pH, pOH and the Logarithmic Acid–Base Scale

What pH actually measures

pH is the negative base-10 logarithm of the hydrogen-ion activity in an aqueous solution. Stripped of the jargon, it is a compressed way of expressing how much H⁺ is floating around per litre of water. The activity is close enough to the molar concentration in dilute solutions that almost every textbook, every classroom, and the pH calculator on this site write the definition as pH = −log₁₀[H⁺] with [H⁺] in mol/L. The "p" prefix means "negative log of"; it is the same operator that turns Ka into pKa and Kw into pKw. The point of the prefix is to make a sprawling concentration range — about fourteen orders of magnitude in ordinary water — fit on a tidy 0-to-14 axis.

That compression is why the scale feels counter-intuitive at first. A drop of one pH unit is a tenfold rise in [H⁺]. Lemon juice at pH 2 has ten times more H⁺ than vinegar at pH 3, a hundred times more than black coffee at pH 5, and ten million times more than household ammonia at pH 11. The visual gap on a pH strip says nothing about how acidic something "feels"; it tells you how many factors of ten separate two solutions. Sørensen introduced the notation in 1909 precisely because spreading the scale linearly would make it unreadable.

The definition in formulas

The pair of equations that drive the entire scale are short:

pH  = −log₁₀[H⁺] pOH = −log₁₀[OH⁻]

and the equilibrium that links them in water,

Kw = [H⁺][OH⁻]

where Kw is the ion-product of water. At 25 °C, Kw = 1.0 × 10⁻¹⁴, so taking −log₁₀ of both sides gives pKw = pH + pOH = 14. That is the canonical relationship that lets a pH calculator compute any one of pH, pOH, [H⁺] or [OH⁻] once you supply the other. Note carefully: Kw depends on temperature. At 0 °C, Kw is about 1.14 × 10⁻¹⁵ and pKw is 14.94; at 100 °C, Kw climbs to roughly 5.5 × 10⁻¹³ and pKw drops to 12.26. The "pH + pOH = 14" identity is a lab-bench convenience, not a universal law. Pure water still has [H⁺] = [OH⁻] at any temperature, so neutral shifts with Kw — at 100 °C, neutral pH is near 6.13 even though the water is still chemically neutral. The calculator on this site is fixed at 25 °C, which is the temperature every published pKa and every standard reference electrode is normalised against.

The IUPAC operational definition is slightly more careful: it uses the activity of hydrogen ions, a thermodynamic quantity that differs from the concentration in concentrated solutions and in the presence of high ionic strength. For everyday work at millimolar to molar concentrations, the activity and the concentration are interchangeable to within the precision of a routine pH meter. The IUPAC Gold Book (doi:10.1351/goldbook.P04524) and the National Institute of Standards and Technology pH reference materials underpin the formal definition; the practical formula in every textbook is the one used here.

Worked examples with real numbers

A strong acid at known concentration. Hydrochloric acid dissociates completely in water, so the molar concentration of HCl equals [H⁺] directly. A 0.01 mol/L HCl solution has [H⁺] = 1.0 × 10⁻² mol/L. Choose "[H⁺] concentration" in the pH calculator and enter 0.01 (or 1e-2 in scientific notation). The result is pH = −log₁₀(0.01) = 2.0. Because pH + pOH = 14 at 25 °C, pOH = 12 and [OH⁻] = 10⁻¹² mol/L. The solution is firmly acidic — about a hundred thousand times more H⁺ than pure water at the same temperature.

Going the other way from a meter reading. A bottle of household ammonia, diluted for cleaning, reads pH 11.0 on a calibrated meter. Choose "pH", enter 11. The calculator returns [H⁺] = 1.0 × 10⁻¹¹ mol/L, pOH = 3.0 and [OH⁻] = 1.0 × 10⁻³ mol/L. The solution is basic and contains a million times more hydroxide than pure water. Comparing these two examples side by side is the easiest way to internalise the logarithmic scale: the HCl example sits four pH units below neutral, the ammonia three units above, and a meter that reports them to one decimal place is resolving differences of factor 1.26 in absolute ion concentration.

