Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

revised: how batteries work



I warned you that the previous version was prepared in haste. This one is
slightly better. One new important concept, and various bug-fixes,
especially regarding the diagrams. I hope the previous version didn't
waste anyone's time too horribly.

Start with a chunk of nickel. Suppose it starts out electrically
neutral; we can just count positive and negative charges and they come out
even. It has some work function.

Now take a second chunk of nickel. It also starts out electrically
neutral. Now imagine that using some powerful neutrino-ray or something,
we cause an inverse-beta (electron capture) reaction that turns the nickel
nuclei into iron nuclei. The crystal structure re-arranges itself
accordingly. The chunk of iron remains, by construction, electrically
neutral. But it has a different work function than the corresponding chunk
of nickel.

The work functions are different because of things like the Pauli exclusion
principle. You have a different number of fermions in a different-sized
box, so the Fermi level will be different. Calculating things like work
functions _ab initio_ is a real tour de force, and we need not discuss the
details here; it suffices to accept the observed work function values:

Work function for aluminum: 4.2 eV
Work function for iron: 4.63 eV
Work function for nickel: 5.2 eV

This is real physics. It's not hard to understand. The electrons want to
be near the metal nuclei. Even if the metal chunk is slightly negatively
charged it will attract electrons. (Indeed even a single neutral hydrogen
atom will attract electrons -- the H- ion in vacuum has lower energy than a
hydrogen atom and electron separately.)

You can even make a connection between the work function (a purely
electrical property) and the elastic properties of the metal: when you
squeeze the chunk of metal you squeeze the electron wavefunctions -- and
that takes energy.


=====

Now we must carefully make the distinction between
-- a test charge, and
-- an electron.

First, make sure both chunks of metal are identical in shape. File one of
them down if necessary. This means they have the same self-capacitance
(also know as capacitance-to-infinity). They are both still electrically
neutral. A test charge placed on the iron chunk creates the same
electrical field pattern and has the same energy as a test charge placed on
the nickel chunk. So the test charge is equally happy either place.

But test charges do not exist in nature. Equilibrium is not established by
the exchange of test charges, or alpha particles, or muons... in all
practical situations equilibrium is established by exchange of electrons.

The Pauli exclusion principle involves electrons excluding other
electrons. It means that electrons are happier being on the nickel chunk
than the iron chunk -- even though a test charge would be equally happy.

So here is a picture of the electron's "unhappiness function" (i.e.
electrochemical potential) when the metal chunks are electrically
neutral. This is NOT the equilibrium situation. Ignore the corresponding
diagram in my previous note, which was wrongly labelled.

Electrochemical potential; electrical neutrality:

................................................................ (zero)
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|_____| | | |_____| | | |_____| | |
Fe | | Fe | | Fe | |
|_____| |_____| |_____|
Ni Ni Ni


Next, we have the diagram for the electrochemical potential when
equilibrium HAS been established by exchange of electrons, as shown below:

Electrochemical potential; electrons in equilibrium:

.............|.....|...............|.....|...............|.....| (zero)
| | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
| | | | | | | | | | | |
|_____| |_____| |_____| |_____| |_____| |_____|
Fe +>>- Ni -<<+ Fe +>>- Ni -<<+ Fe +>>- Ni
field field

Now, in equilibrium, the nickel chunks have a definite excess of electrons,
and the iron chunks have a definite deficit. In the gaps, there will be an
electrical field. In this region, the electric potential is equal to the
electrochemical potential, so there is no ambiguity. That is, in the gaps
electrons and test charges behave the same.

The magnitude of the electric field in the gaps is equal to the
work-function difference divided by the length of the gap, if we assume a
nice parallel-plate geometry. Now the surface charge on each plate is
proportional to the electric field, so we discover that the _amount_ of
charge depends inversely on the gap. (You can wiggle the gap and measure
how much current needs to flow in order to maintain equilibrium; this is a
way to measure work functions. You can even add a potentiometer and make
it a null measurement; this is called a Kelvin bridge.)

Imagine the metal chunks are arranged in a big circle, so we have periodic
boundary conditions on the diagram.

Now suppose we hook up certain pairs using aluminum wire in the usual
clever way. The diagram looks like this:


Electrochemical potential; electrons in equilibrium:

..............|....|................|....|................|....|..
| | | | | |
| | | | | | | | | | |
| || || | | || || | | | | |
| || || | | || || | | || || |
| || || | | || || | | || || |
|____||____||____| |____||____||____| |____||____||____|
-+ +- -<<+ -+ +- -<<+ -+ +-
Fe Al Ni Fe Al Ni Fe Al Ni


The details of the aluminum are not very interesting; its main function is
to ensure that the attached Fe and Ni remain in
electron-equilibrium. Aluminum conducts electrons quite well; it does
not conduct muons or alpha particles or "test charges".

