Thermodynamics and Chemistry

APPENDIXF MATHEMATICALPROPERTIES OFSTATEFUNCTIONS 482

o@f

@y



x

dy

) 

@f

@x



y

dx

9

>>=

>>;dfD



@f

@x



y

dxC



@f

@y



x

dy

x

y

f

dx

dy

slope inf-xplaneD



@f

@x



y

slope inf-yplaneD



@f

@y



x

Figure F.1

form
df DadxCbdyCcdz (F.2.2)

where we can identify the coefficients as

aD



@f

@x



y;z

bD



@f

@y



x;z

cD



@f

@z



x;y

(F.2.3)

These coefficients are themselves, in general, functions of the independent variables and
may be differentiated to give mixed second partial derivatives; for example:

@a
@y



x;z

D

@^2 f

@y@x



@b

@x



y;z

D

@^2 f

@x@y

(F.2.4)

The second partial derivative@^2 f=@y@x, for instance, is the partial derivative with respect
toyof the partial derivative off with respect tox. It is a theorem of calculus that if a
functionf is single valued and has continuous derivatives, the order of differentiation in a
mixed derivative is immaterial. Therefore the mixed derivatives@^2 f=@y@xand@^2 f=@x@y,
evaluated for the system in any given state, are equal:

@a
@y



x;z

D



@b

@x



y;z

(F.2.5)

The general relation that applies to a function of any number of independent variables is

@X
@y



D



@Y

@x



(F.2.6)

wherexandyareanytwo of the independent variables,Xis@f=@x,Y is@f=@y, and
each partial derivative has all independent variables held constant except the variable shown
in the denominator. This general relation is the Euler reciprocity relation, orreciprocity
relationfor short. A necessary and sufficient condition for dfto be an exact differential is
that the reciprocity relation is satisfied for each pair of independent variables.