Page 147 - Thermodynamics of Biochemical Reactions

8.3 Two-Phase System with a Membrane Permeable by a Single Ion 145

tions in the two phases, and an equilibrium relation of the form

(8.2-9)

will be satisfied in each phase and in the system as a whole.

8.3 TWO-PHASE SYSTEM WITH A MEMBRANE
PERMEABLE BY A SINGLE ION

When two different phases are separated by a membrane permeable by a single
ion and that ion has different activities on the two sides of the membrane, an
electric potential difference will be set up at equilibrium (Alberty, 1995a, d, 1997).
We first consider a two-phase system with an aqueous solution of a single salt on
both sides of a membrane that is permeable only to cation C. When electrolytes
are involved, it is necessary that counterions be present because bulk phases are
electrically neutral. When cation C diffuses through the membrane into the fi
phase, the fi phase becomes positively charged with respect to the a phase.
Diffusion stops when a sufficient difference in electric potential has been estab-
lished. When a conductor is charged, the charge migrates to the surface, and for
an aqueous solution of a salt this occurs in the charge relaxation time of about
one nanosecond. Thus a positively charged layer is formed at the surface of the
membrane toward the fi phase and a negatively charged layer is formed at the
surface of the membrane toward the 2 phase. The thickness of the layer in each
aqueous phase is the Debye length of about 1 nm at an ionic strength of 0.1 M.
The amount of charge required to set up a significant potential difference between
the phases is very small. Many biological membranes have capacitances of about
one microfarad per square centimeter (Weiss, 1996). The charge transfer per
square centimeter required to set up a potential difference of 0.1 V is therefore
mol of singly charged ions. As the electric potential of the fi phase increases
due to the diffusion of cation C, the process of diffusion slows and an equilibrium
potential difference is reached.
The fundamental equations for G for the phases on either side of the center
of the membrane are
dG, = - S,dT+ V,dP + ,ucadnC, + +,dQ, (8.3-1)

dG, = - S,dT+ 5dP + p,,dn,, + +,dQB (8.3 -2)
where n,, is the amount of cation C in the a phase and Q is the amount of charge
transferred across the center of the membrane. No term is included for the
monovalent anion A because its concentration in the bulk phase is equal to that
of the monovalent cation C. Since dnAa = dnCa, the inclusion of a term for A in
equation 8.3-1 would yield (pC, + ,uAn)dnCa. However, since the chemical potential
can be defined for an arbitrary reference potential (cf, AfGP), ,uAor can be set equal
to zero.
Since -dQ, = dQ, = dQ, the fundamental equation for G for this two-phase
system prior to the establishment of phase equilibrium is the sum of equation
8.3-1 and 8.3-2:

dG = - SdT+ VdP + ,uCndnCa + ,u,,dn,, + ($, - 4,)dQ (8.3-3)
There is difference in electric potential across the membrane. So there is an electric
field in the membrane, but there are no electric fields in the two bulk phases. The
electrical work required to move charge dQ across the center of the membrane is
(& ~ 4,)dQ. The polarization of the membrane does not change in this process
of charge transport because the potential difference is constant. Since
dnC, = - dn,,, equation 8.3-3 can be used to show that at phase equilibrium,