Part C · How stuff changes

12 · Acids and bases

Red-cabbage juice turns red in lemon juice, purple in water and green in soapy water. Whether something is an acid or a base comes down to one particle, the proton: who has a spare one, and who wants one.

This chapter is mostly chapter 11 applied to a single reaction: a proton (a hydrogen nucleus, H⁺) moving from one molecule to another. Because it involves a free energy, it obeys a Boltzmann factor. Because Boltzmann factors span many powers of ten, chemists measure it on a log scale. That's all pH is.

Water is never just H₂O

Even perfectly pure water reacts with itself. Now and then one water molecule hands a proton to another:

H₂O + H₂O ⇌ H₃O⁺ + OH⁻

The products are a hydronium ion, H₃O⁺, and a hydroxide ion, OH⁻. The reaction is steeply uphill: ΔG° ≈ 80 kJ/mol. Chapter 11's rule says every 5.7 kJ/mol is a factor of 10, and 80 ÷ 5.7 ≈ 14. So the equilibrium constant is about 10⁻¹⁴:

[H₃O⁺] × [OH⁻] = 1.0 × 10⁻¹⁴ (at 25 °C, in mol/L)

In pure water the two are equal, 10⁻⁷ mol/L each. Water itself is 55.5 mol/L, so at any moment about 2 water molecules in a billion have swapped a proton. Few, but in a single drop that's still trillions of ions, constantly forming and recombining. That 14 is where the pH scale gets its range.

"H⁺" in an equation is bookkeeping. A bare proton never floats free in water: it's always attached to water molecules as H₃O⁺ or larger clusters, and it moves by hopping from one water molecule to the next along chains of hydrogen bonds. That hopping is why acid solutions conduct electricity unusually well.

Acids give protons, bases take them

An acid is something that hands protons to water, raising the H₃O⁺ concentration. A base takes protons, lowering it. Vinegar's acetic acid gives one away; baking soda and soap pick them up. Mix an acid and a base and the protons go from one to the other: they neutralise each other. With vinegar and baking soda, one of the products is carbonic acid, which falls apart into water and the CO₂ that fizzes (chapter 9's balloon).

pH: a log scale for protons

Because the concentrations span so many powers of ten, chemists use their logarithm:

pH = −log₁₀[H₃O⁺]

Pure water at 25 °C: pH 7. Each step down is ten times more acidic: lemon juice at pH 2 has about 100,000 times more hydronium than pure water.

The pH scale with some everyday liquids (typical values; real ones vary), the colour red-cabbage juice turns at each pH, and the concentrations of H₃O⁺ and OH⁻. Their product is always 10⁻¹⁴ at 25 °C: when one goes up, the other goes down.

The cabbage colours are approximate. The dye is a family of related molecules, and the exact shades depend on the cabbage, how dilute the juice is, and the lighting.

How hard an acid holds its proton: p`K`_a

Write a general acid as HA. It can give its proton to water and become A⁻:

HA + H₂O ⇌ H₃O⁺ + A⁻

Its equilibrium constant is called K_a, and its negative log is the pK_a. It measures how tightly the acid holds its proton: a low pK_a lets go easily, a high one holds on. Put it together with chapter 11 and you get the most useful formula in acid–base chemistry. At a given pH, the fraction of the acid that has given up its proton is:

fraction as A⁻ = 1 / (1 + 10^{pK_a − pH})

Acetic acid (vinegar) has pK_a 4.76. At what pH is exactly half of it holding its proton and half not?

At pH 4.76: pH = pK_a. That's what pK_a means: the pH of the half-way point. One pH unit below it, about 91% still holds its proton; one unit above, about 91% has let go. Within a couple of units either side, the switch is essentially complete. Slide the pH below and watch the molecules flip.

