Too many potentials to keep in your head
By now you can take any half-reaction, look up its [[standard-reduction-potential|standard reduction potential]], and rank couples in the [[electrochemical-series|electrochemical series]] to predict who oxidizes whom. That works beautifully for a single couple like Fe3+/Fe2+. But many elements are not so tidy. Manganese alone turns up as Mn(VII) in purple permanganate MnO4-, Mn(VI) in green manganate, Mn(IV) in brown MnO2, Mn(III), Mn(II) as the pale pink aqua ion, and Mn(0) as the metal — six accessible states, and a separate reduction potential connecting almost every neighbouring pair. Listing them as a wall of numbers tells you almost nothing at a glance.
What we want is a compact map of an element's whole redox landscape — one picture that shows every [[oxidation-state|oxidation state]] the element can adopt and how willingly each turns into its neighbours. Two such maps were invented for exactly this. The Latimer diagram is the bookkeeper's version: a horizontal chain of species labelled with the potential linking each adjacent pair. The Frost diagram is the same information replotted so that stability literally becomes a height on a graph. Neither contains new chemistry — they just reorganize the standard potentials you already trust into a form your eye can read.
The Latimer diagram: a chain of potentials
A [[latimer-diagram|Latimer diagram]] writes the species of an element in a row, ordered from the highest oxidation state on the left down to the lowest on the right. Between each adjacent pair you draw an arrow and write the standard reduction potential (in volts) for converting the species on the left into the one on the right. Read it like a sentence: each number is just the voltage of that one reduction step. A large positive potential means that step pulls electrons strongly — the left-hand species is a keen [[oxidizing-and-reducing-agents|oxidizing agent]]; a negative potential means the step is reluctant, so the right-hand species is a good reducing agent and resists being oxidized back.
Manganese in acid solution (1 M H+), potentials in volts:
+7 +6 +4 +3 +2 0
MnO4- -- MnO4^2- -- MnO2 -- Mn3+ -- Mn2+ ------- Mn
+0.56 +2.27 +0.95 +1.51 -1.18
\__________________________________/
+1.51 (MnO4- -> Mn2+, the skip)
Note: numbers above each arrow are E-standard for THAT one-step
reduction (left species + electrons -> right species).The one trap to avoid: potentials are not additive — you cannot just add the voltages along a chain to get the potential of a longer jump. The reason is that potential is energy per electron, and different steps move different numbers of electrons. What is additive is the free energy, which is proportional to n times E (where n is the number of electrons). So to combine steps you weight each potential by its electron count: E for the overall jump equals the sum of (n_i times E_i) divided by the total electrons. For permanganate all the way to Mn2+, that weighted average comes out to +1.51 V over 5 electrons. Get into the habit of multiplying by n before you add, and the arithmetic never bites you.
The Frost diagram: turning potentials into heights
A Latimer diagram is precise but flat — it does not show you, at a glance, which state is the comfortable resting place. The [[frost-diagram|Frost diagram]] (also called an oxidation-state diagram) fixes that by plotting a kind of free energy against oxidation state. Along the horizontal axis you put the oxidation state N, from low on the left to high on the right. On the vertical axis you plot the quantity n times E (often written N times E-standard, in volt-electrons) for forming that species from the element. The element itself, at oxidation state zero, sits at height zero by definition. Because that vertical quantity is proportional to free energy, the lower a species sits on the plot, the more thermodynamically stable it is.
Here is the single most useful fact about the diagram: the slope of the line joining any two points is the reduction potential of that couple. Steep downhill slope to the right means a strongly favourable, high-potential reduction; a line tilting uphill means an unfavourable, negative-potential step. So a Frost diagram is really a Latimer diagram with the potentials turned into gradients you can see. You build one straight from the Latimer data — multiply each species' potential-from-the-element by its oxidation-state count to get its height — but the payoff is that thermodynamics is now geometry. Your eye does the analysis.
Reading stability straight off the curve
The whole point of the Frost diagram is that you can read four things by sight, without doing any arithmetic. The lowest point on the curve is the most stable oxidation state — the one an element tends to settle into. For manganese in acid that lowest point is Mn(II), which is why pink Mn2+ is the end of the line for so many manganese reactions. The point sitting highest is the strongest oxidant relative to its neighbours, because reducing it is steeply downhill.
The cleverest reading is about [[disproportionation|disproportionation]] — a single species splitting into a higher and a lower oxidation state of the same element at once. On a Frost diagram this is pure geometry: a species sitting on a convex bump, poking up above the straight line drawn between its two neighbours, is unstable to disproportionation. Why? Because that line between the neighbours sits lower than the bump, and a species can always lower its total free energy by splitting into a mixture whose average lands on that line. The textbook case is Mn(III): Mn3+ perches above the line joining MnO2 and Mn2+, so it spontaneously disproportionates, 2 Mn3+ + 2 H2O -> MnO2 + Mn2+ + 4 H+. Copper(I) does the same — Cu+ sits on a bump above Cu2+ and Cu metal, which is why Cu+ is unstable in water.
The mirror image is comproportionation, the reverse reaction, where a high and a low state of an element meet and merge into a single middle state. That state is favoured exactly when it sits in a concave dip — below the line joining its neighbours. Comproportionation is why mixing permanganate with Mn2+ in the right conditions can deliver brown MnO2: the two extremes collapse down onto the stable intermediate that lies in the valley. So the rule is symmetric and easy to hold: bumps fall apart, dips pull together.
- Find the lowest point: that oxidation state is the most thermodynamically stable, the place the element likes to end up.
- Look for convex bumps: any species poking up above the straight line between its two neighbours is unstable to disproportionation.
- Look for concave dips: a species lying below the line joining its neighbours is stable, and the two flanking states will tend to comproportionate into it.
- Compare slopes: a steeper downward slope to the right means a more positive reduction potential, so that species is the stronger oxidant; a steep upward slope marks a strong reducing agent at the bottom-left.
What these pictures do not tell you
Be honest about three limits, because each one trips people up. First, both diagrams are purely thermodynamic: they tell you which way a reaction wants to go, never how fast. A species can sit on a wild convex bump — thermodynamically desperate to disproportionate — and yet survive for years because the rearrangement is kinetically blocked. This is the same independence of stability and rate you have met before, and it is why permanganate, though a screaming oxidant on paper, is kinetically sluggish enough to be a usable bench reagent. The diagram tells you the destination, not the travel time.
Second, every potential on these diagrams is a standard potential — measured at unit activity, roughly 1 M concentrations, fixed pH and 25 C. Change the concentrations and the real potential shifts according to the Nernst equation; change the pH and, as we noted, you may need an entirely different diagram. Both pictures are snapshots at standard conditions, not guarantees about your actual beaker. Third, oxidation state itself is a bookkeeping device, not a physical charge — Mn(VII) in permanganate does not really carry a +7 charge; the bonds to oxygen are heavily covalent. The diagrams count electrons by that convention so the arithmetic stays clean, but never mistake the label for a literal charge sitting on the atom.