A weak acid using a separate calculation. Acetic acid (the active ingredient in vinegar) has a dissociation constant Ka = 1.8 × 10⁻⁵ at 25 °C. A 0.1 mol/L solution does not fully ionise; the equilibrium gives [H⁺] ≈ √(Ka · C) = √(1.8 × 10⁻⁵ × 0.1) = 1.34 × 10⁻³ mol/L. Plug that into the pH calculator as a [H⁺] concentration and the pH comes out at 2.87, in line with measured values of household vinegar. This is the workflow for every weak acid: derive [H⁺] from Ka and the analytical concentration, then convert.

Why temperature matters more than people expect

Kw is not a constant of nature; it is a temperature-dependent equilibrium quantity. Water self-ionises endothermically, so heating the water pushes the equilibrium toward more H⁺ and more OH⁻ in equal measure. Pure water at 100 °C therefore has more of each ion than at 25 °C but is still neutral — the two concentrations are equal. The values below are from the CRC Handbook of Chemistry and Physics and are worth keeping in mind whenever pH is reported at a non-standard temperature.

Temperature  Kw                    pKw    Neutral pH 0 °C         1.14 × 10⁻¹⁵          14.94  7.47 10 °C        2.93 × 10⁻¹⁵          14.53  7.27 25 °C        1.00 × 10⁻¹⁴          14.00  7.00 37 °C        2.39 × 10⁻¹⁴          13.62  6.81 50 °C        5.48 × 10⁻¹⁴          13.26  6.63 100 °C       5.50 × 10⁻¹³          12.26  6.13

Two consequences follow. First, biological pH is almost always reported at 37 °C, where neutral is 6.81 — not 7.0. Healthy blood at pH 7.4 is therefore about 0.6 units basic relative to neutral water at body temperature, not 0.4 units as the 25 °C scale would suggest. Second, a pH meter that reports the value at the actual sample temperature gives a different number than one with automatic temperature compensation set to a 25 °C reference. For most field work the difference is below the resolution of the meter; for process control, biochemistry and analytical chemistry it can matter.

Factors that affect a measured pH

How much the substance dissociates

Strong acids and bases — HCl, HNO₃, HClO₄, H₂SO₄ (first proton), NaOH, KOH — dissociate essentially to completion. For these, the analytical concentration equals the ion concentration. Weak acids and bases — acetic, citric, carbonic, ammonia — only partially dissociate, and the fraction depends on Ka (or Kb) and the analytical concentration. The same 0.1 mol/L of HCl and acetic acid sit at pH 1 and pH 2.87 respectively; the difference is purely the dissociation behaviour.

Temperature

Both the dissociation of the acid or base and Kw shift with temperature. The shifts are usually small over a typical lab range of 15–30 °C, but they are not zero. Calibration buffers come with temperature tables on the bottle for exactly this reason; an uncompensated reading can drift by 0.05 to 0.1 pH unit per 10 °C of mismatch.

Ionic strength

The IUPAC definition uses activity, not concentration. In dilute solutions the activity coefficient is close to one and the two are interchangeable. In seawater, saturated salt solutions, electrolyte plating baths and concentrated buffers, the activity coefficient deviates from one and the simple −log[H⁺] formula loses precision. Marine chemists use a separate "total scale" pH for this reason, and analytical labs that work near saturation run their own activity corrections.

Atmospheric carbon dioxide

Open water absorbs CO₂ from air and forms a small amount of carbonic acid. Deionised water exposed to laboratory air drifts from pH 7 down to 5.5–5.8 within minutes — not contamination, just equilibrium with the atmosphere at roughly 420 ppm CO₂. Unpolluted rainwater sits naturally near pH 5.6 for the same reason; "acid rain" is the term reserved for water that has additionally absorbed sulfur and nitrogen oxides and falls below pH 5.

