Frost (Oxidation-State) Diagrams

Frost diagrams remain a working tool anywhere multivalent redox chemistry needs fast qualitative triage. In analytical and environmental chemistry, they rationalize why manganese and iron cycle the way they do in natural waters and soils — why Mn$^{2+}$ persists in reducing groundwater while MnO$_2$ coatings form in oxygenated surface water, a question that starts exactly as Example 2 did. In industrial chlor-alkali and bleach chemistry, the ClO$^-$/ClO$_3^-$/Cl$^-$ competition worked through in Example 3 underlies the practical distinction between commercial bleach (cold NaOCl) and chlorate manufacture (hot brine electrolysis), and safety guidance on never mixing bleach with acids or heating it traces directly to the same disproportionation landscape. In battery and electrocatalysis research, Frost-style free-energy-versus-oxidation-state reasoning (often generalized into computational "Pourbaix stability diagrams" built from density-functional total energies rather than tabulated $E°$ values) is used to screen candidate electrode and catalyst materials — for instance mapping out which manganese oxide phases are thermodynamically accessible during battery charge/discharge cycling, or which vanadium oxidation states survive in a redox-flow-battery electrolyte. In bioinorganic chemistry, the diagram helps explain why certain metal cofactors (iron in cytochromes, manganese in the oxygen-evolving complex of Photosystem II, copper in blue-copper proteins) are poised at oxidation states that are kinetically persistent yet thermodynamically metastable — precisely the "hump but not the global minimum" situation flagged in the misconception box above. And pedagogically, the Frost diagram is the direct conceptual bridge to Pourbaix diagrams (a predictable typo aside — see Pourbaix Diagrams) and to the broader program of Predicting Redox Behavior from Diagrams, both of which add further axes (pH, or explicit reaction-coordinate detail) to the same underlying free-energy-landscape idea introduced here.

Ask the tutor: Can you walk me through constructing the full Frost diagram for vanadium in acidic solution and using it to predict which vanadium oxidation states would appear if you added zinc metal to a solution of VO₂⁺?

Start with the quantity that actually deserves to be summed: Gibbs energy. For a half-reaction reducing an element from oxidation state $+n$ down to the standard state ($n = 0$, i.e. the free element),

where $n$ is the number of electrons transferred from oxidation state $0$ to reach state $+n$ (equivalently, the electrons gained going from $+n$ to $0$), $F$ is the Faraday constant, and $E°$ is the standard reduction potential for that couple relative to the free element. Because $\Delta G°$ is extensive (additive along a reaction path) while $E°$ is intensive (it does not add across steps with different electron counts), it is $nE°$ — not $E°$ — that behaves like a proper thermodynamic potential energy for the atom in a given oxidation state. Frost's insight was simply this: plot $nE°$ (in volts, a proxy for $\Delta G°/F$, sometimes labeled $N\bar{E}°$ or "relative free energy per mole of element") on the vertical axis, and formal oxidation state $n$ on the horizontal axis, using the free element as the origin $(0,0)$. Every species of the element — however exotic — becomes a single point on this graph. The free element sits at the origin by construction. Species that release a lot of free energy on formation from the element (very stable relative to it) sit low; species that are high in energy relative to the element sit high. Connect the points with straight-line segments in order of oxidation state, and the resulting jagged curve is the free-energy landscape of the element as a function of formal charge on the central atom. The payoff is that the slope of any chord in this diagram is a potential. Consider two points at oxidation states $n_1$ and $n_2$, with vertical coordinates $n_1 E_1°$ and $n_2E_2°$ (each measured relative to the same $n=0$ reference). The reduction potential for converting the higher state into the lower one is

which is exactly rise-over-run on the Frost plot. This is the same formula used to splice together nonadjacent couples on a Latimer diagram — Frost diagrams simply make the calculation visual: draw the chord, read the slope.

Two geometric facts follow immediately and constitute nearly the entire predictive content of the method. First, lower points are more stable: if species A sits below species B (lower $nE°$ at whatever oxidation states), A is thermodynamically favored, and the potential for converting B into A is positive (spontaneous reduction). Second, convexity governs disproportionation: if an intermediate point lies above the straight line joining its two neighbors — i.e., the curve bulges upward, concave-up, at that point — the intermediate species has higher free energy than a mixture of its neighbors in the right proportions, so it will spontaneously disproportionate into them. Conversely, if a point lies below the chord joining its neighbors — the curve bulges downward, concave-down, a genuine "hump" or valley depending on how you look at it — that species is stable relative to disproportionation into its neighbors; indeed the reverse reaction (comproportionation of the neighbors into the intermediate) is favored.

A third useful reading is about slope as reactivity. The steepest downhill segment leaving a given point, going to the right (toward higher oxidation state) or to the left (toward lower oxidation state), identifies respectively the best oxidizing partner or best reducing partner accessible from that species — because the steepest chord corresponds to the largest magnitude, most favorable $E°$.