So far there is nothing special about this setup. An electron is equally
happy in any of the various chunks of metal. There is an electric field in
the gaps. There is a huge "dipole layer" at the Fe/Al interface and also
at the Al/Ni interface. You can think of it as a near-infinite electric
field over a near-zero distance, resulting in a finite potential difference
(just equal to the work-function difference).

Now let's get down to business. Let's put some alkali goop in the gaps. This
substance has the property that it has lots of ion-pairs in it. When we
put an ion-pair in an electrical field, such as in the Ni-Fe gap, the + ion
will move one way and the - ion will move the other way. This process will
continue until the electrical field in the gap is driven to
zero. (Remember a pair of separated charges have a field-line between
them; the field lines created by the moving ions tend to cancel the
equilibrium field created by the metal chunks.)

The result is shown in the following diagram:

Electrochemical potential; goop in gaps
| | | |
| | | || |x
(A)............|....|(B).|....|......|....|(C).|....||.|x........(A').
| | | || || | | || |x
| | | | | || || | |____||_|x | |
| || || | | || || | x | |
| || || | |____||____||____| x | |
| || || | | || |
|____||____||____| | || |
| || |
|_||____|
-+ +- -+ +- -+ +-
Fe Al Ni Fe Al Ni Fe Al Ni


It is highly ironic that in the guts of the battery where you the chemical
action is taking place, there is a relatively field-free region. The
chemistry happens one place, and the change in voltage happens in another
place! In the previous case (full electron-equilibrium) the potential
difference due to the dipole layers was undone by the field in the
gaps; now, with goop in the gaps, the gaps are field-free and don't undo
anything.

This situation does not represent electron-equilibrium everywhere; there
is some sort of ionic chemical equilibrium in the gaps.

So when an electron flows through our cells from left to right it most
assuredly does gain energy. Each cell is like a step on a staircase. You
wind up with a substantial voltage difference between the right (Fe)
terminal of cell (C) and the left (Ni) terminal of cell (A)/(A') at the
location shown by the "x" symbols. This voltage can easily be measured
with a voltmeter, and can power lamps and motors et cetera.

Also note that as each electron flows through a cell, one molecule of
chemical reaction takes place in the gap where the alkali
lives. Otherwise, this gap would just charge up like a capacitor and the
battery would be orders of magnitude less effective at maintaining its
rated voltage. The voltage stays the same (more or less) until you run out
of chemicals. (This is how the unit of charge was initially defined: the
amount of chemical precipitated in such a cell.)

Here's the essential piece of magic: running a wire between a piece of Fe
and a piece of Ni is very different from putting some reactive ionic goop
between them. If you read the diagram from left to right you get
Fe-Al-Ni-goop-Fe-Al-Ni-goop
which is _not_ a palindrome. There is a definite direction to the
structure, and that determines which end of the battery is + and which is -.

NOTE!!! I didn't say anything about charge being "created". It is OK to
create charge _pairs_ by ionizing some previously-neutral chemical, but no
net charge was created, not even temporarily. It is also OK to speak of
charge flowing _through_ the cells. But charge does not flow "out" of the
cells like water pouring out of a bucket. It just doesn't. If an electron
flows out one terminal, you can be sure that an electron flows in the other
terminal at the very same instant.

And a final word about terminology: Consider the terms "charging a
capacitor" or "charging a battery". Those terms have a perfectly well
defined meaning. Alas, the same words are used with a completely different
meaning when we speak of "charging the terminal of a van de Graaf
generator". This is a problem. I don't see any easy way to fix it.
Explicit example:
meaning 1: "How much charge is on this battery? About 30 amp-hours."
meaning 2: "How much charge is on an electron? About 1.6e-19 coulomb."

Some people advocate using the term "free charge" for meaning 2, but
that doesn't help much, because people like B.C. use "free charge" for
mobile charges carrying ordinary current (i.e. meaning 1).

Perhaps it would help to focus on the prepositions: Meaning 1 speaks of
elementary charges flowing _through_ (not sitting on) the battery. But
students are notoriously poor at discerning subtle technical usage of
prepositions, so maybe that's asking for trouble. Given the clumsiness of
the language, it's a miracle anybody ever communicates anything.

This "charging" problem is even worse than the problems that arise from
having multiple definitions of other terms, for example
-- a technical and a nontechnical definition of the word "elastic"
-- two schools of thought about the technical definition of
the word "heat", not to mention various nontechnical definitions.
Those cases are relatively innocuous, because it is usually possible to
adopt one set of definitions and stick to it. But when we talk about the
"charge" in or on a battery, all hell breaks loose, because we need to use
the word in different senses, sometimes even in the same sentence! Yuuuuck!