Page 29 - Thermodynamics of Biochemical Reactions

2.2 Fundamental Equation for the Internal Energy 23

where N, is the number of different species. This equation really defines the
chemical potential pj of a species. The terms in pidn, are referred to as chemical
work terms. If U for a system can be determined as a function of S, K and {n,},
where {ai> represents the set of amounts of species, then 7; P, and {pi} can be
determined by taking partial derivatives of U. Thus the intensive properties of the
system are obtained by taking derivatives of the extensive property U with respect
to extensive properties. It may be useful to consider the internal energy to be a
function of T, P, and {nj} rather than S, and {n,), but when that is done, it is
not possible to calculate all the other thermodynamic properties of the system by
taking partial derivatives.
When U is expressed as a function of S, and {a,}, calculus requires that the
total differential of U is given by

Comparison of equations 2.2-8 and 2.2-9 indicates that

(2.2- 1 0)

(2.2-11)

(2.2- 12)

The intensive variables T, P, and {pi) can be considered to be functions of S, K
and {q) because U is a function of S, r! and {q}. If U for a system can be
determined experimentally as a function of S, r/; and {a,}, then 7; P, and (pi} can
be calculated by taking the first partial derivatives of U. Equations 2.2-10 to
2.2-12 are referred to as equations of state because they give relations between
state properties at equilibrium. In Section 2.4 we will see that these N, + 2
equations of state are not independent of each other, but any N, + 1 of them
provide a complete thermodynamic description of the system. In other words, if
N, + 1 equations of state are determined for a system, the remaining equation of
state can be calculated from the N, + 1 known equations of state. In the preceding
section we concluded that the intensive state of a one-phase system can be
described by specifying N, + 1 intensive variables. Now we see that the determi-
nation of N, + 1 equations of state can be used to calculate these N, + 1 intensive
properties.
The beauty of the fundamental equation for U (equation 2.2-8) is that it
combines all of this information in one equation. Note that the N, + 2 extensive
variables S, V and {ai) are independent, and the N, + 2 intensive variables 7; P,
and [pi} obtained by taking partial derivatives of U are dependent. This is
wonderful, but equations of state 2.2-10 to 2.2-12 are not very useful because S
is not a convenient independent variable. Fortunately, more useful equations
of state will be obtained from other thermodynamic potentials introduced in
Section 2.5.
The fundamental equation for U is in agreement with the statement of the
preceding section that for a homogeneous mixture of N, substances, the state of
the system can be specified by N, + 2 properties, at least one of which is extensive.
The total number of variables involved in equation 2.2-8 is 2N, + 5. N, + 3 of
these variables are extensive (U, S, r/; and {ni}), and N, + 2 of the variables are
intensive (7; P, {,ui}). Note that except for the internal energy, these variables
appear in pairs, in which one property is extensive and the other is intensive; these
are referred to as conjugate pairs. These pairs are given later in Table 2.1 in
Section 2.7. When other kinds of work are involved, there are more than 2N, + 5
variables in the fundamental equation for U (see Section 2.7).