Page 107 - Thermodynamics of Biochemical Reactions

102 Chapter 5 Matrices in Chemical and Biochemical Thermodynamics

Table 5.1 Numbers of Natural Variables and Numbers of Intensive Degrees of
Freedom for One-Phase Reaction Systems at Equilibrium

Numbers of Natural Numbers of Intensive Degrees
Variables of Freedom

Chemical reaction system D=C+2 F=C+1
Biochemical reaction system D’ = C’ + 3 F‘ = C‘ + 2
=c+2 =c+1
Biochemical reaction system D’ = C’ + 2 F‘ = C‘ + 1
after specification of pH =c+1 =c

system at equilibrium. The first line of the table describes a chemical reaction
system at equilibrium (see equations 5.4-13 and 5.4-14). The second line of the
table describes a biochemical reactin system at equilibrium before the pH is held
constant (see equations 5.5-4 and 5.4-5). Since C‘ = C - 1, the number of natural
variables and intensive degrees of freedom are not changed in making the
Legendre transform and separating out a term in dpH. However, after the
specification that the pH is constant, the third line of the table shows that the
number D’ of natural variables and the number F’ of intensive degrees of freedom
have each been reduced by one. Since C’ = C - 1, the number of extensive degrees
of freedom is reduced to C by holding the pH constant. In the next chapter we
will see that it may be useful to make further Legendre transforms, but the
maximum number of these further Legendre transforms is C - 1, because at least
one component must remain.
When the chemical potentials of several species are held constant, it may be
useful to write the Legendre transform in terms of matrices. For example, when
the chemical potentials of several species are specified, the Legendre transform can
be written as

G’ = G - p,N,n (5.5-8)

where p, is a 1 x C row matrix of the components for which the chemical
potentials are specified. The number of components for which chemical potentials
are specified has to be at least one less than the number of components in the
reaction system. N, is a C x N, matrix of the numbers of specified components in
the N, species. This N, matrix is like the conservation matrix A, except that the
rows correspond to the specified Components, and not all of the components. The
N, x 1 column matrix n gives the amounts of all species in the system. The
differential of G‘ is given by
dG’ = dG - pcN,dn - (dp,)N,n (5.5-9)

Equation 4.1-6 can be written

dG = - SdT+ VdP + picdn,, + p,dn, (5.5- 10)
where p;,dn,, is the term for noncomponents after the chemical potentials pc have
been specified for certain components. Substituting equation 5.5-1 1 in 5.5-10
yields

dG’ = - SdT+ VdP + pAcdnnc - (dp,)Ncn (5.5-1 1)
since dn, = Ncdn (see equation 5.1-2). The transformed chemical potentials pLc of
noncomponents are given by
PAC = Ph, - PcNc (5.5-12)

Note that pc is 1 x C and N, is C x N,,.