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Autoionisation of water and pH

T-050Home CU-204Threads equilibrium
Statement

Kw and the definition of the pH scale.

Why it matters

bronsted-lowry establishes proton transfer as the basic currency of acid-base chemistry; applying that framework to water reacting with itself gives the single most important quantitative reference point in all of aqueous acid-base chemistry — the pH scale — which henderson-hasselbalch, titration-curves, and polyprotic-acids all build directly on top of.

Because \(K_w\) links \([\text{H}_3\text{O}^+]\) and \([\text{OH}^-]\) reciprocally in every aqueous solution, not only pure water, it is the single relationship that lets a chemist compute one ion's concentration from the other in any acid, base, salt, or buffer solution, making it the quantitative backbone underlying essentially every calculation elsewhere in this unit.

Hypotheses
Pure water undergoes a genuine, if extremely slight, autoionisation equilibrium.This is a specific Bronsted-Lowry acid-base reaction in which one water molecule acts as proton donor and another as proton acceptor simultaneously; without this self-ionisation, pure water would contain no \(\text{H}_3\text{O}^+\) or \(\text{OH}^-\) at all, and the entire pH scale would have no natural reference point. \(K_w\) is treated as a true equilibrium constant, with pure water's own concentration folded into the constant, and its value, \(1.0\times10^{-14}\) at \(25^\circ\text{C}\), is specific to that one fixed temperature.This mirrors the general convention for any pure liquid or solid appearing in a heterogeneous or solvent equilibrium; using the \(25^\circ\text{C}\) numerical value at a different temperature would give an incorrect result. \(K_w\) is itself temperature-dependent (autoionisation is endothermic, so \(K_w\) increases with rising temperature), meaning "neutral" pH is exactly 7 only at \(25^\circ\text{C}\); at other temperatures, neutral water still has \([\text{H}^+]=[\text{OH}^-]\), but not at pH 7 exactly.
Proof
1
2\text{H}_2\text{O}(l) \rightleftharpoons \text{H}_3\text{O}^+(aq)+\text{OH}^-(aq)
Water's autoionisation is a specific Bronsted-Lowry acid-base reaction in which water plays both the proton-donor and proton-acceptor roles simultaneously. A
2
K_w=[\text{H}_3\text{O}^+][\text{OH}^-]
Pure liquid water's own concentration is omitted from the expression by the same convention used for any pure liquid or solid in a heterogeneous equilibrium constant (Hypotheses), leaving \(K_w\) defined purely in terms of the two dissolved ionic species. A
3
K_w=1.0\times10^{-14}\ (25^\circ\text{C}) \Rightarrow [\text{H}_3\text{O}^+]=[\text{OH}^-]=\sqrt{K_w}=1.0\times10^{-7}\,\text{M}
In pure water, stoichiometry (Step 1) requires equal concentrations of the two ions; substituting the measured value of \(K_w\) at \(25^\circ\text{C}\) and taking the square root gives this well-known reference concentration exactly at that one temperature. A
4
\text{pH}=-\log_{10}[\text{H}_3\text{O}^+], \quad \text{pOH}=-\log_{10}[\text{OH}^-] \Rightarrow \text{pH}+\text{pOH}=\text{p}K_w=14.00
Taking \(-\log_{10}\) of both sides of Step 2's \(K_w\) expression gives this fixed relationship, holding for any aqueous solution at \(25^\circ\text{C}\), not only for pure water. A
5
\text{Since } K_w \text{ is fixed, adding any acid or base shifts } [\text{H}_3\text{O}^+] \text{ and } [\text{OH}^-] \text{ in exactly compensating directions.}
Because \(K_w\) is fixed at a given temperature regardless of what else is dissolved, their product must always remain equal to \(K_w\) — the single relationship letting pH be computed from either ion's concentration interchangeably in any aqueous solution. A
Result
K_w=[\text{H}_3\text{O}^+][\text{OH}^-]=1.0\times10^{-14}\ (25^\circ\text{C}), \qquad \text{pH}+\text{pOH}=14.00

Reading. A single, temperature-fixed equilibrium constant for water's self-ionisation defines the entire pH scale and links \([\text{H}^+]\) and \([\text{OH}^-]\) reciprocally in every aqueous solution, not merely pure water.

Scope. The numerical value \(14.00\) for \(\text{p}K_w\), and hence "neutral pH = 7," holds specifically at \(25^\circ\text{C}\) (Hypotheses); at other temperatures \(K_w\) itself changes.

