Maxwell Relations

Equations

(these are the Maxwell relations)
(
∂T
∂v
)
 
 
s
 =  −(
∂P
∂s
)
 
 
v
(
∂T
∂P
)
 
 
s
 =  (
∂v
∂s
)
 
 
P
(
∂P
∂T
)
 
 
v
 =  (
∂s
∂v
)
 
 
T
(
∂v
∂T
)
 
 
P
 =  −(
∂s
∂P
)
 
 
T


Nomenclature
Ttemperature
vspecific volume
Ppressure
sentropy
Uinternal energy
uspecific internal energy
Henthalpy
hspecific enthalpy
Sentropy
sspecific entropy
AHelmholtz function
GGibbs function


Details

Consider a simple control mass of fixed chemical composition. The Maxwell relations, which can be written for such a system, are four equations relating the properties P, v, T, and s. These will be found to be useful in the calculation of entropy in terms of other, measurable properties.

The Maxwell relations are most easily derived by considering the different forms of the thermodynamic property relation. The two forms of this expression, rewritten here are:

(Eq1)     du = T ds – P dv

and

(Eq2)     dh = T ds + v dP

from the equation:

dz = M dx + N dy

the equations can be rewritten:

u = u(s, v)andh = h(s, P)

in both of which entropy is used as one of the two independent properties. This is an undesirable situation in that entropy is one of the properties that cannot be measured. However, entropy can be eliminated as an independent property by introducing two new properties and thereby two new forms of the thermodynamic property relation. The first of these is the Helmholtz function A:

A = U – TSanda = u – Ts

Differentiating and substituting Eq1:

(Eq3)     da = du – T ds – s dT = -s dT – P dv

It can be noted that this is a form of the property relation utilizing T and v as the independent properties. The second new property is the Gibbs function G:

G = H – TSandg = h – Ts

Differentiating and substituting Eq2:

(Eq4)     dg = dh – T ds – s dT = -s dT – v dP

a fourth form of the property relation, this form using T and P as the independent properties.

Since Eq1 through Eq4 are all relations involving only properties, it can be concluded that these are exact differentials and, therefore, are of the general form of

dz = M dx + N dy

from which the following equation relates the coefficients M and N:

(
∂ M
∂ y
)
 
 
 x
 =  (
∂ N
∂ x
)
 
 
 y
     click here for reference to this equation

From this, the Maxwell relations follow:

(
∂T
∂v
)
 
 
s
 =  −(
∂P
∂s
)
 
 
v
(
∂T
∂P
)
 
 
s
 =  (
∂v
∂s
)
 
 
P
(
∂P
∂T
)
 
 
v
 =  (
∂s
∂v
)
 
 
T
(
∂v
∂T
)
 
 
P
 =  −(
∂s
∂P
)
 
 
T

which are the same as those at the top of the lesson. These equations, the Maxwell relations, are for a simple compressible mass and possess great utility. These equations will allow for the calculation of entropy changes in terms of the measurable properties: pressure, temperature, and specific volume.

Other useful relations can be derived from the Maxwell relations as follows:

(
∂ u
∂ s
)
 
 
 v
 =  T
(
∂ u
∂ v
)
 
 
 s
 =  −P
(
∂ h
∂ s
)
 
 
 P
 =  T
(
∂ h
∂ P
)
 
 
 s
 =  v
(
∂ a
∂ v
)
 
 
 T
 =  −P
(
∂ a
∂ T
)
 
 
 v
 =  −s
(
∂ g
∂ P
)
 
 
 T
 =  v
(
∂ g
∂ T
)
 
 
 P
 =  −s

As these are for a simple compressible substance, similar Maxwell relations may be written for substances which take into consideration other effects, such as surface or electrical. For example, any equation that is of the form:

dz = M dx + N dy

can be rewritten in the following form:

(
∂ M
∂ y
)
 
 
 x
 =  (
∂ N
∂ x
)
 
 
 y
     click here for reference to this equation

This approach could be extended to system having multiple effects. This matter becomes more complex when the application of the property relation to a system of variable composition is considered.

Note: (big letter) = (mass)*(small letter), ex: W = mw, this is applicable for work (W, w), heat (Q, q), enthalpy (H, h), entropy (S, s). The pattern here is the energy terms with the exception of volume/specific volume (V, v)

⇐ Prev Lesson: Relations for a Homogeneous Phase

Related
▪ P - Property Relation From Gibbs and Maxwell Relations