APPENDIXF MATHEMATICALPROPERTIES OFSTATEFUNCTIONS 483
F.3 Integration of a Total Differential
If the coefficients of the total differential of a dependent variable are known as functions
of the independent variables, the expression for the total differential may be integrated to
obtain an expression for the dependent variable as a function of the independent variables.
For example, suppose the total differential of the state functionf .x; y; z/is given by
Eq.F.2.2and the coefficients are known functionsa.x; y; z/,b.x; y; z/, andc.x; y; z/.
Becausefis a state function, its change betweenf .0; 0; 0/andf .x^0 ; y^0 ; z^0 /is independent
of the integration path taken between these two states. A convenient path would be one with
the following three segments:
1.integration from.0; 0; 0/to.x^0 ; 0; 0/:Rx 0
0 a.x; 0; 0/dx
2.integration from.x^0 ; 0; 0/to.x^0 ; y^0 ; 0/:Ry 0
0 b.x(^0) ; y; 0/dy
3.integration from.x^0 ; y^0 ; 0/to.x^0 ; y^0 ; z^0 /:
Rz 0
0 c.x
(^0) ; y (^0) ; z/dz
The expression forf .x; y; z/is then the sum of the three integrals and a constant of inte-
gration.
Here is an example of this procedure applied to the total differential
df D.2xy/dxC.x^2 Cz/dyC.y 9z^2 /dz (F.3.1)
An expression for the functionfin this example is given by the sum
f D
Zx 0
0
.2x0/dxC
Zy 0
0
å.x^0 /^2 C0çdyC
Zz 0
0
.y^0 9z^2 /dzCC
D 0 Cx^2 yC.yz 9z^3 =3/CC
Dx^2 yCyz 3z^3 CC (F.3.2)
where primes are omitted on the second and third lines because the expressions are supposed
to apply to any values ofx,y, andz.Cis an integration constant. You can verify that the
third line of Eq.F.3.2gives the correct expression forf by taking partial derivatives with
respect tox,y, andzand comparing with Eq.F.3.1.
In chemical thermodynamics, there is not likely to be occasion to perform this kind
of integration. The fact that it can be done, however, shows that if we stick to one set of
independent variables, the expression for the total differential of an independent variable
contains the same information as the independent variable itself.
A different kind of integration can be used to express a dependent extensive property
in terms of independent extensive properties. Anextensiveproperty of a thermodynamic
system is one that is additive, and anintensiveproperty is one that is not additive and has
the same value everywhere in a homogeneous region (Sec.2.1.1). Suppose we have a state
functionfthat is an extensive property with the total differential
df DadxCbdyCcdzC: : : (F.3.3)
where the independent variablesx; y; z; : : : are extensive and the coefficientsa; b; c; : : :
are intensive. If the independent variables include those needed to describe an open system
(for example, the amounts of the substances), then it is possible to integrate both sides of