16.3 Thermodynamic equilibrium relations

The ¬rst law of thermodynamics is the conservation of energy. If the number

of particles and the composition does not change, the change in internal

energy dU is due to absorbed heat dq and to work exerted on the system

dw. In equilibrium, the former equals T dS and the latter ’p dV , when

430 Review of thermodynamics

other types of work as electrical work, nuclear reactions and radiation are

disregarded. Hence

dU = T dS ’ p dV. (16.24)

With the de¬nitions given in Table 16.1 and (16.8), we arrive at the following

di¬erential relations:

dU = T dS ’ p dV + μi dni , (16.25)

i

dH = T dS + V dp + μi dni , (16.26)

i

dA = ’S dT ’ p dV + μi dni , (16.27)

i

dG = ’S dT + V dp + μi dni . (16.28)

i

Each of the di¬erentials on the left-hand side are total di¬erentials, de¬ning

12 partial di¬erentials such as (from (16.28)):

‚G

= ’S, (16.29)

‚T p,ni

‚G

= V, (16.30)

‚p T,ni

‚G

= μi . (16.31)

‚ni p,T

These are among the most important thermodynamic relations. The reader

is invited to write down the other nine equations of this type. Note that

the entropy follows from the temperature dependence of G. However, one

can also use the temperature dependence of G/T (e.g. from an equilibrium

constant), to obtain the enthalpy rather than the entropy:

‚(G/T )

= H. (16.32)

‚(1/T )

This is the very useful Gibbs“Helmholtz relation.

Being total di¬erentials, the second derivatives of mixed type do not de-

pend on the sequence of di¬erentiation. For example, from (16.28):

‚2G ‚2G

= , (16.33)

‚p ‚T ‚T ‚p

16.3 Thermodynamic equilibrium relations 431

implies that

‚S ‚V

’ = . (16.34)

‚p ‚T

T,ni p,ni

This is one of the Maxwell relations. The reader is invited to write down

the other 11 Maxwell relations of this type.

16.3.1 Relations between partial di¬erentials

The equations given above, and the di¬erential relations that follow from

them, are a selection of the possible thermodynamic relations. They may

not include a required derivative. For example, what is the relation between

CV and Cp or between κS and κT ? Instead of listing all possible relations,

it is much more e¬ective and concise to list the basic mathematical relations

from which such relations follow. The three basic rules are given below.

Relations between partial di¬erentials

f is a di¬erentiable function of two variables. There are three variables x, y, z,

which are related to each other and all partial di¬erentials of the type (‚x/‚y)z

exist. Then the following rules apply:

Rule 1:

‚f ‚f ‚f ‚y

= + . (16.35)

‚x ‚x ‚y ‚x

z y x z

Rule 2:

’1

‚x ‚y

= (inversion). (16.36)

‚y ‚x

z z

Rule 3:

‚x ‚y ‚z

= ’1 (cyclic chain rule). (16.37)

‚y ‚z ‚x

z x y

From these rules several relations can be derived. For example, in order to

relate general dependencies on volume with those on pressure, one can apply

Rule 1:

‚f ‚f ± ‚f

= + , (16.38)

‚T V ‚T p κT ‚p T

or Rule 3:

‚f 1 ‚f

=’ . (16.39)

‚V κT V ‚p

T T

432 Review of thermodynamics

Useful relations are

±2 V T

Cp = CV + , (16.40)

κT

±2 V T

κT = κS + , (16.41)

Cp

‚p ‚U

’

p=T . (16.42)

‚T V ‚V T

Equation (16.42) splits pressure into an ideal gas kinetic part and an internal

part due to internal interactions. The term (‚U/‚V )T indicates deviation

from ideal gas behavior. The proofs are left to the exercises at the end of

this chapter.

16.4 The second law

Thus far we have used the second law of thermodynamics in the form dS =

dq/T , valid for systems in equilibrium. The full second law, however, states

that

dq

dS ≥ (16.43)

T

for any system, including non-equilibrium states. It tells us in what direction

spontaneous processes will take place. When the system has reached full

equilibrium, the equality holds.

This qualitative law can be formulated for closed systems for three di¬er-

ent cases:

• Closed, adiabatic system: when neither material nor heat is exchanged

with the environment (dq = 0), the system will spontaneously evolve in

the direction of maximum entropy: