Figure 1: Electrochemical potential; electrical neutrality, not equilibrium
The purpose of this section is to introduce the idea of an equilibrium electric field and to quantify it in terms of the work function. This will be a plausibility argument in the form of a Gedankenexperiment.
Start with a chunk of nickel. Make sure it is electrically neutral.
Also take a corresponding chunk of iron. It has the same FCC crystal structure as the nickel, although the lattice spacing will be slightly different. It also starts out electrically neutral.
As we shall see, these two electrically-neutral pieces of metal attract electrons differently. We give a name to this: we say they have different work functions. (For more about the microscopic origin of the work function, see reference 1.) The work function measures the energy required to take an electron from the surface of the metal to some far-away place.1
The work functions are different because of things like the Pauli exclusion principle.2 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. We will accept the observed work function values and then make some plausibility arguments.
| Element | Work function | |
| silver | 4.26 eV | |
| aluminum | 4.28 eV | |
| zinc | 4.33 eV | |
| iron | 4.5 eV | |
| copper | 4.65 eV | |
| nickel | 5.15 eV | |
| gold | 5.1 eV |
The fact that a metal has some sort of nonzero work function is not particularly 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.)
The fact that different materials have different work functions is perfectly understandable also. Different materials have a different spacing between nuclei. So think of it as a simple quantum-mechanical particle-in-a-box problem: The smaller the box, the higher the kinetic energy the electrons must have. 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 raises their kinetic energy.
Let’s look at this idea in more detail. The following is accurately true for one metal compared to a compressed piece of the same metal, and true to a first approximation for nickel compared to iron: Since the two pieces of metal have the same number of conduction electrons, and similar shapes, there must be a one-to-one correspondence between their conduction-band wavefunctions. The wavefunctions in the smaller box will (to a first approximation) look the same as those in the larger box, just scaled to a smaller wavelength so they fit in the box. Smaller wavelength λ means higher momentum p = h/λ, hence higher kinetic energy Ek = p2/2m.
To obtain a better approximation, you would need to take into account other contributions to the energy. For starters, there is a contribution to the potential energy from the fact that a nickel atom has more protons than an iron atom. That difference is mostly compensated by an equal difference in the number of non-conduction electrons, but then those extra electrons affect the kinetic energy of the conduction electrons (via the exclusion principle).
Remember, the point of this section was just to make it plausible that different materials have different work functions. For our purposes, we do not need to calculate accurate work function values from first principles. It suffices to accept the experimentally observed values.
To make progress, we must carefully make a distinction: a test charge is not the same as a real electron.
First, arrange for both chunks of metal to be identical in size and shape. File one of them down if necessary. This won’t change the work function. The objective is to make both chunks have the same self-capacitance. 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 practical situations (with a few important exceptions, as we shall see), equilibrium is established by exchange of electrons. Real 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.
Figure 1 shows the electron’s "unhappiness function" (i.e. electrochemical potential) when the metal chunks are electrically neutral. This is not the equilibrium situation.
The black line in the diagram represents the energy level of the electron, as a function of position. The green and yellow shaded regions are meant to represent the Fermi sea within the chunks of metal. The bottom of the Fermi sea is not interesting, because it is impossible to inject an electron into this level (by the exclusion principle), and it is energetically unfavorable to extract an electron from this level. Figure 1 does not attempt to accurately portray the bottom of the Fermi sea.
What matters is the top of the Fermi sea. The top of the Fermi sea for iron is labelled t in the diagram. The most easily-added electron will go in just above this level, and the most easily-extracted electron will come out from just below this level. (This corresponds to the LUMO and HOMO in molecular physics – but if that doesn’t mean anything to you, don’t worry about it.)
The length of the vertical steps in the black line represent the work function of the associated chunk of metal, i.e. the energy that the electron loses if it gets ejected from the metal, via the photoelectric effect or whatever. The workfunction for iron is labelled Φ in the figure.
In the figure I have indicated a “zero level” for the potential This choice is arbitrary and has does not affect the physics in any way. In this case I have chosen the energy of an isolated far-away electron in the vacuum to be the reference point for zero potential.
Next, figure 2 shows the electrochemical potential when equilibrium has been established by exchange of electrons:
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 electric field, as indicated by the blue arrows. In the gaps – but not within the metals – the energy per electron can be equated to the conventional electric potential (with the usual minus sign). Within the metals, we won’t even talk about the electric potential, because it is irrelevant. What matters is the electrochemical potential, which contains a huge contribution from the kinetic energy of the electrons.