Corollaries & converses
  • henderson-hasselbalch's buffer equation and titration-curves' equivalence-point predictions both rest on the same fixed pH scale and \(K_w\) relationship established here.
  • polyprotic-acids' successive dissociation steps are each still governed by the same overall aqueous pH scale, even though each step has its own distinct equilibrium constant.
  • The pH scale's familiar zero-to-fourteen range is a direct, simple consequence of \(K_w\)'s specific numerical value at room temperature, not an independently chosen convention.
  • For any conjugate acid-base pair, \(K_aK_b=K_w\): multiplying the acid dissociation constant of a weak acid by the base hydrolysis constant of its own conjugate base always returns exactly \(K_w\), a direct consequence of adding the two corresponding proton-transfer equilibria together and recognising that water itself cancels — a relationship used repeatedly in henderson-hasselbalch and polyprotic-acids to move between \(K_a\) and \(K_b\) for the same species.
Fails without
  • Ignore water's own autoionisation in an extremely dilute strong acid or base solution (Step 5): at concentrations comparable to or below \(10^{-7}\,\text{M}\), water's own contribution to \([\text{H}_3\text{O}^+]\) or \([\text{OH}^-]\) is no longer negligible, and a calculation that omits it can predict a physically impossible pH on the wrong side of 7 for a genuine acid or base.
  • Apply the \(25^\circ\text{C}\) value of \(K_w\) at a different temperature (Hypotheses' \(t3\)): since autoionisation is endothermic and \(K_w\) rises with temperature, using \(1.0\times10^{-14}\) at, say, body temperature gives an incorrect neutral-pH reference point and mis-locates where "neutral" actually falls on the pH scale at that temperature.
Common errors
  • Treating pH = 7 as the universal definition of "neutral," rather than recognising that neutral specifically means \([\text{H}_3\text{O}^+]=[\text{OH}^-]\), which only corresponds to pH 7.00 at exactly \(25^\circ\text{C}\).
  • Omitting water's own autoionisation contribution when computing the pH of an extremely dilute strong acid or base solution (Fails without).
  • Forgetting that \(K_w\), like any equilibrium constant, is specific to a stated temperature, and applying the \(25^\circ\text{C}\) value unthinkingly at a different temperature.
  • Confusing pH and pOH, or forgetting the negative-logarithm sign convention, so that a smaller \([\text{H}_3\text{O}^+]\) is mistakenly thought to give a smaller, rather than a larger, pH.
Discussion

The modern pH scale, and the underlying negative-logarithm convention, was introduced by the Danish chemist Søren Sørensen in 1909, originally in the context of quality-control work in brewing, where precise, convenient tracking of acidity was of direct practical importance. The scale's logarithmic character reflects the enormous range of \([\text{H}_3\text{O}^+]\) values encountered in real aqueous chemistry, which a simple linear concentration scale would represent far less conveniently.

Because \(K_w\) rises with temperature (Hypotheses), physiological fluids at human body temperature (\(\approx37^\circ\text{C}\)) have a neutral pH measurably below 7.00, even though they remain, by the physically meaningful definition \([\text{H}_3\text{O}^+]=[\text{OH}^-]\), genuinely neutral; blood itself is normally maintained slightly basic relative to this warmer neutral point, not relative to the familiar \(25^\circ\text{C}\) value of 7.00, a frequent source of confusion when comparing physiological pH values against the room-temperature reference scale used throughout the rest of this page.

Common misconception: that pH values are always confined to the familiar 0-14 range. Since pH is simply defined as \(-\log_{10}[\text{H}_3\text{O}^+]\), a sufficiently concentrated strong acid or base solution can genuinely give a pH below 0 or above 14; the 0-14 range is only the range spanned by dilute solutions whose ion concentrations do not exceed roughly 1 M, not a hard mathematical bound on the scale itself.

Worked examples
1
0.0025\,\text{M HCl (strong acid, fully dissociated)}: [\text{H}_3\text{O}^+]=2.5\times10^{-3}\,\text{M}
Since this concentration is far above water's own autoionisation contribution of \(\sim10^{-7}\,\text{M}\), that contribution is safely neglected; \(\text{pH}=-\log(2.5\times10^{-3})\approx2.60\). A
2
\text{pOH}=14.00-2.60=11.40 \Rightarrow [\text{OH}^-]=10^{-11.40}\approx3.98\times10^{-12}\,\text{M}
Applying Step 4's fixed relationship converts the computed pH directly into pOH, and hence into the corresponding, extremely small hydroxide concentration, illustrating the reciprocal link between the two ions in this strongly acidic solution. A
\text{pH}\approx2.60, \quad \text{pOH}\approx11.40, \quad [\text{OH}^-]\approx3.98\times10^{-12}\,\text{M}

Reading. A single given concentration of strong acid converts, via two direct applications of the Result, into the full set of pH, pOH, and hydroxide-ion concentration.

Scope. The identical two-step procedure (pH from \([\text{H}_3\text{O}^+]\); pOH and \([\text{OH}^-]\) from \(K_w\)) applies to any strong acid or base solution concentrated enough that water's own autoionisation can be neglected.

Problems
  1. Compute the pH of a \(0.020\,\text{M}\) NaOH solution.
    SolutionNaOH is a strong base, fully dissociated: \([\text{OH}^-]=0.020\,\text{M}\), \(\text{pOH}=-\log(0.020)\approx1.70\). By Step 4, \(\text{pH}=14.00-1.70=12.30\).
  2. A solution has pH \(=9.45\). Find its pOH and \([\text{OH}^-]\).
    Solution\(\text{pOH}=14.00-9.45=4.55\). \([\text{OH}^-]=10^{-4.55}\approx2.82\times10^{-5}\,\text{M}\).
  3. Explain why the pH of a \(1\times10^{-8}\,\text{M}\) HCl solution is not simply \(-\log(1\times10^{-8})=8.00\), and describe qualitatively what must be included instead.
    SolutionA pH of 8.00 would be basic, which is impossible for a solution of a genuine acid, however dilute. At this extremely low concentration, the added HCl's contribution to \([\text{H}_3\text{O}^+]\) is comparable to water's own autoionisation contribution (\(\sim10^{-7}\,\text{M}\)), so that contribution can no longer be neglected (Fails without); a correct treatment must solve for \([\text{H}_3\text{O}^+]\) accounting for both sources simultaneously, which gives a pH slightly below 7, not above it.