The pH Scale

Clinical medicine depends on pH constantly: arterial blood gas analysis reports pH alongside $P\text{CO}_2$ and bicarbonate to diagnose acidosis or alkalosis, and a pH swing of even 0.1–0.2 units outside the 7.35–7.45 range signals a medical emergency because so many plasma proteins and enzymes have activity profiles that peak sharply near physiological pH. Gastric pH (typically 1.5–3.5, driven by the H⁺/K⁺-ATPase of parietal cells) activates pepsinogen to pepsin and is the pharmacological target of proton pump inhibitors like omeprazole, which block acid secretion to treat ulcers and reflux. Lysosomes maintain an internal pH near 4.5–5.0, maintained by V-type ATPases, so that their hydrolytic enzymes — optimized for that acidic environment — remain nearly inert if they leak into the neutral cytosol, a safety feature exploited and subverted by various pathogens and by lysosomal storage disease pathology. Industrial biotechnology relies on pH control bioreactors to keep fermentation organisms and recombinant protein-expressing cell lines in their productive pH window; agriculture uses soil pH measurements (often 4.5–8.5 depending on region) to determine nutrient availability, since phosphate and micronutrient solubility are steep functions of soil pH. Modern pH meters use glass electrodes whose membrane potential obeys the Nernst equation, converting a measured voltage directly into a pH reading through the same logarithmic relationship derived here — meaning every benchtop pH measurement in a lab is, mechanistically, an electrochemical instantiation of the algebra in this article. Finally, ocean acidification research tracks a global pH drop of roughly 0.1 units since preindustrial times as dissolved CO₂ shifts the carbonate equilibrium, a shift that — per the logarithmic definition — corresponds to roughly a 30% increase in oceanic hydrogen ion concentration, threatening calcifying marine organisms.

:::quiz[Pure water at 25 °C has pH = 7.00. What is $[\text{OH}^-]$ in this water?]

• 1.0×10⁻⁷ M ✓
• 1.0×10⁻¹⁴ M
• 7.0 M
• 1.0×10⁷ M

:::quiz[A solution has pH = 5.70. What is its approximate $[\text{H}^+]$?]

• 5.70×10⁻⁷ M
• 2.0×10⁻⁶ M ✓
• 7.0×10⁻⁶ M
• 5.0×10⁻¹ M

:::faq[Why is pH defined with a negative sign instead of just using $\log[\text{H}^+]$ directly?] Hydrogen ion concentrations in aqueous solution are almost always less than 1 M, meaning $\log_{10}[\text{H}^+]$ is a negative number (since the logarithm of a value less than 1 is negative). Sørensen's negative sign convention converts these negative logarithms into positive, easy-to-read numbers: pure water's $[\text{H}^+]=10^{-7}$ M becomes the tidy positive value pH 7 rather than the equivalent but less intuitive $\log[\text{H}^+] = -7$. The negative sign is purely a bookkeeping convenience — all the physical content is already present in the raw logarithm — but it has become so universal that "pH" and "the negative log of hydrogen ion activity" are treated as synonymous.

Ask the tutor: How does the pH scale connect to the Henderson–Hasselbalch equation used for buffers?

Water self-ionizes in a proton-transfer equilibrium best written as two water molecules exchanging a proton:

which is conventionally abbreviated $\text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^-$, with the understanding that the "free proton" is always solvated as hydronium and a chain of hydrogen-bonded waters (the Grotthuss mechanism lets protons hop from molecule to molecule far faster than any single ion could physically diffuse). At equilibrium, this process is characterized by the ion product $K_w = [\text{H}^+][\text{OH}^-]$, a quantity already established as approximately $1.0\times10^{-14}$ at 25 °C. Pure water at this temperature therefore contains equal, tiny concentrations of $\text{H}^+$ and $\text{OH}^-$: each is $\sqrt{K_w} = 1.0\times10^{-7}\ \text{M}$. The problem this creates is one of scale. Biological hydrogen ion concentrations range from about $0.03\ \text{M}$ in gastric juice down to about $10^{-8}\ \text{M}$ or lower in mildly alkaline pancreatic fluid — six orders of magnitude. Writing these numbers directly, comparing them, and plotting them on ordinary graph paper is unwieldy, and — more importantly — it obscures the physically relevant fact that equilibrium constants and Gibbs energies depend on concentration through a logarithm, not linearly. A change in $[\text{H}^+]$ by a factor of 10 corresponds to a constant increment of Gibbs energy for any proton-linked equilibrium, exactly analogous to how a tenfold change in equilibrium constant corresponds to a constant $-RT\ln 10$ increment in $\Delta G^{\circ}$. The Danish chemist Søren Sørensen recognized this in 1909 while studying enzyme activity in brewing and proposed the logarithmic index "pH" (from the French pouvoir hydrogène, "hydrogen power," or possibly the Latin pondus hydrogenii), defined precisely so that a linear pH axis corresponds to an exponential concentration axis. Formally,

and, symmetrically, since $\text{OH}^-$ is the other product of water's autoionization,

