Page 65 - Thermodynamics of Biochemical Reactions

60 Chapter 4 Thermodynamics of Biochemical Reactions at Specified pH

The number D = N, + 2 of natural variables of G has not been changed by
separating the term for the hydrogen component.
The differential of the transformed Gibbs energy (equation 4.1-1) is
dG' = dG - n,(H)dp(H') - p(H+)dn,(H) (4.1-8)

and substituting equation 4.1-7 into this relation yields a form of the fundamental
equation for G':

N,- 1
dG' = -SdT+ VdP + 1 ,Li;drzi - n,(H)dp(H+) (4.1-9)
j- 1
Note that the Legendre transform has interchanged the roles of the conjugate
intensive p(H+) and extensive n,(H) variables in the last term of equation 4.1-9.
The number D' of natural variables of G' is N, + 2, just as it was for G, but the
chemical potential of the hydrogen ion is now a natural variable instead of the
amount of the hydrogen component (equation 4.1-7).
Since the chemical potential ,u(H+) depends on both the temperature and the
concentration of hydrogen ions, it is not a very convenient variable when the
temperature is changed. The hydrogen ion concentration can be made an
independent intensive variable in the fundamental equation for G' by use of the
expression for the differential of the chemical potential of H+:

The first partial derivative in this equation is equal to -S,(H+), where S,(H+)
is the molar entropy of the hydrogen ion. To evaluate the second partial derivative
in equation 4.1-10, we need to recall that the chemical potential of species Bj is
given by
/ij = ,LL~ + RT In [Bj] (4.1-11)

where ,uy is the standard chemical potential of species j. Thus the chemical
potential of Bj in a 1 M solution at the specified ionic strength is given by kip.
Since the thermodynamic properties are taken to be functions of the ionic
strength, we do not have to deal with activity coefficients explicitly. Equation
4.1-1 1 indicates that dp(H')jd[H+] = RT/[H+], and since dpH/d[H+] =
- l/(ln(lO)[H+]), equation 4.1-10 can be written
dp(H+) = -S,(H+)dT- RTln(l0)dpH (4. I - 12)

Substituting this in equation 4.1-9 yields

N,- 1
dG' = -S'dT+ VdP + 1 pidrz, + RTln(lO)n,(H)dpH (4.1-13)
j= 1
where the transformed entropy S' of the system at a specified pH is given by
S' = S - n,(H)S,(H+) (4.1 - 14)

Since the enthalpy H of the system is defined by H = G + TS, substituting
equations 4.1-1 and 4.1-14 in this expression for H yields

H = G' + n,(H)p(H+) + T(S' + n,(H)S,(H+)) = H' + n,(H)H,(H+) (4.1-15)
where the transformed enthalpy H' of the system is given by

H' = G' + TS' (4.1 - 16)
and the molar transformed enthalpy of hydrogen ions is given by
Hk(H+) = p'(H+) + TSk(H+) (4.1 -1 7)