Time and biological activity

Samples that contain microbes, enzymes, or reactive metals do not have a stable pH. Milk souring, beer fermenting, blood drawn into the wrong tube — all drift over minutes to hours as biochemistry proceeds. For any sample that is not chemically inert, "the pH" means the pH at the moment of measurement.

How pH is measured in practice

Four methods cover almost everything. Litmus paper is qualitative — red for acid, blue for base, no resolution between. Universal indicator paper with a printed colour chart resolves to about half a pH unit and is the standard primary-school tool. A pocket pH meter using a combination glass electrode, calibrated against two or three NIST-traceable buffer standards (the usual set is pH 4.01, 7.00 and 10.01), reaches about ±0.1 pH unit on clean aqueous samples. A research-grade benchtop meter with temperature compensation and a freshly conditioned electrode reaches ±0.01 pH unit under controlled conditions.

Whichever method, the work flow is the same: warm the buffers and the sample to the same temperature, calibrate, rinse, measure, rinse, store the electrode in storage solution (never deionised water — that leaches ions out of the glass bulb and ruins the response). For samples below pH 2 or above pH 12, accuracy degrades and the activity correction matters; for those ranges, double-junction electrodes and standard-addition methods are the right tools. The pH calculator on this site assumes the input is a clean aqueous measurement at 25 °C — outside that envelope, treat its output as a guide rather than a calibrated result.

Common mistakes

Treating pH like a percentage. pH 4 is not "twice as acidic" as pH 8; it is ten thousand times. The scale is logarithmic and the only meaningful comparisons between two pH values are the ratios in [H⁺] you compute by exponentiating the difference. Use the exponent calculator or a quick mental "10 to the power of the difference" to recover the linear ratio.

Forgetting the temperature. Quoting pH without a temperature is sloppy but tolerable around 25 °C. Doing it for blood at 37 °C, hot process water, refrigerated samples or hydrothermal systems hides a real shift in the neutral point. Every published result that matters states the temperature.

Confusing analytical concentration with [H⁺]. A 0.1 mol/L vinegar solution has [H⁺] of about 1.3 × 10⁻³ mol/L, not 0.1 mol/L. Strong acids dissociate completely; weak acids do not. Plugging the bottle concentration of a weak acid into a pH calculator as [H⁺] overestimates acidity by orders of magnitude.

Skipping calibration. A glass electrode drifts with age, with storage, and with the previous sample. A single-point calibration at pH 7 is not enough — at least two points spanning the expected sample range are needed, and most manufacturers recommend three for clinical or food-quality work. A meter that has not seen calibration buffer in months should not be trusted within half a pH unit.

Reading too many decimal places. A field meter with a quoted ±0.1 accuracy reporting "pH 7.34" is overstating its resolution. Round to the precision the instrument actually delivers.

When the simple formula is not enough

The −log[H⁺] formula is the workhorse for routine chemistry, but it is a model with limits. At [H⁺] above about 0.1 mol/L the activity coefficient drops noticeably below one and pH measured by a glass electrode no longer matches −log of the analytical concentration; for concentrated battery acid (about 4 mol/L H₂SO₄) the operational pH can sit well below −1 but the formula gives a misleading picture. Below [OH⁻] of 1 mol/L the same problem appears at the basic end. Buffers with high ionic strength, marine and biological systems where multiple equilibria operate simultaneously, and any kinetically unstable sample all require more than a single pH reading and a textbook formula. For those, the right tool is a proper analytical chemistry workflow: titration curves, activity coefficient models such as Debye–Hückel or Davies, and reference standards traceable to NIST or an equivalent national metrology institute.

Frequently asked questions

Detailed answers to the most common pH questions — temperature dependence, measurement accuracy, pKa versus pH, weak acid calculations, and the bicarbonate buffer that keeps blood near 7.4 — are listed in the FAQ section on this page. For the underlying maths, see the logarithm calculator and the exponent calculator; for the numerical workflow, return to the pH calculator and step through pH, pOH and the two ion concentrations in one pass.