |
\mu | |
i |
(T,p,n)=RTlnZ
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
\left( | \partialA |
\partialV |
\right)T,=\left(
\partialAid | |
\partialV |
\right)T,+\left(
\partialAres | |
\partialV |
\right)T,\iff\left(
\partialAres | |
\partialV |
\right)T,=\left(
\partialA | |
\partialV |
\right)T,-\left(
\partialAid | |
\partialV |
\right)T,=-p(T,V,n)-(-pid(T,V,n))
such thatfurther, because any fluid reduces to an ideal gas in the limit of infinite volume,
A(T,V=infty,n)=Aid(T,V=infty,n)\iffAres(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
Ares(T,V,n)=
V | |
\int | |
infty |
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
Ares