Top: 60 molecules of the chosen acid, each flipping at random between holding its proton (blue, with a small grey proton) and having given it away (orange). Charges are shown: acetic acid gives up its proton to become negative, while ammonium starts positive (NH₄⁺) and ends up neutral (NH₃). Bottom: the fraction given away against pH for each acid. The current acid is drawn bold; the dot is the current pH. The CO₂ curve is simplified: it uses the combined "apparent" pKa of dissolved CO₂ and ignores bicarbonate losing a second proton above about pH 9.

This curve explains a lot of everyday chemistry:

Indicators. A dye whose HA and A⁻ forms have different colours changes colour over about two pH units around its pK_a. Red cabbage contains anthocyanin dyes with several protons and several pK_as, so you get a whole rainbow instead of a single switch.
Buffers. Near pK_a, a mix of HA and A⁻ soaks up added acid (A⁻ takes the protons) and added base (HA supplies them), so the pH barely moves. Your blood is held at pH 7.4 ± 0.05 largely by the dissolved CO₂ / bicarbonate pair (effective pK_a about 6.1 at body temperature), with your lungs adjusting the CO₂ and your kidneys the bicarbonate.
Strong versus weak. Hydrochloric acid's pK_a is far below zero, so in water it gives up essentially every proton. Acetic acid is weak: at 0.1 mol/L its pH is about 2.9, not 1, because most of the molecules keep their proton.

Common trap

"Strong acid" doesn't mean "concentrated acid". Strong is about pK_a, how completely each molecule gives up its proton. Concentrated is about how much is dissolved. Dilute hydrochloric acid is strong but mild; concentrated acetic acid is weak but corrosive. Also: pH 7 is neutral only at 25 °C. The water reaction above gives more ions when warmer, so at body temperature neutral is about pH 6.8. And bases aren't the "safe" opposite of acids: drain cleaner (pH 13–14) burns skin as badly as strong acid.

ML bridge

The fraction formula is a logistic sigmoid. Write it as σ(z) with logit z = ln(10) × (pH − pK_a). It's chapter 7's two-state Boltzmann distribution with the free energy of proton transfer as the logit: ΔG° = RT ln(10) × pK_a, so one pK_a unit is 5.7 kJ/mol at 25 °C. The pH acts as a bias term set by the environment: it fixes the proton's chemical potential (its "price"), and every acid in the solution answers with its own sigmoid. A molecule with several acidic groups is a softmax over its protonation states.

The formula really uses activities (effective concentrations), which differ from plain concentrations in salty or concentrated solutions. pK_a values are for water; in other solvents they shift a lot, sometimes by many units. Roald Hoffmann lists the acid–base concept itself among chemistry's productive ideas that don't reduce neatly to physics.

Tell it to a 6-year-old

Look: Purple red-cabbage juice (a grown-up chops red cabbage, pours hot water over it, waits, then strains it) poured into six small clear cups. Next to them: lemon juice, vinegar, plain water, baking soda, a little washing-up liquid, and (grown-up only) a pinch of washing soda.
Wonder: "What colour do you think each one will make? Let's guess before we pour."
Try: Add one thing to each cup. Line them up from red to green. You've made a rainbow, and a measuring scale.
True sentence: "Lemon and vinegar are acids. Baking soda and soap are the opposite kind, called bases. The colour pieces in cabbage juice change their shape when they meet an acid or a base, and each shape shows a different colour."

Each colour is a different protonation state of the cabbage's anthocyanin dyes. Washing soda is a skin and eye irritant: the grown-up handles it, and nothing gets tasted. Don't use bleach: it destroys the dye rather than changing its colour. Expect pink-red with lemon, purple with water, blue with baking soda, green with washing soda; your shades may differ.

Play more

PhET: pH Scale (and pH Scale: Basics, good with the 6-year-old). Dilute everyday liquids and watch the pH.
PhET: Acid–Base Solutions. Strong versus weak acids, with molecule views.
Crick Kids: kitchen chemistry. The red-cabbage experiment, written for children.