Both are dimensionless numbers (logarithms of a numerical value, with concentration implicitly expressed in mol/L, technically as activity relative to the standard state of 1 M). Because $[\text{H}^+]$ and $[\text{OH}^-]$ are coupled through $K_w$, pH and pOH are coupled too — and that coupling is the central quantitative relationship of this article, derived explicitly below.

where: $[\text{H}^+]$ and $[\text{OH}^-]$ are molar concentrations (strictly, activities) of hydrogen and hydroxide ion, and $K_w$ is the temperature-dependent ion product of water, $\approx 1.0\times10^{-14}\ \text{M}^2$ at 25 °C. Reading: this is the equilibrium constant for water's autoionization, and it is the single empirical fact from which everything else in this article follows.

where: the logarithm is base 10 by convention (chosen for human readability, not any deep physical necessity — natural log would work with a different constant). Reading: each unit drop in pH represents a tenfold increase in $[\text{H}^+]$; each unit rise represents a tenfold decrease.

where: $\text{p}K_w \equiv -\log_{10}K_w$, equal to 14.00 at 25 °C but shifting with temperature (13.995 at 25 °C precisely, 13.26 at 37 °C, roughly 14.94 at 0 °C). Reading: pH and pOH are not independent; fixing one at a given temperature fixes the other.

where: neutrality is defined as $[\text{H}^+] = [\text{OH}^-]$, not as pH = 7 — the latter is only true at 25 °C where $\text{p}K_w = 14.00$. Reading: at 37 °C (body temperature), neutral pH is about 6.63, not 7.00, because $K_w$ is larger at higher temperature (autoionization is endothermic).

where: this is simply the inverse of the defining logarithm. Reading: to recover concentration from a measured pH, raise 10 to the negative pH power — the step every student must perform correctly and the step most prone to sign errors.

Example 1 (foundational) — Computing pH from a stated hydrogen ion concentration. A dilute solution of hydrochloric acid has $[\text{H}^+] = 4.5\times10^{-3}\ \text{M}$ (an asymmetric, non-round value, deliberately not $1\times10^{-3}$). Find the pH and pOH at 25 °C. Apply the defining formula:

Split the logarithm using $\log(a\times10^n) = \log a + n$:

Since $\log_{10}(4.5) \approx 0.653$:

Therefore:

Now find pOH using the invariant derived above (valid at 25 °C):

Check by computing $[\text{OH}^-]$ directly: $[\text{OH}^-] = K_w/[\text{H}^+] = (1.0\times10^{-14})/(4.5\times10^{-3}) = 2.22\times10^{-12}\ \text{M}$, and $-\log_{10}(2.22\times10^{-12}) = 11.65$ — consistent to the precision shown. This solution is strongly acidic, roughly comparable to gastric juice diluted somewhat, sitting well below neutral pH 7.

Example 2 (applied) — Blood plasma alkalosis threshold. Human arterial blood plasma is normally buffered to pH 7.40 by the bicarbonate system (a preview of Buffers Resist pH Change). A patient hyperventilating blows off CO₂, raising blood pH to 7.52 (respiratory alkalosis, clinically dangerous above ~7.45–7.50). By what factor has $[\text{H}^+]$ changed, and what is the new $[\text{H}^+]$ in nanomolar units? First find the baseline hydrogen ion concentration:

Write $10^{-7.40} = 10^{-8}\times10^{0.60}$. Since $10^{0.60} \approx 3.98$:

Now the alkalotic value:

Since $10^{0.48} \approx 3.02$:

The ratio of the two concentrations:

So a pH rise of only 0.12 units corresponds to a roughly 32% drop in free hydrogen ion concentration — clinically enormous for enzyme active sites and hemoglobin oxygen affinity (the Bohr effect), even though the pH numbers themselves look like a small change. This is the pedagogical point of logarithmic compression: a seemingly modest pH shift of a tenth of a unit is a large fractional change in the actual reactive species concentration, exactly the reason clinicians treat pH deviations of ±0.05 as physiologically serious.

Example 3 (exam/competition level) — Mixing two strong-acid solutions and finding resultant pH. 100.0 mL of a strong acid solution at pH 2.00 is mixed with 250.0 mL of a different strong acid solution at pH 3.30. Assuming volumes are additive and both acids dissociate completely (strong acids), find the pH of the resulting mixture at 25 °C. This problem requires converting pH to concentration, applying conservation of moles (not of pH — pH values cannot simply be averaged, only concentrations and mole amounts obey linear conservation), and converting back.

Step 1. Convert each stock to $[\text{H}^+]$:

(using $10^{0.70}\approx 5.01$, from $\log_{10}5 = 0.699$).

Step 2. Compute moles of H⁺ in each aliquot:

Step 3. Total moles and total volume:

Step 4. New concentration and pH:

Note the resulting pH (2.49) is much closer to the more concentrated stock's pH (2.00) than a naive volume-weighted average of the pH numbers would give — $(2.00\times100 + 3.30\times250)/350 \approx 2.93$, which is wrong and differs from the correct answer by nearly half a pH unit. This is the central trap of pH arithmetic: pH values are not additive or averageable; only concentrations (or moles) are. The correct procedure always detours through $[\text{H}^+]$.