The defining relationship of the diagram is the free-energy proxy plotted on the vertical axis:

where: $n$ = oxidation state of the species (relative to the element, at $n=0$); $E°(n\to 0)$ = the standard reduction potential of the couple that takes the species all the way down to the free element; $y(n)$ has units of volts but represents $-\Delta G°/F$ referenced to the element.

Reading: every point's height is (up to the constant $F$) minus its standard free energy of formation from the element via electron transfer. It is not simply "$n$ times some tabulated single-step $E°$" unless the Latimer diagram happens to give you the direct $n\to 0$ arrow; more often $y(n)$ must be built up additively from the Latimer chain, step by step, which is exactly the content of the first derivation below. The chord-slope relationship, already stated, is the second core equation:

where: $y(n_A)$, $y(n_B)$ are the Frost-diagram heights of species A and B; $n_A, n_B$ their oxidation states; $E°_{B\to A}$ is the potential for the half-reaction reducing B to A.

Reading: any two points on the diagram, no matter how far apart in oxidation state, define a valid reduction potential — the diagram lets you read off any Latimer-type potential, not just adjacent ones, by simple slope, without redoing the electron-weighted averaging by hand. The disproportionation/comproportionation criterion is the sign of a second-difference (discrete curvature):

for three consecutive states $n_A < n_B < n_C$; equivalently, disproportionation of B is spontaneous exactly when

where: the right-hand side is the height of the chord from A to C evaluated at $n_B$. If $y(n_B)$ (the actual point) exceeds that chord height, B is above the line and disproportionates.

Reading: this is just "is the point above or below the straight line joining its neighbors," stated algebraically; convex-up (point above chord) means unstable to disproportionation, convex-down (point below chord) means stable and prone to the reverse (comproportionation).

Example 1 (foundational) — building the Frost diagram for nitrogen in acid from a short Latimer chain. Take a simplified acidic Latimer fragment for nitrogen restricted to three species: $\text{NO}_3^-$ (N = +5), $\text{HNO}_2$ (N = +3), and $\text{N}_2$ (N = 0), with tabulated adjacent potentials

(these are illustrative simplified values chosen to be non-round on purpose, for a clean asymmetric check). Set $y(0) = 0$ for N$_2$. Step from N$_2$ ($n=0$) to HNO$_2$ ($n=+3$): the arrow's potential of $+1.45$ V corresponds to a 3-electron reduction of HNO$_2$ to $\tfrac12$N$_2$, so

Step from HNO$_2$ ($n=+3$) to NO$_3^-$ ($n=+5$): this is a 2-electron couple with $E° = 0.93$ V for the reduction NO$_3^- \to$ HNO$_2$. Its free-energy contribution adds onto the HNO$_2$ value:

So the three Frost points are $(0,0)$, $(3, 4.35)$, $(5, 6.21)$. Check the chord-slope rule against the given data: the slope from $n=3$ to $n=5$ is

which reproduces the tabulated $E°$ for NO$_3^-\to$HNO$_2$ exactly, as it must (this pair of points was that arrow). Now compute the nonadjacent potential for NO$_3^- \to \tfrac12$N$_2$ directly from the diagram — something the Latimer chain alone would require a separate weighted-average calculation to get:

Limiting check: if the two arrows had happened to carry the same potential (say both equal to some $E$), the three points would be exactly collinear, and the nonadjacent slope would trivially also equal $E$ — confirming that collinearity ⇒ equal slopes ⇒ no hidden curvature, consistent with the general chord formula reducing correctly in that degenerate case.

Example 2 (applied) — is Mn(III) stable in acidic solution, or does it disproportionate? Manganese in acid has (approximate, standard) adjacent Latimer potentials:

with oxidation states $+7, +6, +4, +3, +2, 0$ respectively. Fix $y(0)=0$ for Mn metal and build outward.

(reduction of Mn$^{2+}$ to Mn has $E°=-1.18$ V, i.e. this reduction is unfavorable, so Mn$^{2+}$ sits at negative $y$ — deeply stable relative to the metal, since a very negative reduction potential to the element means Mn$^{2+}$ strongly resists being reduced, i.e. it's a low, stable point).

So the Frost points, in V, are: Mn(0,0), Mn$^{2+}$(2,−2.36), Mn$^{3+}$(3,−0.85), MnO$_2$(4,0.10), MnO$_4^{2-}$(6,4.62), MnO$_4^-$(7,5.18). Now test Mn$^{3+}$ for disproportionation using its neighbors Mn$^{2+}$ (n=2) and MnO$_2$ (n=4). The chord height at $n=3$ (halfway, since $3$ is the midpoint of $2$ and $4$) is

The actual point is $y(3) = -0.85$ V, which is above the chord ($-0.85 > -1.13$), so Mn$^{3+}$ lies above the line joining Mn$^{2+}$ and MnO$_2$ — it is thermodynamically unstable to disproportionation:

with a driving force given by the slope difference, $E°_{\text{disp}} = E°_{3\to2} - E°_{4\to3} = 1.51 - 0.95 = +0.56$ V — comfortably spontaneous. This matches the well-known laboratory fact that Mn$^{3+}$(aq) is not isolable in simple acidic media except when kinetically stabilized (e.g. by complexation with pyrophosphate or fluoride, which lowers its effective potential and changes the local slopes).

