Page 30 - Thermodynamics of Biochemical Reactions

24 Chapter 2 Structure of Thermodynamics

Equation 2.2-8 indicates that the internal energy U of the system can be taken
to be a function of entropy S, volume V, and amounts {n,} because these
independent properties appear as differentials in equation 2.2-8; note that these
are all extensive variables. This is summarized by writing U(S, V, (nl)). The
independent variables in parentheses are called the natural variables of U. Natural
variables are very important because when a thermodynamic potential can be
determined as a function of its natural variables, all of the other thermodynamic
properties of the system can be calculated by taking partial derivatives. The
natural variables are also used in expressing the criteria of spontaneous change
and equilibrium: For a one-phase system involving PV work, (dU) < 0 at
constant S, U; and ini}.
Fundamental equation 2.2-8 has been presented as the equation resulting
from the first and second laws, but thermodynamic treatments can also be based
on the entropy as a thermodynamic potential. Equation 2.2-8 can alternatively be
written as
1 P N, p.
dS = -dU + -dV- c Adn, (2.2-1 3)
T T i=l T
This fundamental equation for the entropy shows that S has the natural variables
U, U; and {n,). The corresponding criterion of equilibrium is (dS) 2 0 at constant
U, U; and {n,). Thus the entropy increases when a spontaneous change occurs at
constant U, V, and (ni>. At equilibrium the entropy is at a maximum. When LJ,
and {n,) are constant, we can refer to the system as isolated. Equation 2.2-13
shows that partial derivatives of S yield 117; PJT and ,uI/T which is the same
information that is provided by partial derivatives of U, and so nothing is gained
by using equation 2.2-13 rather than 2.2-8. Since equation 2.2-13 does not provide
any new information, we will not discuss it further.
Equation 2.2-8 can be integrated at constant values of the intensive properties
7: P, and in,} to obtain

(2.2-14)

This is referred to as the integrated form of the fundamental equation for U.
Alternatively, equation 2.2-14 can be regarded as a result of Euler's theorem.
A function f(x 1, x2,. . . , xN) is said to be homogeneous of degree n if

f(kx,, kx,, . . . , kx,) = k"f(x,, x2,. . . ,x,) (2.2-15)
For such a function, Euler's theorem states that

(2.2- 1 6)

The internal energy is homogeneous of degree 1 in terms of extensive ther-
modynamic properties, and so equation 2.2-8 leads to equation 2.2-14. All
extensive variables are homogeneous functions of the first degree of other
extensive properties. All intensive properties are homogeneous functions of the
zeroth degree of the extensive properties.
Since integration introduces a constant, the value of U obtained from
equation 2.2-14 is uncertain by an additive constant, but that is not a problem
because thermodynamic calculations always involve changes in U, that is, AU.

2.3 MAXWELL EQUATIONS

If the differential of a function f(x, y) given by

df = Mdx + Ndy (2.3-1)