Chapter 1 A ruler of powers of ten
The pH scale
A ruler of powers of ten for hydrogen-ion concentration: what it measures, where everyday substances sit on it, and why neutral water is not always pH 7.
pH measures the concentration of hydrogen ions (strictly, their activity) and writes it as a power of ten. A low number means many ions per liter and a high number few; the beaker below turns the concentration back into a count you can see. Each step of one unit on the scale is a factor of ten (pH 3 has ten times the hydrogen ions of pH 4, and a hundred times those of pH 5), which is why a scale that runs from 0 to 14 can cover a hundred-trillion-fold range of acidity.
Written as a formula, pH is the negative base-10 logarithm of the hydrogen-ion concentration: pH = −log10[H⁺], with [H⁺] in moles per liter. Pure water at 25 °C has [H⁺] = 1.0 × 10⁻⁷ mol/L and so pH 7.00. Lemon juice, around pH 2.3, has about 50,000 times more.
The definition is short, and nearly everything interesting about pH is hidden in it. The logarithm hides how many ions there are. The minus sign means more acid gives a smaller number. The “1 × 10⁻⁷” for water is not a law of nature but a property of water at one temperature. This chapter takes the definition apart, one piece at a time, with the beaker above: add acid to it and watch the count.
What the number measures
In water a free proton never floats alone. It attaches at once to a water molecule to make the hydronium ion, H3O⁺, which is itself constantly handing the proton on to neighbours. Chemists write H⁺ and H3O⁺ interchangeably, and so will we. What pH measures is their concentration, the number of these ions per volume, on a logarithmic scale:
A mole is 6.022 × 10²³ particles[1], so concentrations in moles per liter hide very large counts. A 100 mL glass of pure water holds about 6.0 × 10¹⁵ hydronium ions. That sounds like a lot until you set it against the 3.3 × 10²⁴ water molecules in the same glass[2]: only one water molecule in about 550 million is carrying an extra proton at any moment. Try changing that in the beaker at the top of the page.
Three things are worth noticing. First, one drop of dilute acid, 0.05 mL of 0.1 M HCl, takes the glass from pH 7.00 to about 4.30. Water on its own has nothing to resist the change, a point we return to in the chapter on buffers. Second, to move from 4.3 to 3.3 you need ten times as many drops as you used to get to 4.3, and to reach 2.3 a hundred times as many. Equal steps on the pH scale are equal ratios of hydrogen ions, not equal amounts.
Third, look at the loupe at high pH. At pH 7, a cubic micrometre of water contains on average 60 hydronium ions. At pH 9 the average is 0.6, so most such volumes contain none at all and some contain one or two. Concentration is an average over enormous numbers of molecules; in very small volumes it turns into a statement about probabilities. The loupe draws a fresh random count, from the Poisson distribution with the right average, each time the pH changes.
Why a logarithmic scale
Hydrogen-ion concentrations in ordinary life run from about 0.02 mol/L in an empty stomach to 10⁻¹³ mol/L in a 0.1 M solution of sodium hydroxide. No linear ruler can show both ends. If you drew one so that stomach acid sat 10 cm from zero, pure water would sit half a micrometre from zero, and the sodium hydroxide solution would be a million times closer still, far less than the width of an atom. The logarithm turns that range into fourteen evenly spaced steps. It also matches how chemistry behaves: many reaction rates and equilibria respond to the ratio of concentrations, so equal steps in pH tend to have comparable effects.
The price is that the numbers stop meaning what they look like. A drop from 5.6 to 4.3 looks like a modest change. Before you check, make a guess.
Switching the axis to a linear scale makes the point better than any argument. On it, stomach acid and lemon juice are visible and everything else, from tomato juice to sodium hydroxide, is squashed against the left edge. The logarithm is a magnifying glass that magnifies small numbers more than large ones.
The values in the figure come from measurements, and they are ranges, because real lemons and real rain vary. The table gives them with their sources. Where this site computed a value from its own model, it says so.
| Substance | pH | Source |
|---|---|---|
| Stomach contents, fasting (median of healthy adults) | 1.7 | Dressman et al. 1990[5] |
| Lemon juice | 2.00–2.60 | FDA[6] |
| Vinegar | 2.40–3.40 | FDA |
| Orange juice (California) | 3.30–4.19 | FDA |
| Tomato juice | 4.10–4.60 | FDA |
| Acid rain, usual range | 4.2–4.4 | US EPA[7] |
| Bananas | 4.50–5.20 | FDA |
| Normal rain | about 5.6 | US EPA |
| Cow’s milk | 6.40–6.80 | FDA |
| Pure water, 25 °C | 7.00 | computed from Kw[8] |
| Blood | 7.35–7.45 | Shaw & Gregory 2022[9] |
| Surface seawater, today’s average | about 8.1 | NOAA[10] |
| 0.1 M sodium bicarbonate (baking soda) | 8.34 | computed (pKas from Goldberg et al.[11]) |
| 0.1 M ammonia | 11.12 | computed |
| 0.1 M sodium hydroxide | 12.99 | computed |
A note on the computed rows. A “household ammonia” bottle is a mixture of unknown strength, so instead of quoting a vague range we give the exact equilibrium pH of a defined solution. For 0.1 M ammonia the calculation, explained in Chapter 2, gives 11.12.
Neutral, acidic, basic, and pOH
Where do the hydronium ions in pure water come from? From water itself. A tiny fraction of water molecules hand a proton to a neighbour:
The product Kw, the ion product of water, is fixed at a given temperature. Its current reference value at 25 °C is 10−13.994, from the formulation adopted by the International Association for the Properties of Water and Steam in 2024[8][12]. (Textbooks round it to 1.0 × 10⁻¹⁴, and so the pure-water pH to 7.00; the exact figure is 6.997.) Because the product is fixed, hydrogen and hydroxide ions behave like the two ends of a seesaw. Add acid and [H⁺] rises, so [OH⁻] must fall by the same factor. Take logarithms of both sides and you get the rule printed in every textbook:
A solution is neutral when [H⁺] = [OH⁻], acidic when there are more hydrogen ions than hydroxide ions, and basic (or alkaline) when there are fewer.
