230 Chapter 7. Entropic forces at work[[Student version, January 17, 2003]]
the electrostatic potential energy of such a shell (also called itsBorn self-energy)is
1
2 qV(r), orq(^2) /(8πε 0 r). In this formulaε 0 is a constant describing the properties of
air, thepermittivity.Appendix B gives e^2 /(4πε 0 )=2. 3 · 10 −^28 J·m.The chargeq
on the drop equals the number density of water molecules, times the drop volume,
times the charge on a proton, times 1%. Squaring gives
(q
e
) 2
=
(
103 kg
m^36 · 1023
0. 018 kg·
4 π
3
(10−^3 m)^3 · 0. 01) 2
=1. 9 · 1036.
Multiplying by 2. 3 · 10 −^28 J·mand dividing by 2ryields about 2· 1011 J.Two hundred billion joules is a lot of energy—certainly it’s much bigger thankBTr!And indeed,
macroscopic objects really are electrically neutral (they satisfy the condition of “bulk electroneu-
trality”). But things look different in the nanoworld.
Your Turn 7d
Repeat the calculation for a droplet of radiusr=1μmin water. You’ll need to know that the
permittivityεof water is about eighty times bigger than the one for air used above, or in other
words that thedielectric constantε/ε 0 of water is about 80. Repeat again for ar=1nmobject
in water.Thus itis possible for thermal motion to separate a neutral molecule into charged fragments.
Forexample, when we put an acidic macromolecule such as DNA in water, some of its loosely
attached atoms can wander away, leaving some of their electrons behind. In this case the remaining
macromolecule has a net negative charge: DNA becomes a negativemacroion.This is the sense in
which DNA is charged. The lost atoms are then positively charged. They are calledcounterions,
since their net charge counters (neutralizes) the macroion. Positive ions are also calledcations,
since they’d be attracted to acathode; similarly the remaining macroion is calledanionic.
The counterions diffuse away because they were not bound by chemical (covalent) bonds in the
first place, and because by diffusing away they increase their entropy. Chapter 8 will discuss the
question of what fraction detach, that is, the problem of partial dissociation. For now let us study
the simple special case of fully dissociated macroions. One reason this is an interesting case is
because DNA itself is usually nearly fully dissociated.
The counterions, having left the macroion, now face a dilemma. If they stay too close to home,
they won’t gain much entropy. But to travel far from home requires lots of energy, to pull them
awayfrom the opposite charges left behind on the macroion. The counterions thus need to make
acompromise between the competing imperatives to minimize energy and maximize entropy. This
subsection will show that for a large flat macroion, the compromise chosen by the counterions is
to remain hanging in a cloud near the macroion’s surface. After working Your Turn 7d, you won’t
besurprised to find that the cloud can be a couple of nanometers thick. Viewed frombeyond
the counterion cloud, the macroion appears neutral. Thus, a second approaching macroion won’t
feel any attraction or repulsion until it gets closer than about twice the cloud’s thickness. This
behavior is quite different from the familiar behavior of charges in a vacuum. In a vacuum Gauss’s
Law (Equation 7.20 below) implies that, for a planar charged surface, the electric field doesn’t fall