Page 37 - Thermodynamics of Biochemical Reactions

2.6 Thermodynamic Potentials for a Single-Phase Systems with One Species 31

U[7; P] = G = - RTln A, where A is the isothermal-isobaric partition function;
and U[7;p] = -RTlnE, where E is the grand canonical ensemble partition
function. When a system involves several species, but only one can pass through
a membrane to a reservoir, U(T,pl] = -RTlnY, where Y is the semigrand
ensemble partition function. The last chapter of the book is on semigrand partition
functions.
Taking the differentials of the seven thermodynamic potentials defined above
and substituting equation 2.6-1 yields the fundamental equations for these seven
additional thermodynamic potentials:
dH = TdS + VdP + pdn (2.6-7)

dA = -SdT- PdV+ pdn (2.6-8)
dG = -SdT+ VdP + pdn (2.6-9)
dU[p] = TdS - PdV- ndp (2.6- 1 0)

dU[P,p] = TdS + VdP - ndp (2.6-1 1)
dU[7;p] = -SdT- PdV- ndp (2.6-1 2)

dU[T,P,p] = -SdT+ VdP- ndp = 0 (2.6- 1 3)
This last equation is the Gibbs-Duhem equation for the system, and it shows that
only two of the three intensive properties (7; P, and p) are independent for a
system containing one substance. Because of the Gibbs-Duhem equation, we can
say that the chemical potential of a pure substance substance is a function of
temperature and pressure. The number F of independent intensive variables is
F = 1 - 1 + 2 = 2, and so D = F + p = 2 + 1 = 3. Each of these fundamental
equations yields D(D - 1)/2 = 3 Maxwell equations, and there are 24 Maxwell
equations for the system.
The integrated forms of the eight fundamental equations for this system are
U(S, vn) = TS - PV+ pn (2.6-14)

H(S, P, n) = TS + pn (2.6-15)
A(7; vn) = -PV+ pn (2.6-16)

G(7: P, n) = pn (2.6- 17)
U[p](S, Kp) = TS ~ PV (2.6-18)
UCP, PI(S, p, = TS (2.6-19)

WTpI(7; vp) = -PV (2.6-20)
u CT,P,PI(7; P,p) = 0 (2.6-2 1)
where the natural variables are shown in parentheses.
The basic question in all of thermodynamics is: A certain system is under such
and such constraints, what is the equilibrium state that it can go to spontaneous-
ly? The amazing thing is that this question can be answered by making
macroscopic measurements. Thermodynamics does not deal with the question
as to how long it will take to reach equilibrium. We now have seven criteria
for equilibrium in a one-phase system with one species and only PV work.
The criteria of equilibrium provided by these thermodynamic potentials are
(dU)s,",n d 0, (dH)S,P,n 0, (dA)T,",, d 0, (dG)*,P,n d 0, (d~CPl)s,v,fi d 0,
(dWP, PI)S,P,@ d 0, and (d~CT,P1)T,v,fi d 0.
The reason for going into this much detail on all of the thermodynamic
potentials that can be defined for a one-phase, one-species system and the
corresponding criteria for spontaneous change is to illustrate the process by which
these thermodynamic potentials are defined and how they provide criteria for