384
9 Bioenergetics
−
D
C
∂C
∂x
= (μq)
∂V
∂x
.
(9.161)
This differential equation for C is solved by
C = C 0 exp (−qqV / (μD)) .
(9.162)
We now use Einstein’s relation, D = μk B T , and solve for the potential difference
across the membrane, giving the Nernst–Planck equation
V 2 − V 1 =
k B T
q
ln
C 1
C 2
(9.163)
This relation is often expressed in the form:
V 2 − V 1 =
RT
N A |e| z
ln
C 1
C 2
(9.164)
=
RT
F
1
z
ln
C 1
C 2
(9.165)
=
RT ln 10
F
1
z
log
C 1
C 2
(9.166)
= 61.54
millivolts
z
log
C 1
C 2
at 37
◦ C
(9.167)
= 57.97
millivolts
z
log
C 1
C 2
at 19
◦ C
(9.168)
in terms of the gas constant R = 8.3145 J/deg, Faraday’s constant F = 96,485 coul
(the charge of a mole of protons), and z the number of electronic charges on the ion
(with z negative if a negative ion).
Biological cell membranes control the ions Na + , Cl − , K + , and Ca ++ . Their
interior concentration are important to the functioning of the cell. For each of these
ions, the voltages inside the cell due to any one of these ions are given by
V Na =
RT
F
ln
[Na
+
] O
[Na
+
] I
V K =
RT
F
ln
[K
+
] O
[K
+
] I
V Cl =
RT
F
ln
[Cl
−
] I
[Cl
−
] O
V Ca =
RT
F
1
2
ln
[Ca
++
] O
[Ca ++ ] I
(9.169)