How to calculate pH
Take the hydronium ion concentration, in moles per litre, and hit it with a negative base-10 logarithm: pH = -log10[H3O+]. That is the entire calculation. A solution with 0.010 mol/L of hydronium has pH = -log10(0.010) = 2.00, because 0.010 is 10-2 and the log just reads off the exponent with its sign flipped. Going the other way is the same move in reverse: [H3O+] = 10-pH.
Everything else on this page is one of two companions to that line. The first is pOH, the same idea applied to hydroxide, joined to pH by the fact that water autoionizes: at 25 C the two concentrations always multiply to 1.0 × 10-14, so pH + pOH = 14.00 and knowing either one hands you the other. The second is the strong acid shortcut: HCl, HBr, HI, HNO3 and HClO4 dissociate completely in water, so the hydronium concentration simply equals the acid concentration and no equilibrium table is needed. Strong bases work the same way through pOH, with one wrinkle worth respecting: Ca(OH)2 and Ba(OH)2 release two hydroxides per formula unit, so their concentration gets doubled first.
The formula
pOH = -log10[OH-]
pH + pOH = pKw = 14.00 (at 25 C)
[H3O+] × [OH-] = Kw = 1.0 × 10-14 (at 25 C)
[H3O+] is the hydronium ion concentration in mol/L, [OH-] is the hydroxide ion concentration, and Kw is water's autoionization constant. The last two lines are the same fact written twice: take the negative log of the product identity and it becomes the sum identity. Strictly, pKw at 25.0 C is 13.995; the whole world rounds it to 14.00, a convention worth 0.005 pH, which is smaller than the calibration drift of any electrode you will ever meet.
Worked example
What is the pH of 0.010 M hydrochloric acid? HCl is a strong acid, so every formula unit hands its proton to water: [H3O+] = 0.010 mol/L. Then pH = -log10(0.010) = 2.00, pOH = 14.00 - 2.00 = 12.00, and [OH-] = 10-14 ÷ 10-2 = 1.0 × 10-12 mol/L.
Now a base with the wrinkle. 0.050 M calcium hydroxide releases two hydroxides per formula unit, so [OH-] = 2 × 0.050 = 0.10 mol/L. That gives pOH = -log10(0.10) = 1.00, and pH = 14.00 - 1.00 = 13.00. Forgetting the 2 is the most common lost point on this kind of problem, which is why the calculator renders it as its own step.
And a check you can recognise. A hydronium concentration of 1.8 × 10-5 mol/L comes out at pH 4.74. If that number looks familiar, it should: it is the classic acetate-buffer pH from every textbook, and reproducing a figure the world already agreed on is how you know the arithmetic here can be trusted on the figures you cannot check.
What pH actually measures
pH is a count of hydronium ions, compressed by a logarithm so the numbers stay walkable. Real concentrations span an absurd range: lemon juice carries about a hundred billion times the hydronium of bleach. Write those as raw concentrations and you drown in zeros; write them as pH and the whole of everyday chemistry fits between about 0 and 14. The price of the compression is that each step of one pH unit is a factor of ten. pH 5 coffee is ten times as acidic as pH 6 milk, and a hundred times as acidic as pH 7 water. Two units is 100x, three is 1,000x. The calculator renders this comparison for your own number, because it is the single best defence against reading pH like a linear scale.
| Substance | pH (approximate) |
|---|---|
| Lemon juice | 2 |
| Black coffee | 5 |
| Milk | 6.5 |
| Pure water at 25 C | 7.00 |
| Human blood | 7.35 to 7.45 |
| Baking soda solution | 9 |
| Household bleach | 12.5 |
Every entry except pure water is approximate: real lemons and real coffees vary, and that is fine, because on a log scale being off by 0.3 units is only a factor of two. If you need the concentration behind any rung of the ladder, type its pH into the calculator and read off the hydronium.
The exam trap: 1.0 × 10-8 M HCl is not pH 8
Here is the question professors love, and it deserves its reputation. Take the log of 1.0 × 10-8 and you get pH 8.00, which is basic. Read that answer back slowly: it claims that adding acid to neutral water made the water basic. An acid cannot do that, ever, and the number is trying to tell you the method broke.
