pH Calculator

Ask for the pH of a very dilute acid and the obvious formula gives a silly answer. Hydrochloric acid at one hundred millionth of a mole per litre should, by the rule every student learns, have a pH of 8.

That would make a solution of acid basic, and it is not. The real figure is 6.98, a shade on the acid side of neutral, and the gap between 8 and 6.98 is the subject here. The rule did not fail because the chemistry is exotic. It failed because it had quietly left something out, and at that dilution the missing piece was the largest thing in the beaker.

A scale built on a logarithm

The idea of pH is a little over a century old. The U.S. Geological Survey credits it to the Danish biochemist Søren Sørensen, who introduced it in two scientific papers in 1909. He was head of the Carlsberg Laboratory in Copenhagen from 1901 to 1938, and brought in the scale there as a simple way of expressing a hydrogen ion concentration. The papers appeared in German and French together, and the p stands for Potenz and puissance, both meaning power. Stephen Lower, whose open chemistry text Chem1 is drawn on throughout this piece, puts it more plainly: power in the sense of an exponent.

The definition is short. The pH is the negative base ten logarithm of the hydrogen ion concentration, so each step of one unit is a factor of ten. The Geological Survey gives the scale of that: water draining from an abandoned coal mine at pH 2 is 100,000 times more acidic than neutral water. A logarithm is the only sensible way to hold such a range on one ruler.

Water sets the other end of the scale. A few of its molecules always split into hydrogen and hydroxide ions, and the product of the two concentrations is a constant, written Kw. At 25 degrees Celsius the classroom value is 1.00 times ten to the minus fourteen, which is why neutral water sits at pH 7 and why pH and pOH add up to 14.

That 7 is a fact about one temperature. Lower notes that Kw varies with temperature, and a reference table in the Journal of Physical and Chemical Reference Data shows by how much. It puts pKw, the negative logarithm of Kw, at 13.99 at 25 degrees, 13.26 at 50 degrees and 12.25 at 100. Halve the middle figure and pure water at 50 degrees has a pH of 6.63, and is still exactly neutral. Hot water is not acidic. Its neutral point has moved.

Where the one-line formulas come from

Acids divide into two camps. A strong acid ionizes essentially completely, so its hydrogen ion concentration is simply the concentration of acid you dissolved, and the pH is the negative logarithm of that. A weak acid ionizes only a little, and how little depends on its ionization constant, Ka. Lower counts hundreds of thousands of weak acids against no more than a few dozen strong ones.

The classroom treatment of a weak acid is a small table. Start with a concentration C of acid and nothing else. Let x of it ionize. At equilibrium there is C minus x of the acid left and x each of hydrogen ion and of the acid’s partner ion, so Ka equals x squared over C minus x. That is a quadratic equation, and nobody wants to solve one by hand.

So the textbooks make a bargain. If x is small beside C, then C minus x is nearly C, and x is just the square root of Ka times C. One line, one button on a calculator. The bargain comes with a test: work out x, compare it with C, and accept the answer if x is under 5 percent of the starting concentration. Lower calls this the five percent rule and describes it as the usual advice, which is a fair description of a rule of thumb.

It is a good bargain most of the time. It turns bad exactly where nobody thinks to run the check: in dilute solutions.

Take acetic acid, for which a review of buffer constants by chemists at the National Institute of Standards and Technology selects a pK of 4.756. At 0.1 molar the square root gives x as 0.00132 molar and a pH of 2.878. The check passes easily, since x is 1.32 percent of C. Solve the quadratic properly and the pH is 2.881, which rounds to 2.88. The shortcut cost three thousandths of a pH unit, which no classroom instrument would notice. You can reproduce all of this by choosing acetic acid at 0.1 molar and reading the steps.

Now dilute it a thousandfold, to 0.0001 molar. The square root says pH 4.38. But x is now 42 percent of C, the 5 percent test fails loudly, and the quadratic gives 4.47. Nothing about the acid changed. A weak acid ionizes more as it is diluted, a behaviour Lower ties to the Ostwald dilution law, and the more it ionizes the less true it is that x is small.

What the shortcuts leave out

Both formulas share a deeper omission. The little table starts with no hydrogen ion at all, as though the water were not there. Water always supplies some, about a ten millionth of a mole per litre when left alone. Beside 0.1 molar acid that is nothing. Beside a hundred millionth molar acid it is ten times larger than the acid itself.

Lower sets out the method that does not leave anything out. Write every equilibrium expression, including the one for water. Add a mass balance for each substance, which says that the forms of an acid must add up to what was put in. Then add one charge balance, because a solution is electrically neutral and its positive and negative ions must cancel. The result is one equation in one unknown, the hydrogen ion concentration, and a computer can find its root however awkward it looks.