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 way, the way batteries are hooked up, as shown in figure 3.
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 a dipole layer as a near-infinite electric field over a near-zero distance. Let x be the effective thickness of the metal/metal interface. As x becomes small, the electrical field grows like 1/x, so the potential difference remains independent of x and remains equal to the work-function difference. The Fe and Ni pieces are at the same potential here as they were back in figure 2.
Now let’s get down to business. Let’s put an electrolyte in the gaps. Let’s use potassium hydroxide for the electrolyte. We keep the iron electrode from the discussion above. Rather than using a plain nickel electrode, we use nickel coated with nickel oxyhydroxide, NiOOH, because that gives us a nice rechargeable cell.
A good electrolyte has several interesting properties; for one thing 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 positive ion will tend to drift one way and the negative ion will tend to drift the other way. This process will continue until the electrochemical field in the interior of the gap becomes a constant, independent of position, which is the equilibrium condition. There could be some electrical potential gradient; I don’t think there is much but there could be some. And there could be some concentration gradient; again I don’t think there is much, but there could be some. In any case, when you consider the concentration gradient and the electrical gradient together, in equilibrium there is no net motion of ions. In the simplest case, there is no electrical field (hence no drift) and no concentration gradient (hence no net diffusion). In the more general case, drift due to the electrical potential gradient is counterbalanced by diffusion along the concentration gradient. The two gradients point in opposite direction, and when we add the two effects (electrical and concentration) we find that the electrochemical potential has zero gradient.
All that applies to the interior of the gaps, in the bulk electrolyte. At the ends of each gap, there will, in general, be some accumulation of ions. This produces complicated dipole layers there. The strength of the dipole layer depends on the properties of the electrolyte, as well as on the properties of the adjacent metal, as discussed in section 1.6. The strength of the dipole layer determines how the potential in the interior of the gap is related to the potential in the interior of the adjacent metal.
The result is shown in figure 4.
In the previous case, when we had electron-equilibrium as shown in figure 3, the potential difference due to the work-function difference was undone by the field in the gaps; now, with electrolyte in the gaps, the gaps are field-free except in the boundary layers. Note that the potential in the gap at A’ lines up with the potential in the gap at A, as it should, since A and A’ represent the same point (since we have periodic boundary conditions), and the gap at A≡A’ is one of the field-free electrolyte-filled regions.
This situation does not represent electron-equilibrium everywhere; the ions in the gaps establish a field-free region in the interior of each gap, which is not the equilibrium condition.
Let’s cut the aluminum wire between cell C and cell A’ and insert a voltmeter there, at the location marked “external circuit” figure 4.
Each cell is like a step on a staircase. An electron in the leftmost Fe electrode has less energy than an electron in the next Fe electrode, which in turn has less energy than an electron in the rightmost Fe electrode. You wind up with a substantial voltage indicated on the voltmeter.
We can figure out the sign of the voltage from figure 4. Electrons are more unhappy just to the left of the external circuit than they are just to the right of it. So somebody outside the battery, treating the battery as a two-terminal black box, will see electrons trying to leave via the Fe terminal, flow through the external circuitry (including the voltmeter) and thence enter the Ni terminal. The Fe terminal is therefore appropriately labelled the "-" terminal. Naturally the Ni terminal is appropriately labelled the "+" terminal.
Here’s one essential piece of the magic: running a wire between a piece of Fe and a piece of Ni is very different from putting some reactive ionic electrolyte between them. If you read the diagram from left to right you get
Fe-Al-Ni-electrolyte-Fe-Al-Ni-electrolyte
which is not a palindrome. There is a definite direction to the structure, and that determines which end of the battery is positive and which is negative. If you replaced the electrolyte with an aluminum wire, you would get a palindrome, which couldn’t possibly produce a useful voltage.
Work functions are part of the story, but not the whole story. The work functions measure the energy required to take an electron out of the metal into the vacuum. This is not quite what’s happening in the battery; there are chemical reactions taking place that change the energy budget. This is discussed in section 1.5 and especially section 1.6.
In figure 4. the battery is open-circuited, so no current of any kind is flowing anywhere. But if we attach an electrical load to the terminals of the battery, the voltage-difference across the terminals sags a bit, as shown in figure 5.
In this case, the electrolyte-filled gaps are no longer field-free. The electrolyte carries a current, but it is not a current of electrons, but rather a current of drifting ions. (There may also be some leakage of electrons, but this is an unimportant nuisance effect.) Ion-pairs are pulled apart by the field. Positive ions drift left, and negative ions drift right. The positive ions are neutralized by electrons that come out of the Ni electrode; the negative ions are neutralized by donating an electron to the Fe electrode. The result is a steady flow of negative electrical current from left to right, or to say it the other way, a steady flow of conventional positive current from right to left.
