Superacids: Definition and Hammett Acidity

Superacid chemistry is not a museum piece; it underlies real industrial and analytical processes. Solid superacids (sulfated zirconia, sulfated titania, and related sulfate-promoted metal oxides, which mimic the local $H_0$ environment of liquid superacids on a surface) are used industrially as heterogeneous catalysts for alkane isomerization and alkylation, replacing older liquid HF- or $\text{H}_2\text{SO}_4$-based processes in some refinery contexts because solids are easier to separate, recycle, and contain safely. Superacid-mediated alkylation and isomerization chemistry underlies the production of high-octane branched alkanes for gasoline blending, a direct descendant of the same Brønsted–Lewis conjugate-acid chemistry described here. In fundamental and preparative inorganic chemistry, fluoroantimonic-acid-type media remain the tool of choice for generating and stabilizing exotic cations that are too reactive to exist in any conventional solvent: protonated noble-gas compounds, halogen and interhalogen cations, and a wide range of main-group Lewis-acid-stabilized cations used to probe the limits of chemical bonding. Superacid systems are also central to modern mechanistic organic chemistry, where stable-ion NMR studies in $\text{SbF}_5$-based media continue to test computational predictions of nonclassical carbocation structures (bridged, "non-classical" cations analogous to the norbornyl cation problem) against experiment. The conceptual machinery built here — a differentiating acidity scale detached from water, and a Lewis-acid/Brønsted-acid synergy that both strengthens the acid and shrinks the basicity of its conjugate base — recurs directly in the chemistry of anhydrous sulfuric acid and of halogen fluorides such as $\text{BrF}_3$, both of which are self-ionizing, highly differentiating, non-aqueous solvent systems where the same Hammett-style logic and the same "weakly coordinating anion" strategy reappear in new structural clothing.

Ask the tutor: How does the Hammett acidity function get extended and calibrated when no single indicator's protonation ratio is measurable across the whole range from water to fluoroantimonic acid?

Ordinary acid strength in water is bounded above by the strength of $\text{H}_3\text{O}^+$: any acid HA with $\text{p}K_a$ more negative than about $-1.7$ transfers its proton to water essentially quantitatively, and the solution's acidity is capped at whatever $[\text{H}_3\text{O}^+]$ the concentration allows. This is the leveling effect, and it means pH — defined via water activity and hydronium concentration — cannot distinguish HClO$_4$ from HI in water: both look "100% dissociated." To go further, chemists move to a non-leveling, non-aqueous solvent (concentrated $\text{H}_2\text{SO}_4$, $\text{HF}$, $\text{FSO}_3\text{H}$) and define acidity through the equilibrium behavior of a dissolved neutral indicator base B that protonates to BH$^+$:

Because the indicator's protonation state can be read spectrophotometrically (B and BH$^+$ typically have very different UV-visible spectra, usually a color change), and because a chain of overlapping indicators with known aqueous $\text{p}K_{\text{BH}^+}$ values can be used to "walk" the scale from water into the superacid regime, the acidity of the medium can be quantified far beyond where pH electrodes or water-based indicators would work at all. This walk is exactly how Louis Hammett built the $H_0$ function in the 1930s for concentrated sulfuric acid solutions, and it was later extended by Ronald Gillespie and George Olah into the superacid regime. Structurally, the "magic" of magic acid and fluoroantimonic acid is a Lewis-acid/Brønsted-acid handshake. $\text{SbF}_5$ is a strong fluoride-ion acceptor because antimony(V) is a small, highly electrophilic $d^0$ center that becomes octahedral (or forms fluorine-bridged oligomers) upon accepting F$^-$, releasing considerable lattice/coordination stabilization. When $\text{SbF}_5$ is dissolved in HF or FSO$_3$H, it abstracts a fluoride (or fluorosulfate) from the solvent's conjugate base pathway, producing large, delocalized, weakly basic anions such as $\text{SbF}_6^-$, $\text{Sb}_2\text{F}_{11}^-$, or $\text{Sb}_2\text{F}_{10}(\text{SO}_3\text{F})^-$. The proton left behind in solution is now paired with an anion that has essentially no desire to take it back — hence "superacid."