Put that tiny amount of hydrochloric acid through the full equation and the answer is 6.98. The acid nudges water’s own balance rather than overwhelming it, and as Lower says, no amount of dilution can make a strong acid alkaline. The same equation handles an acid with several protons to give. Carbonic acid at 0.010 molar, with the two constants from the same review, comes out at a hydrogen ion concentration of 0.000067 molar. The second ionization contributes almost nothing, because its constant is nearly ten thousand times smaller than the first.

Buffers are where the full equation earns its keep in a different way. A buffer is a weak acid mixed with its partner base, and it resists a change in pH when acid or base is added. Its pH is usually found from the Henderson-Hasselbalch equation, which adds the logarithm of the ratio of the two to the acid’s pKa. By Lower’s account Lawrence Henderson, a physician studying the acidity of blood, worked out the relationship in 1908, and the physiologist Hasselbalch put it in logarithmic form in 1916.

That equation, Lower warns, is an approximation of an approximation: the exact buffer equation with the hydrogen and hydroxide ion concentrations dropped as negligible. For acetic acid and acetate at 0.1 molar each they are negligible, and the equation and the full calculation agree to the third decimal, both about 4.76. Make the same buffer at 0.00001 molar each and the equation still says 4.76, because a ratio does not care about dilution. The full calculation says 5.27. Half a unit of error, from a formula that gave no warning.

Reading a titration curve

A titration turns all of this into a picture. Run sodium hydroxide into acetic acid and plot pH against the volume added. Follow 25.00 millilitres of 0.100 molar acid with base of the same strength: the full calculation starts at 2.88, sits at 4.76 when half the acid is used up, and reaches 8.73 at the equivalence point, where the amounts of acid and base match.

Each landmark is one of the earlier calculations. The start is the weak acid alone. The long gentle slope is a buffer, and at its midpoint the two forms are equal, so the pH equals the pKa. The equivalence point is not 7, because what remains is acetate, a weak base in its own right. Past it, the pH is set by the leftover hydroxide.

The steep jump is what makes the method practical, since an indicator only has to change color somewhere on the cliff. An indicator is itself a weak acid whose two forms differ in color, and Lower puts the change at about 1.5 to 2 pH units wide. Wikipedia’s table of common indicators gives phenolphthalein a range of 8.3 to 10.0, which sits on the cliff for acetic acid, while methyl orange at 3.1 to 4.4 would finish changing long before the right volume. For a strong acid the cliff is taller, and almost any indicator serves.

Phenol red is a closer call. Wikipedia’s article on the dye describes it as passing from yellow to red between pH 6.8 and 8.2 and turning bright pink beyond. One percent short of the equivalence point the acetic acid curve stands near pH 6.75, so the whole change fits on the cliff, with five hundredths of a unit to spare.

Two finer points show up on a computed curve. The steepest point falls a hair before true equivalence, at 24.99999 millilitres in the example above. And an acid with two protons shows two clear jumps only when its two constants differ by several orders of magnitude, which is why Lower’s curve for sulfurous acid has two and his curve for succinic acid has one.

Concentration is not quite what a meter reads

Everything above uses concentrations, and that is an idealization. Ions in real solutions crowd and pair with one another, so each behaves as if slightly less of it were present. Chemists call the effective amount the activity. Lower writes it as the concentration times a coefficient that approaches one only as the solution becomes very dilute, and notes that pH is properly defined through activity. The U.S. Environmental Protection Agency agrees: what is actually measured is hydrogen ion activity, not concentration. Lower’s example is 0.10 molar hydrochloric acid, where the ion’s activity is only 81 percent of its concentration. Take the logarithm of that and the pH is 1.09, not the 1.00 the simple rule gives.

Measurement has its own machinery for that reason. NIST calls pH the most often measured chemical quantity, and ties every reading back to reference buffers certified in a special electrochemical cell under an agreed convention. A calculation cannot stand in for that chain.

It also inherits whatever constants it is fed. Most of the tool’s list comes from the NIST review. Six more families, formic, hydrofluoric and nitrous acid and the bases dimethylamine, trimethylamine and aniline, carry the figures given in Wikipedia’s articles on those substances, and the tool tags them so. Such figures differ in the second decimal from one table to the next: the article on hydrofluoric acid gives 3.17 in its summary box and 3.18 in its text. Lower’s advice is to treat a tabulated Ka as good to about 5 percent at best, which leaves little reason to quote a calculated pH to more than two significant figures.

So a calculated pH is a statement about a model: a dilute, ideal solution at a stated temperature, with constants from a named table. Within that model the full calculation is exact and the shortcuts are approximations whose errors can be measured. The useful habit is to compute both and look at the gap. When it is small, the one-line formula has told you something true about which species dominates. When it is large, the formula has told you something too, about what it assumed was not there.