It is important that electrons (and not ions) can move through the aluminum wire, while in contrast ions (and not free electrons) can move through the electrolyte.
Consider cell (B). An electron will not naturally flow left-to-right from inside the Ni electrode to the left of point (B) to inside the Fe electrode to the right of that point. An electron that magically appeared in the gap would flow the correct way, left-to-right, but when you take the work functions into account, it would be energetically unfavorable for an electron to hop out of the Ni into the gap. When the battery is in operation, we are not asking any electrons to flow like that. Net negative charge yes, electrons no.
But you may well ask, doesn’t it come to the same thing? Well, not quite, because the electron that comes out of the Ni electrode to neutralize the positive electrolyte ion gets help from the energy of the chemical reaction that takes place on the electrode. The physics goes like this:
The reaction at the Fe electrode does not undo that energy gain, because it is a different reaction. Different chemical species are involved. We say that there are two half-cell reactions. In this case there is a half-cell reaction between the positive ions and the Ni electrode, and a different half-cell reaction between the negative ions and the Fe electrode.
So we see that the energy of the battery comes from the chemical reactions occurring at the surfaces of the electrodes. That’s not a surprise.
For each unit charge that flows through a cell, one unit of chemical reaction takes place at each plate, at the plate/electrolyte boundary. (We are assuming that the electrolyte was 100% ionized to begin with. In accordance with convention, we don’t count the ion-forming reaction.) Without these reactions, the cell would just charge up like a capacitor and the battery would not be effective at maintaining its rated voltage when placed under load. 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.)
Section 1.5 mentioned that the energy of the chemical reaction "helped" the electron overcome the work function. We don’t expect the half-cell reaction to have exactly the right energy to match a particular work function, which is why the potential in the gap in figure 4. has a nontrivial relationship to the potential in the adjacent metal. Let’s see how this relationship arises.
Start with the picture in figure 4. where there is no field in the gaps. Then let a current flow through the battery. The gap will charge up like a capacitor. There will develop a field in the gap, roughly as shown in figure 5. The field will cause ions to drift. Positive ions will congregate near the Ni electrode. So far, none of them have been reacted with the electrode, because they haven’t been able to get the necessary electron. The drift will get rid of the field in most of the gap, but there will be a tremendous field right near the electrode, where all the ions are congregating. (The average field in the gap will be unaffected by the drift; it will just be the "capacitor" voltage divided by the gap distance.)
Before long, the concentrated field near the plate will become so strong that the chemical reaction plus this field will provide enough energy to pull an electron up and over (or through) the work function barrier. One unit of chemical reaction will take place.
During discharge, the reaction that takes place at the nickel electrode is:
| (1) |
The net energy that an electron gains in this process is called the half-cell potential.
Meanwhile, at the other electrode, another half-cell reaction is taking place.
| (2) |
The overall reaction in the cell consists of two copies of equation 1 and one copy of equation 2, for a total of:
| (3) |
A large table of half-cell reactions and the associated half-cell potentials can be found in reference 2.
Many people learn about “chemistry” in one class and learn about “electricity” in another class. That’s unfortunate, to the extent that the connections between the two subjects are not clearly explained.
What’s worse is that many chemistry texts introduce electrochemistry in terms of the “oxidation number” scheme, which is – to put it politely – unnecessarily complicated, unclear, and unreliable. I have never seen any application of the “oxidation number” scheme that could not be handled more conveniently by other methods, as we now discuss.
By way of background: The process of balancing chemical reactions, – i.e. the topic of stoichiometry – can be considered nothing more or less than the application of certain conservation laws. If you are considering 92 different chemical elements, there are 92 different conservation laws, since in chemical reactions each element is separately conserved.3
If you understand stoichiometry, you can understand oxidation-reduction reactions with no additional conceptual effort. You just need to add a 93rd conservation law, namely conservation of charge. Then to write a balanced electrochemical equation, you just need to balance it with respect to atoms and with respect to charge.