The picture that emerges is that superacidity is not primarily about a stronger O–H or F–H bond being easier to break; it is about what happens to the fragment left behind. Removing basicity from the conjugate base is the lever, and $\text{SbF}_5$'s exceptional fluoride-ion affinity — among the highest of any simple molecule — is what pulls that lever hardest.

The central quantitative object is the Hammett acidity function, defined for a neutral indicator base B that protonates according to $\text{B} + \text{H}^+ \rightleftharpoons \text{BH}^+$:

where:

• $\mathrm{p}K_{\mathrm{BH^+}}$ is the (aqueous-referenced) acid dissociation constant of the protonated indicator, $-\log K_{\mathrm{BH^+}}$, determined in dilute aqueous acid where normal pH applies,
• $[\text{BH}^+]/[\text{B}]$ is the ratio of protonated to neutral indicator measured spectrophotometrically in the medium of interest,
• $H_0$ reduces to ordinary pH in dilute aqueous solution, since there $[\text{BH}^+]/[\text{B}] \to K_{\mathrm{BH^+}}[\text{H}^+]/\gamma$-type ideal behavior and the activity coefficient terms cancel to leading order.

Reading it: $H_0$ measures the proton-donating power per unit indicator conversion, referenced back to water through the indicator's known $\mathrm{p}K_{\mathrm{BH^+}}$. A more negative $H_0$ means the medium can push a much weaker base (more electron-poor B, more negative intrinsic $\mathrm{p}K_{\mathrm{BH^+}}$) over to the protonated form. Because indicators overlap in usable range, a "ladder" of nitroanilines and related weak bases lets $H_0$ be extended continuously from water's mild acids down to $-25$ or beyond. The underlying thermodynamic definition, from which the operational formula is derived, is

where:

• $a_{\mathrm{H^+}}$ is the (formally undefined-in-isolation, but thermodynamically consistent) proton activity in the medium,
• $\gamma_{\mathrm{B}}$ and $\gamma_{\mathrm{BH^+}}$ are activity coefficients of the neutral base and its conjugate acid,
• the ratio of activity coefficients is assumed approximately constant across a family of structurally similar indicators (the "Hammett postulate"), which is what allows different indicators to be strung together onto one continuous scale.

Reading it: $H_0$ is explicitly not $-\log[\text{H}^+]$; it folds in the non-ideality of a concentrated, often highly structured medium (ion pairing, hydrogen-bonded networks, extensive solvation change) that makes naive concentration-based pH meaningless in these solvents. The second governing relationship connects Lewis acidity of the activator to the acidity gain, via fluoride-ion affinity (FIA), the enthalpy of the gas-phase reaction $\text{MF}_n + \text{F}^- \to \text{MF}_{n+1}^-$:

where:

• larger positive $\Delta H_{\mathrm{FIA}}$ means the Lewis acid MF$_n$ binds F$^-$ more strongly (a more negative reaction enthalpy, reported as a positive affinity by convention),
• $\text{SbF}_5$ has one of the largest known fluoride-ion affinities among simple molecular fluorides (of order 100 kJ mol$^{-1}$ larger than typical group 13/14 fluorides like BF$_3$ or PF$_5$),
• this affinity is the thermodynamic driving force that pulls F$^-$ away from HF or FSO$_3$H and traps it as $\text{SbF}_6^-$ or $\text{Sb}_2\text{F}_{11}^-$.

Reading it: whichever Lewis acid has the largest FIA will, all else equal, generate the strongest conjugate superacid when paired with a given Brønsted acid, because it most effectively suppresses the basicity of the leaving conjugate-base fragment.

Example 1 (foundational): computing H0 from an indicator ratio. Suppose an indicator base B (a substituted nitroaniline) has aqueous $\mathrm{p}K_{\mathrm{BH^+}} = -0.94$ (a typical Hammett indicator value), and in a given sulfuric acid mixture the spectrophotometrically measured ratio is $[\mathrm{BH^+}]/[\mathrm{B}] = 6.3$. Then

This corresponds to roughly 20–25 wt% aqueous sulfuric acid, well within the range where the Hammett scale was originally calibrated against real solution composition data — far from superacid territory ($H_0 < -12$), illustrating that most of the useful "acid strength differentiation" work happens long before superacidity is reached. As a limiting-case check: if the ratio were exactly 1 (equal populations of B and BH$^+$), $H_0 = \mathrm{p}K_{\mathrm{BH^+}}$ exactly, which is the defining calibration point for any indicator — the acidity at which that particular indicator is half-protonated.