What the naive log forgets is the water. Pure water already carries 1.0 × 10-7 mol/L of hydronium from its own autoionization, ten times more than this acid brings to the party. Below roughly 10-6 M the two sources compete, and the honest move is to solve the charge balance, which gives a small quadratic: [H3O+] = (C + √(C2 + 4Kw)) ÷ 2. For 1.0 × 10-8 M HCl that returns pH 6.98: just barely on the acid side of neutral, which is the only side an acid can ever land on. This calculator detects dilute inputs, runs the correction automatically, and shows you both numbers so the trap teaches instead of bites. At sensible bench concentrations the correction vanishes: by 0.001 M the two answers agree to well within 0.01 pH, which is why the shortcut is safe everywhere except the exam question built to break it.
One small delight buried in the algebra: at exactly 1.0 × 10-7 M, where the acid's contribution matches water's own, the quadratic returns [H3O+] = 1.618... × 10-7 mol/L. That is the golden ratio times the naive answer. Nothing mystical is happening: phi is (1 + √5) ÷ 2, and that is what the formula collapses to when C = √Kw. It is simply a pleasing place for the math to land.
Neutral is 7 only at 25 C
Kw moves with temperature, because autoionization is an equilibrium and heat pushes it forward. At body temperature, 37 C, pKw is about 13.6, so neutral water there has [H3O+] near 1.6 × 10-7 mol/L and a pH of about 6.8. That water is not acidic. Neutral has never meant pH 7; it means the hydronium and hydroxide concentrations are equal, and at 37 C they are equal at 6.8. The number 7 is just where equality happens to sit at 25 C, the temperature the tables are printed for. This page computes at 25 C and says so, because carrying a temperature dial would mean maintaining a Kw table; what you should carry instead is the idea, which never goes stale.
Sig figs in a logarithm
Logs have their own significant-figure rule, and it trips people because it looks backwards: the decimal places of a pH correspond to the significant figures of the concentration. In pH 4.74, the 4 in front of the decimal point is the characteristic; it only places the power of ten and carries no measurement information at all. The 74 after the point is the mantissa, and that is where the two sig figs of 1.8 × 10-5 live. So a concentration known to two figures earns a pH with two decimals, and a pH quoted to two decimals can only give back a concentration to two figures. This calculator applies the rule both directions and defaults to two decimal places, which is also about what a well-calibrated pH meter honestly delivers. Our significant figures calculator covers the ordinary, non-logarithmic rules.
Negative pH and pH above 14
The 0 to 14 scale on the classroom poster is a habit, not a law. pH is a logarithm, and logarithms keep going as long as concentrations do: 10 M hydrochloric acid works out to about pH -1, and 10 M sodium hydroxide to about pH 15. The calculator accepts these happily and will tell you what they mean. The honest caveat at such strengths is that ions crowd each other, so a pH electrode reads activity rather than concentration and the measured value drifts from the calculated one. Treat the answer here as the concentration answer, which is what the formula defines, and expect a real meter in a real 10 M solution to disagree with it somewhat. The most acidic water ever reliably measured in the field, at California's Iron Mountain mine, came in around pH -3.6, so even the far end of the scale is not hypothetical.
Where this page stops: weak acids need Ka
Acetic acid, citric acid, ammonia and nearly everything else in a kitchen or a cell are weak: they only partly dissociate, and how far they go depends on their equilibrium constant Ka and on the concentration, solved through an ICE table. Running vinegar through the strong acid mode above would overstate its acidity about a hundredfold, so this page refuses to pretend. It does strong acids, strong bases and the four-way conversion, exactly and with the working shown, and it hands the weak-acid problem off honestly rather than doing it badly. If you are preparing solutions, our molarity calculator turns a target concentration into a weighing instruction and our dilution calculator handles the C1V1 = C2V2 step; and if logarithms are the part you are warming up to, the same tool is doing different work in our half-life calculator, where it counts decays instead of ions.