Electroneutrality
of bulk solutions
Bulk
solutions, i.e. aqueous salt solutions of substantial volume, have essentially
the same number of cations and anions regardless of
the value of the electrical potential.
This is true to a high degree of accuracy.
-70mV ?Cation+ + ?Anion-
Consider
an animal cell with a typical electrical potential. What difference in charge
is necessary to produce this potential?
The percent charge of the total ions is
minuscule. In addition all excess charge
is relegated to a thin layer next to the membrane. Why? Because the electromagnetic force is extremely strong.
Born Self Energy
In Vacuum:
To explore this concept of bulk electroneutrality, we can use the property of self energy
to relate charge to energy. Electrostatic Self Energy can be written as
or
Where:
q is charge in Coulombs
ε0 is the
permittivity of free space:
8.85 x 10-12 C2 J-1 m-1
r is the radius of
the charged sphere
Ψ
is the voltage
Note:
= 2.3 * 10-28 J*m
According to this equation, self energy is a
function of charge and radius. As an example, we can suppose there is a water
droplet with a radius of 1mm. If just 1% is charged, the self energy of the
droplet becomes 2*1011 J. That is equivalent to 2*1012 V
or 1032 kJ/mol. Thermal energy (RT) is only 2.4 kJ/mol. Considering
that the Boltzmann distribution is 
, the probability of this example occurring is practically
impossible. Electroneutrality applies as the
percentage of charge must be much smaller than 1%.
Permittivity of Water:
However,
the permittivity of water is greater than that of a vacuum or free space.
Permittivity is associated with the ability of a substance to polarize as an
electric field passes through it. Aqueous solutions are conductive like metals,
though to a lesser extent. An analogy can be drawn between the traditional
example of charge on a metal sphere and the charge on an aqueous body. To take
this into account, an additional constant (ε) is introduced. The Born Equation
becomes:
Where:
ε is
the relative permittivity or dielectric constant
ε of
water is ~ 80
This form of the equation is closer to that of
living systems, but the resultant energy is still too high. The self energy is
slightly reduced to 1030 kJ/mol. Electroneutrality
still applies.
|
# Charges Moved |
E/mole (kJ/mol) |
Probability Ratio |
Φ (mV) |
|
1 |
9 x 10-4 |
.9996 |
2 x 10-2 |
|
100 |
8.6 |
.027 |
1.8 |
|
1000 |
865 |
10-157 |
18 |
We
can improve the situation by incorporating fewer charges. The following table shows the self-energy for
different numbers of added charges, the probability of this occurring
spontaneously at room temperature and the electrical potential of the water
droplet.
Scale:
However, the scale of living systems is much
smaller than a 1mm radius water droplet. If the radius is reduced to 1nm, the
self energy becomes 1.6 kJ/mol. This is within the
order of thermal energy and has a probability ratio of about 0.5.
Consider the first
example of a bacterial cell. The number
of cations and anions inside is essentially the same
despite 180 mV negative potential in the cytosol.
Note: This model is based on self energy and
neglects membrane capacitance.
See posted spreadsheet called electroneutrality.
Electric field across a membrane and its effect
on the distribution of cations and anions next to the
membrane
There is a net charge on each side of the
membrane and this is responsible for the change in electrical potential. Actually the electric field, the slope of the
voltage trace, changes only if there is a net charge. A net negative charge causes an increase in
slope and a net positive charge causes a decrease in slope.
Some conclusions
When the scale is very small electroneutrality
can be violated, but at larger sizes this is highly improbable.
In the case of a membrane about 5 nm in
thickness, charge can collect on both sides of the membrane. This creates a
force near the surface of the membrane, but it quickly dissipates about 1 to 2
nm from the surface.