Example 2 (applied): estimating the acid strength needed to protonate a very weak base, and comparing HF to HF·SbF5. Suppose we want to protonate a very weakly basic ketone-type substrate with intrinsic $\mathrm{p}K_{\mathrm{BH^+}} = -8.2$ to the extent of 95% conversion (i.e. $[\mathrm{BH^+}]/[\mathrm{B}] = 19$). The required medium acidity is

Pure water obviously cannot reach $H_0 = -9.48$ (it levels out near pH 0–1 at best for concentrated strong acid, since $\text{H}_3\text{O}^+$ is the strongest acid species present). Concentrated sulfuric acid, with $H_0 \approx -12$ at 100% concentration, comfortably clears this threshold, achieving well over 95% conversion. But now consider a substrate with $\mathrm{p}K_{\mathrm{BH^+}} = -21.5$ (representative of extremely weak bases such as simple alkanes being protonated toward carbonium-ion-like transition states, or CH$_4$ itself) requiring the same 95% conversion:

Sulfuric acid ($H_0 = -12$) fails by ten orders of magnitude in the underlying activity ratio. Magic acid, with $H_0$ typically in the $-21$ to $-24$ range depending on SbF$_5$ loading, sits right at the edge, while fluoroantimonic acid, capable of $H_0$ below $-25$, clears the threshold decisively — consistent with the experimental observation that superprotonation of alkanes (giving nonclassical carbonium ions such as protonated methane, $\text{CH}_5^+$) is achieved specifically in $\text{HF}\cdot\text{SbF}_5$ and not in ordinary strong acids.

Example 3 (exam/competition-level): fluoride-ion affinity ranking and predicting relative H0 shift. Given approximate gas-phase fluoride-ion affinities (kJ mol$^{-1}$, illustrative literature-consistent values): $\text{BF}_3$: 350; $\text{PF}_5$: 410; $\text{AsF}_5$: 435; $\text{SbF}_5$: 495. A superacid chemist wants to activate FSO$_3$H with the cheapest Lewis acid that still yields $H_0 < -18$. Using the (empirically observed, roughly linear over this narrow range) trend that each 50 kJ mol$^{-1}$ increase in FIA shifts $H_0$ by approximately $-1.3$ units relative to a BF$_3$-activated baseline of $H_0 \approx -15.0$: For AsF$_5$ ($\Delta\text{FIA}$ relative to BF$_3$ $= 435 - 350 = 85$ kJ mol$^{-1}$):

This falls just short of the $-18$ target. For SbF$_5$ ($\Delta\text{FIA} = 495 - 350 = 145$ kJ mol$^{-1}$):

SbF$_5$ clears the target; AsF$_5$ does not, despite AsF$_5$ being a respectable Lewis acid in its own right — illustrating why SbF$_5$, not AsF$_5$ or PF$_5$, is the industry-standard activator even though it is more expensive and more corrosively reactive. (These proportionality constants are illustrative pedagogical estimates for teaching the scaling trend, not a literature-fitted regression; the qualitative ordering BF$_3$ < PF$_5$ < AsF$_5$ < SbF$_5$ in both FIA and resulting $H_0$ is the experimentally robust result.) As a limiting check: setting FIA equal to the BF$_3$ reference value recovers $\Delta H_0 = 0$ exactly, i.e. $H_0 = -15.0$, consistent by construction with using BF$_3$ as the zero-point of the relative scale.

Olah's work rested on foundations laid by Ronald Gillespie, who characterized the fluorosulfuric acid/antimony pentafluoride and related systems structurally and quantified their Hammett acidities in the 1960s, and by Louis Hammett himself, whose 1930s acidity function work in concentrated sulfuric acid provided the entire measurement framework that superacid chemistry would later extend by two more orders of magnitude in $H_0$.