Page 28 - Thermodynamics of Biochemical Reactions

22 Chapter 2 Structure of Thermodynamics

obtained by use of the operations of calculus. The third law states that the entropy
of each pure element or substance in a perfect crystalline form is zero at absolute
zero.
The entropy provides a criterion of spontaneous change and equilibrium at
constant U and V because (dS),., 3 0. Thus the entropy of an isolated system
can only increase and has its maximum value at equilibrium. The internal energy
also provides a criterion for spontaneous change and equilibrium. That criterion
is (dU),,, 6 0, which indicates that when spontaneous changes occur in a system
described by equation 2.2-1 at constant S and V; U can only decrease and has its
minimum value at equilibrium.
The inequalities of the previous paragraph are extremely important, but they
are of little direct use to experimenters because there is no convenient way to hold
U and S constant except in isolated systems and adiabatic processes. In both of
these inequalities, the independent variables (the properties that are held con-
stant) are all extensive variables. There is just one way to define thermodynamic
properties that provide criteria of spontaneous change and equilibrium when
intensive variables are held constant, and that is by the use of Legendre
transforms. That can be illustrated here with equation 2.2-1, but a more complete
discussion of Legendre transforms is given in Section 2.5. Since laboratory
experiments are usually carried out at constant pressure, rather than constant
volume, a new thermodynamic potential, the enthalpy H, can be defined by
H=U+PV (2.2-2)

The differential of the enthalpy is given by
dH = dU + PdV+ VdP (2.2-3)

Substituting equation 2.2-1 yields
dH = TdS + VdP (2.2-4)

The use of a Legendre transform has introduced an intensive property P as an
independent variable. It can be shown that the criterion for spontaneous change
and equilibrium is given by (dH),,, 3 0.
The temperature can be introduced as an independent variable by defining
the Gibbs energy G with the Legendre transform
G=H-TS (2.2-5)

The differential of the Gibbs energy is given by
dG = dH - TdS - SdT (2.2-6)

Substituting equation 2.2-4 yields
dG = -SdT+ VdP (2.2-7)

The use of this Legendre transform has introduced the intensive property T as an
independent variable. It can be shown that the criterion for spontaneous change
and equilibrium is given by (dG),,, 3 0. The Gibbs energy is so useful because T
and P are convenient intensive variables to hold constant and because, as we will
see shortly, if G can be determined as a function of T and P, then S, V, H, and U
can all be calculated.
Gibbs (1873) showed how to include the contributions of added matter to the
fundamental equation by introducing the concept of the chemical potential p, of
species i and writing the fundamental equation for the internal energy of a system
involving PV work and changes in the amounts n, of species as
v<
dU = TdS - PdV+ 2 pidni (2.2-8)
i= 1