Can pH be below 0 or above 14?
Yes. The scale is not bounded at 0 and 14; those are just where ordinary solutions of about 1 mol/L of strong acid or strong base land. A 2 M solution of hydrochloric acid has a nominal pH of −0.3 by equation (1.1). At such concentrations the ions crowd and interact, and the activity no longer follows the concentration in any simple way. For hydrochloric acid near 2 mol/kg the two effects happen to cancel (its mean activity coefficient is close to 1), so the activity-based pH really is near −0.3; by 10 mol/kg the coefficient is about 10 and the pH is near −2, a full unit below what the concentration says[13]. What fails first at such strengths is the measurement: a glass electrode and its liquid junction are calibrated with dilute buffers. Negative pH values are real but the simple formula does not describe them, which is why the calculators on this site shade everything above about 0.1 mol/L as outside the ideal range[3].
Diluting an acid forever
Here is a classic trap. Take 0.1 M hydrochloric acid, pH 1. Dilute it tenfold: pH 2. Again: pH 3. Each dilution adds one to the pH, because hydrochloric acid is a strong acid and gives up all its protons, so [H⁺] = C.
The rule “pH = −log C” forgets that water makes hydrogen ions too. Once the acid is as dilute as 10⁻⁶ mol/L, water’s own contribution starts to matter, and by 10⁻⁷ mol/L it is comparable. The exact statement is a charge balance: in a solution of HCl, every positive charge is a hydrogen ion and every negative charge is either a chloride or a hydroxide ion, so
At 10⁻⁷ M, the naive rule says pH 7.00; the exact answer is 6.79, and 38 % of the hydrogen ions come from water. At 10⁻⁸ M the naive rule gives 8, a basic solution made of acid, which is absurd; the exact answer is 6.98. However much you dilute an acid, its pH approaches neutral from below and never crosses it. This is the first appearance of the idea that runs through the whole site: the pH is whatever value makes the charges balance, and the shortcuts work only when one term in that balance dwarfs the others.
Is boiling water acidic?
Look again at the definition: neutral means equal, not “pH 7”. At 25 °C equality happens at pH 7.00, but that is a consequence of the value of Kw, and Kw changes with temperature. The ionization of water absorbs heat, so warmer water ionizes more. Between 0 °C and 100 °C, Kw rises roughly five-hundredfold, from 10−14.947 to 10−12.254[8]. Because neutral means [H⁺] = [OH⁻] = √Kw, the neutral pH is half of pKw, and it moves with temperature.
At body temperature, 37 °C, neutral is pH 6.81. Blood at pH 7.40 is therefore more alkaline, relative to neutral, than the same number would suggest at room temperature. At 0 °C neutral is 7.47; at 100 °C, 6.13. This is not a curiosity for specialists. A pH meter reading taken at the wrong temperature, or a “pH 7 means neutral” rule applied to a hot process stream, is a real source of error, which is why good meters correct for temperature and why IUPAC’s standard buffers are certified at each temperature[3].
Where pH came from
Sørensen’s paper, received by the Biochemische Zeitschrift on 29 May 1909 and published simultaneously in French by the Carlsberg Laboratory, set out to measure that concentration properly[14]. documented He used two methods. The first was electrometric: a hydrogen electrode, whose voltage against a calomel reference depends on the hydrogen-ion concentration, calibrated with dilute hydrochloric acid of known strength. The second was colorimetric: he characterized a long list of indicator dyes by the hydrogen-ion concentration at which each changes color. Both are still the basis of pH measurement.
Then he proposed a notation. Hydrogen-ion concentrations, he wrote, are best expressed as a negative power of ten, and he gave the name Wasserstoffionenexponent, “hydrogen-ion exponent”, and the symbol pH, to the number in the exponent. A solution 0.01 normal in hydrogen ions has pH 2. He also measured the ionization of water at 18 °C, finding 0.72 × 10⁻¹⁴ = 10−14.14, so that, in his words, pure water and truly neutral solutions have a hydrogen-ion concentration of 10−7.07. That historical point sits on the temperature figure above; the modern value at 18 °C is 10−14.24, a difference of 0.1 that reflects how much better the measurement has become[8].
What does the “p” stand for? You will read that it means power, Potenz or puissance, or “potential of hydrogen”. Sørensen never said. Jens Nørby, reading the original papers, pointed out that in Sørensen’s derivation of the electrometric method the two solutions being compared are labelled p and q, and the unknown concentration is called Cp; the letter may be no more than a label[15]. contested The honest answer is that nobody knows.
The modern definition replaces concentration with activity: pH = −log10 aH. IUPAC’s 2002 recommendations make pH a conventional quantity, measured through a chain of comparisons that runs back to primary standard buffers whose pH is assigned with a Harned cell, a hydrogen electrode in a cell without a liquid junction[3]. established The pH meter on a laboratory bench uses a glass electrode instead of Sørensen’s hydrogen electrode, but it is calibrated against those buffers, so every reading still descends from the method he described in 1909.
What comes next
So far the only acids have been strong ones, which hand over every proton. Most acids in kitchens, bodies and oceans are weak: they hand over only some of their protons, and how many depends on an equilibrium constant. That is where the charge balance stops being a quadratic and becomes something you have to solve, and where pH starts to tell you about the chemistry instead of only about the amount. Chapter 2 takes that step.