Thermodynamics and Chemistry

CHAPTER 5 THERMODYNAMIC POTENTIALS

5.4 CLOSEDSYSTEMS 140

the dependent variable on the left side varies as a function of changes in two independent
variables (the natural variables of the dependent variable) on the right side.
By identifying the coefficients on the right side of Eqs.5.4.1–5.4.4, we obtain the fol-
lowing relations (which again are valid for a closed system of one component in one phase
with expansion work only):

from Eq.5.4.1:



@U

@S



V

DT (5.4.5)



@U

@V



S

Dp (5.4.6)

from Eq.5.4.2:



@H

@S



p

DT (5.4.7)



@H

@p



S

DV (5.4.8)

from Eq.5.4.3:



@A

@T



V

DS (5.4.9)



@A

@V



T

Dp (5.4.10)

from Eq.5.4.4:



@G

@T



p

DS (5.4.11)



@G

@p



T

DV (5.4.12)

This book now uses for the first time an extremely useful mathematical tool called the
reciprocity relationof a total differential (Sec.F.2). Suppose the independent variables are
xandyand the total differential of a dependent state functionf is given by

df DadxCbdy (5.4.13)

whereaandbare functions ofxandy. Then the reciprocity relation is

@a
@y



x

D



@b

@x



y

(5.4.14)

The reciprocity relations obtained from the Gibbs equations (Eqs.5.4.1–5.4.4) are called
Maxwell relations(again valid for a closed system withCD 1 ,PD 1 , and∂w^0 D 0 ):

from Eq.5.4.1:



@T

@V



S

D



@p

@S



V

(5.4.15)

from Eq.5.4.2:



@T

@p



S

D



@V

@S



p

(5.4.16)

from Eq.5.4.3:



@S

@V



T

D



@p

@T



V

(5.4.17)

from Eq.5.4.4: 



@S

@p



T

D



@V

@T



p

(5.4.18)