Limiting check: notice the huge jump in slope around MnO$_4^{2-}$ (n=6): the chord from MnO$_2$(4,0.10) to MnO$_4^{2-}$(6,4.62) has slope $(4.62-0.10)/2 = 2.26$ V, matching the tabulated arrow exactly (as it must, since these are adjacent Latimer species and the "chord" is that arrow) — a useful internal consistency check whenever adjacent Latimer species are used directly as Frost points.

Example 3 (exam/competition level) — chlorine, comparing hot and cold base disproportionation. Aqueous chlorine chemistry is the textbook case where a Frost diagram earns its keep, because there are genuinely competing disproportionation channels and temperature changes which one wins. Standard basic-solution Latimer potentials (approximate) for chlorine oxidation states 0 ($\text{Cl}_2$), +1 ($\text{ClO}^-$), +3 ($\text{ClO}_2^-$), +5 ($\text{ClO}_3^-$), +7 ($\text{ClO}_4^-$), and −1 ($\text{Cl}^-$):

Build Frost heights with $y(0) = 0$ for Cl$_2$.

Walking the other direction (oxidizing side), each arrow above is written as a reduction potential for the higher state going to the next-lower one, so we accumulate free energy by subtracting as $n$ increases (since forming a higher, less favorable oxidation state costs free energy — algebraically, $y(n_{i+1}) = y(n_i) + (n_{i+1}-n_i)E°_i$ using the reduction arrow exactly as before):

Frost points (base): Cl$^-$(−1,−1.36), Cl$_2$(0,0), ClO$^-$(1,0.42), ClO$_2^-$(3,1.74), ClO$_3^-$(5,2.40), ClO$_4^-$(7,3.12). First test whether Cl$_2$ itself disproportionates in base — this is the famous "cold dilute base gives hypochlorite, hot concentrated base gives chlorate" question. Compare Cl$_2$ (n=0) against the chord joining Cl$^-$ (n=−1) and ClO$^-$ (n=+1):

Actual $y(0) = 0 > -0.47$, so Cl$_2$ lies above the chord: disproportionation to Cl$^-$ and ClO$^-$ is spontaneous,

confirming the cold-base reaction (this is why household bleach, made in cold dilute NaOH, is a Cl$^-$/ClO$^-$ mixture). Now test whether ClO$^-$ (n=+1) itself disproportionates further, comparing it to the chord joining Cl$^-$(−1,−1.36) and ClO$_2^-$(3,1.74):

Actual $y(1) = 0.42 > 0.19$: ClO$^-$ is also above its chord, so it too is thermodynamically unstable and should disproportionate further, to Cl$^-$ and ClO$_2^-$ (or, checking against ClO$_3^-$ directly at n=5, an even more favorable disproportionation, since the Frost curve for chlorine keeps curving the same way out to n=5 before flattening toward ClO$_4^-$). This is exactly the thermodynamic basis for hot concentrated base driving hypochlorite on to chlorate: $3\,\text{ClO}^- \to 2\,\text{Cl}^- + \text{ClO}_3^-$ is thermodynamically favorable at all temperatures; only kinetics (slow at room temperature, fast when heated) determines whether cold-base bleach (kinetically trapped at ClO$^-$) or hot-base chlorate is the isolated product. The Frost diagram tells you the thermodynamic destination; it does not, by itself, tell you the rate — a crucial and frequently tested distinction.

Limiting check: if disproportionation of ClO$^-$ were not favorable, the point at n=1 would need to sit at or below the chord value 0.19 V; since real ClO$^-$ sits substantially above it (0.42 V), the reaction is robustly, not marginally, favorable — consistent with the qualitative fact that hypochlorite bleach solutions genuinely do decompose (slowly at room temperature) toward chlorate and chloride even though they are commercially useful as kinetically persistent products.

The diagram appeared in a short 1951 paper, "Oxidation Potential-Free Energy Diagrams," published while Frost was a professor at Northwestern University. The context was pedagogical necessity as much as research: inorganic chemists working with elements like manganese, chlorine, nitrogen and vanadium needed a fast way to reason about which of a dozen-plus oxidation states would actually persist in solution, without re-deriving nonadjacent potentials from Latimer chains every time. Frost's contribution was not new thermodynamics — every number on the diagram comes straight out of standard $\Delta G° = -nFE°$ relationships already used in constructing Latimer diagrams — but a new representation, exploiting the fact that $nE°$, unlike $E°$ itself, is additive and therefore plottable as a genuine potential-energy surface.