Consider the comparison:
| Consider a non-electrochemical reaction, such as the simple reaction of carbon with oxygen to form carbon dioxide. It doesn’t make sense to talk about this reaction except in terms of the balanced reaction equation: C + O2 → CO2. You must account for all the atoms. | The same applies to electrochemical equations, i.e. oxidation-reduction reactions. You must account for all the atoms and account for all the charge. |
| Even if you consider side-reactions, such as the possibility of reacting carbon with oxygen to form carbon monoxide, it doesn’t make sense to talk about the side-reaction except in terms of the balanced reaction equation, C + 0.5 O2 → CO. | In each and every reaction, including side reactions, you must account for all the atoms and account for all the charge. |
For details (including examples) of how to balance reaction equations with respect to atoms and charge, see reference 3. Tangential remark: The idea that reactions must be balanced with respect to atoms and with respect to charge is useful in many contexts, not limited to batteries. For example, it allows you to understand why you can do things with aqua regia that you can’t do with nitric acid or hydrochloric acid separately.
While we’re on the subject: Reactions must be balanced with respect to energy, not just atoms and charge. This three-way linkage of chemistry, charge, and energy is what makes batteries possible. That’s because ultimately, the energy of the battery is the energy of chemical reactions.
In a battery under load, i.e. when it is functioning as a battery, the electric field in almost all of the electrolyte compartment is in the “forward” direction, i.e. the direction that causes any mobile charges to drift in the direction of overall current flow in the circuit. This can be seen in figure 5.
By Kirchhoff’s law, we know that the total voltage drop around the circuit is zero. All of the reverse voltage drops occur within very thin dipole layers. This is also where all of the interesting chemistry takes place. There are tremendously high electrical fields in these regions, and high chemical concentration gradients. Analyzing the details of what goes on in these regions is not easy.
According to a modern microscopic understanding of metals, the relevant electrons (i.e. the ones that give the metal its metallic properties) all have a tremendous amount of kinetic energy. They are zooming around like crazy. If no net current is flowing, it’s because an equal number are zooming in all directions, so the electron wavefunction is a standing wave. If a current is flowing, it’s because of a slight increase in the number zooming to the right and a slight decrease in the number zooming to the left, with no significant change in the kinetic energy.
The amount of such kinetic energy depends on the lattice-spacing in the metal and on the number of electrons, as discussed in section 1.1. This is a big contribution to the electrochemical potential, as discussed in section 1.2.
You may be wondering, which is it: kinetic or potential? If it’s called a potential, how can it be kinetic? The answer is that the defining property of a potential is that it has a value that depends on position, independent of how you got there. The relevant electrons in a metal do in fact have this property: they must have a huge kinetic energy, otherwise they could not exist inside the metal. This energy is independent of how the electrons got there, so it is not wrong to call it a potential. It acts like a potential. In fact, people have been calling it a potential for over a hundred years, since long before anyone could explain it microscopically, in terms of kinetic energy and the exclusion principle.
The full electrochemical potential includes to contributions. In addition to the "chemical" contribution just discussed, there is the plain old electrostatic potential. If you put a net charge on a piece of metal, you change its potential, according to the usual notions of capacitance.
Note that 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. In normal operation, if an electron flows out one terminal, an electron flows in the other terminal at the very same instant.
Also, a related point: 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 some people 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
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!
In this document, we have discussed several examples of battery chemistry, not including lead-acid batteries. They are obviously very important in practice. They are not, however, easy to understand. See reference 4.
There are some things in physics that can be explained just fine by treating atoms as classical particles. The ideal gas law PV = NRT is often cited as an example. But there are other things that seem just as elementary but cannot be explained in terms of 19th-century physics. The Gibbs "paradox" is an example. It makes sense if you know how quantum mechanics deals with identical particles, and it is severely paradoxical otherwise.
Similarly, there are some parts of chemistry that can be described just fine using ball-and-stick models of atoms and bonds that snap together to make molecules. But there are other things that cannot be understood this way. For starters, if you insist on a strictly 19th-century analysis, atoms are unstable. The electron will spiral in towards the nucleus and the atom will shrink away to nothing. If you insist on a purely classical analysis, there will be no atoms, no molecules, no metals, no work functions, no batteries, and no people around to complain about it.
You need quantum mechanics if you want atoms to even exist. You need more quantum mechanics if you want the various chemical elements to not behave all the same. Without the exclusion principle, helium would behave just like a heavy isotope of hydrogen. And lithium would be the same, only even heavier. And oxygen would be the same again, only heavier still.
If you want to understand where work functions come from and/or where battery voltages come from, you will have to accept the fact that electrons stick to some atoms more than others. You can accept this as an observed fact without explanation, or you can seek an explanation in terms of quantum mechanics.
We live in a world governed by the laws of quantum mechanics. Get used to it. Sometimes there is a ball-and-stick model that provides a passable approximation to the real (quantum) physics; sometimes there isn’t.
