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battery : ball-and-stick model



Here is a ball-and-stick model of how a battery works. It
embodies a number of simplifications, but it's not grossly
wrong. We start with a mechanical analogy:


m m m m m m m m
rrrrrrrrrrrrrrrr mC C
A \ /
\ /
mC_____\/_____C
/\s
/ \ s
/ \ s
mC C s
B m s
dddddddddddddddd s
L


A number of marbles (m) sit in a high reservoir (rrr). One
by one they enter a cup (C) on a wheel and ride down until
they are emptied out into a drain area (ddd). The wheel
turns a shaft (sss) which does work on the load (L).

If the wheel isn't turning too quickly,
-- the device produces a more-or-less constant torque on
the shaft, until you run out of marbles, and
-- the device is more-or-less reversible.

In contrast, if you try to make it go fast, i.e. demand a
lot of power, there will be all sorts of nonidealities.

You must also make sure there are no internal short
circuits. If marbles could sneak directly from point A to
point B without riding in the cups, the device would have no
shelf life.

This is a pretty tight analogy to how a battery works.
Actually, a battery has two such processes, one at each
terminal.

At the + terminal, there is a chemical reaction that wants
to take place, but it cannot take place unless somebody
supplies an electron. This is called a half-cell reaction.
(The reaction has already proceeded to some extent, gobbling
up all the electrons it can find in the vicinity, which is
why that terminal already carries a + charge.)

Similarly at the - terminal, there is a chemical reaction
that wants to take place, but it cannot take place unless
somebody supplies some net positive charge, i.e. by taking
away an electron. This is another half-cell reaction.
(Again, the reaction has already proceeded to some extent,
which is why the terminal already carries a - charge.)

It is important that the electrolyte between the terminals
does not conduct electrons. Otherwise the battery would
have no shelf life. The only way electrons can flow from
one terminal to the other is by flowing through the external
load.

When the battery is not under load, there is no electrical
field in the electrolyte, except in very thin layers near
each electrode. You can think of this field being zero
because of the sum of several contributions:
a) The two electrodes are like the plates of a capacitor.
They are charged, so they produce a field between them.
b) There are lots of mobile ions in the electrolyte. In
the bulk, they move under the influence of the
aforementioned field, until they get near one electrode or
the other. This effect dominates at almost all locations on
all interesting timescales. The point to remember is that
because the ions are mobile, there cannot be any electrical
field in the bulk electrolyte in open-circuit equilibrium.
c) Near each electrode, the energy landscape is greatly
influenced by the half-cell reactions. The details are
beyond the scope of this discussion. [This affects the
height of the electric potential in the bulk, but does not
affect its slope, which must be zero for reason (b) above.]

It is worth discussing the details of what happens when an
electron flows through the load, from one terminal to the
other. For every electron that flows, one unit of half-cell
reaction takes place at each terminal. Let's assume that
the product molecules preciptate out and are never heard
from again.

Because of this electron that flowed, the + terminal is a
little less positive than it was in open-circuit
equilibrium, and the - terminal is a little less negative.
Since there was no field in the bulk electrolyte in
equilibrium, this change creates a field. Somewhere in the
bulk, an ion-pair separates, and the - member drifts toward
the - electrode, and the + member drifts toward the +
electrode. This restores everything to the status quo
ante. Therefore the cell maintains a more-or-less constant
voltage, until you run out of chemicals.

It is worth emphasizing that when the battery is under load,
+ ions are drifting through the electrolyte toward the +
terminal. A smallish field pushes them in that direction.
This is opposite to the direction of the field outside the
electrolyte, the field that pushes current through the
external load from + to -. At some point the + charge on
this ion is hoisted into the + terminal. The energy for
this comes from the half-cell reaction. In more physical
terms, what happens is that an electron in the + terminal
plus a neutral atom from the + terminal react with the ion
to form a neutral product. The electron is very tightly
bonded in this product molecule. The electron has been
taken from a high-energy reservoir and ends up in a
low-energy drain area. If it weren't for this chemical
reaction, you would never be able to coax electrons to leave
the + terminal.

==================================================

Tangential remark:

Note that the marble-machine operates entirely by use of
conservative forces. All the forces involved are the
gradient of a potential. Non-potential energies (such as
one might get from a time-varying magnetic field) are not
involved.

But you may ask, how is it that the shaft can go around and
around and around, doing work on each cycle and repeatedly
returning to a seemingly-similar situation, when only
potentials are involved? The marbles are held in the cups
by a force of constraint, and supposedly a constraint cannot
do any work, right?

Well, there are to parts to the answer:

1) The true theorem is that a !!nonmoving!! constraint
cannot do any work. Here we've got moving constraints.
They do work.

2) Timescales matter. Over some unreasonably-long
timescale, the marbles will simply evaporate from the cups
and condense in the drain area. So we have to operate the
device fast compared to that timescale. That's not hard.
But on the other hand, when the cups are at the special
places where they are supposed to get filled and emptied, we
need that to happen fairly quickly. So we need to have a
hierarchy of timescales:
evaporation >> normal transit >> fill/empty
time time process time

Similar remarks apply to batteries. All the forces involved
are purely electrostatic. The slightly-tricky bit is to
realize that electrons cannot move through the electrolyte.
The only way they can get from the - terminal to the +
terminal on any reasonable timescale is to flow through the
external load.