Thermodynamics and Chemistry

APPENDIXF MATHEMATICALPROPERTIES OFSTATEFUNCTIONS 484

the equation from a lower limit of zero for each of the extensive functions while holding the
intensive functions constant:
Zf 0

0

dfDa

Zx 0

0

dxCb

Zy 0

0

dyCc

Zz 0

0

dzC: : : (F.3.4)

f^0 Dax^0 Cby^0 Ccz^0 C: : : (F.3.5)

Note that a term of the formcduwhereuisintensivebecomeszerowhen integrated with
intensive functions held constant, because duis this case is zero.

F.4 Legendre Transforms

ALegendre transformof a state function is a linear change of one or more of the indepen-
dent variables made by subtracting products of conjugate variables.
To understand how this works, consider a state functionf whose total differential is
given by
df DadxCbdyCcdz (F.4.1)

In the expression on the right side,x,y, andzare being treated as the independent variables.
The pairsaandx,bandy, andcandzareconjugate pairs. That is,aandxare conjugates,
bandyare conjugates, andcandzare conjugates.
For the first example of a Legendre transform, we define a new state functionf 1 by
subtracting the product of the conjugate variablesaandx:

f 1

def

D fax (F.4.2)

The functionf 1 is a Legendre transform off. We take the differential of Eq.F.4.2

df 1 Ddf adxxda (F.4.3)

and substitute for dffrom Eq.F.4.1:

df 1 D.adxCbdyCcdz/adxxda

DxdaCbdyCcdz (F.4.4)

EquationF.4.4gives the total differential off 1 witha,y, andzas the independent variables.
The functionsxandahave switched places as independent variables. What we did in order
to letareplacexas an independent variable was to subtract fromf the product of the
conjugate variablesaandx.
Because the right side of Eq.F.4.4is an expression for the total differential of the state
functionf 1 , we can use the expression to identify the coefficients as partial derivatives of
f 1 with respect to the new set of independent variables:

xD



@f 1

@a



y;z

bD



@f 1

@y



a;z

cD



@f 1

@z



a;y

(F.4.5)

We can also use Eq.F.4.4to write new reciprocity relations, such as



@x

@y



a;z

D



@b

@a



y;z

(F.4.6)