Free pH Calculator with Worked Steps

What pH actually measures

In any sample of water a few molecules have fallen apart into H+ and OH− ions, and acidity is a count of the H+ ions: the more of them per litre, the more acidic the solution. The trouble is the range — [H+] runs from around 1 mol/L in strong lab acids down to 1e-14 mol/L in concentrated lye, fourteen powers of ten. Nobody wants to compare 0.00072 against 0.0000000034 by eye, so the scale is compressed with a logarithm: pH = −log10[H+].

Read it backwards and every pH unit is a factor of ten: pH 3 has ten times the hydrogen-ion concentration of pH 4 and a hundred times that of pH 5. The minus sign is there so everyday solutions land on positive numbers — the log of a concentration like 1e-5 is −5, and flipping the sign turns it into the familiar pH 5.

pOH is the same compression applied to the other ion: pOH = −log10[OH−]. Acidic solutions have high [H+] and low pH; basic solutions have high [OH−] and low pOH. At 25 °C the two scales are locked together by the water constant — next sections — which is why entering any one of the four values into the calculator above fixes the other three.

How to calculate pH from [H+]: the log step, worked

Take [H+] = 3.6 × 10⁻⁴ M, a typical dilute-acid problem. The definition says pH = −log10(3.6e-4), and the log splits cleanly across the two factors of the scientific notation: log10(3.6e-4) = log10(3.6) + log10(1e-4). The first piece is a number between 0 and 1, because 3.6 sits between 10⁰ = 1 and 10¹ = 10; a calculator gives log10(3.6) = 0.556. The second piece is exactly the exponent: log10(1e-4) = −4.

Add them and flip the sign: log10(3.6e-4) = 0.556 − 4 = −3.444, so pH = 3.44. Notice the shortcut hiding in that arithmetic — the exponent of the concentration hands you the whole-number neighbourhood of the pH before you touch a calculator. A concentration of 10⁻⁴-something always lands between pH 3 and 4; one of 10⁻⁹-something between 8 and 9. Counting the exponent first is the fastest sanity check there is for any pH answer.

Significant figures work differently for logarithms, and graders check it: the digits before a pH's decimal point only place the power of ten, so they never count toward precision. The decimal places carry it, and their number should match the significant figures of the concentration. 3.6e-4 M has two sig figs, so pH 3.44 gets two decimals — writing 3.4437 claims precision the measurement never had. The calculator deliberately prints a couple of extra decimals so nothing is lost mid-chain; trimming to your measurement's sig figs is the step you do when you write the answer.

pH to pOH and back: what Kw = 1.0e-14 buys you

Water itself ionizes a little: H2O ⇌ H+ + OH−. At 25 °C that equilibrium settles wherever the product of the two ion concentrations equals the water constant, Kw = [H+][OH−] = 1.0 × 10⁻¹⁴ — the product, always, in pure water and in any dilute aqueous solution alike. Add acid and [H+] climbs, so [OH−] must fall to keep the product fixed; the two concentrations ride a seesaw.

Take −log10 of both sides of [H+][OH−] = 1.0e-14 and the product becomes a sum: pH + pOH = 14.00. That one line is the entire pH-to-pOH conversion — subtract from 14 — and it is why one input is enough here. pH 3.60 means pOH = 14.00 − 3.60 = 10.40, and undoing the log recovers the concentration: [OH−] = 10^−10.40 = 4.0e-11 M.

The fine print is temperature. Kw = 1.0e-14 holds at 25 °C; warm the water and it ionizes more — at 50 °C, Kw is about 5.5 × 10⁻¹⁴ and pKw drops to 13.26. Hot pure water then sits near pH 6.6 while being perfectly neutral, because [H+] and [OH−] are still equal. Neutral means equal ions, not pH 7; pH 7 is merely where equal lands at room temperature. That is why this page prints the 25 °C assumption next to every result instead of hiding it inside the formula.

From molarity to pH: the strong acid and base path

For a short list of acids, pH follows from molarity alone. HCl, HBr, HI, HNO3, HClO4 — and the first proton of H2SO4 — dissociate completely in water: every dissolved molecule gives up its H+. That is the whole method: for 0.0025 M HCl, [H+] = 0.0025 M, so pH = −log10(0.0025) = 2.60. Strong bases mirror it through pOH: 0.050 M NaOH puts [OH−] = 0.050 M, so pOH = 1.30 and pH = 14.00 − 1.30 = 12.70.

The phrase "dissociates completely" is an assumption, and the calculator prints it as step one instead of burying it — because it is exactly where the method breaks. Acetic acid, HF, and every other weak acid keep most of their protons: 0.10 M acetic acid is only about 1.3% dissociated, so its real pH is 2.87, while the strong-acid shortcut would claim 1.00 — nearly two full units too acidic. A weak acid needs its Ka and an equilibrium calculation, which is why this page has no Ka input and says so plainly rather than returning a confident wrong number.

There is a subtler trap at the dilute end. For 1.0e-8 M HCl the shortcut says pH = 8 — a basic reading for an acid, which is impossible. At concentrations that low, water's own autoionization supplies more H+ than the acid does, so the calculator switches to the full charge balance, [H+] = (C + √(C² + 4Kw)) ÷ 2, and reports the true answer: pH 6.98, a hair under neutral, exactly where a vanishing trace of acid belongs.

Mistakes that cost marks, and how to check any answer

• Treating a weak acid as strong. If the problem names acetic acid, formic acid, HF, or ammonia and hands you a Ka or Kb, the molarity shortcut does not apply — 0.10 M acetic acid is pH 2.87, not 1.00.
• Using pH + pOH = 14 away from 25 °C. The 14 is pKw at room temperature, not a law of nature; a heated-water problem that ignores the temperature can be off by more than half a pH unit.
• Doubling H2SO4's molarity. Sulfuric acid gives up its first proton completely but its second only partially (Ka2 ≈ 0.012), so 0.010 M H2SO4 is not simply [H+] = 0.020 M. The strong-acid mode here is strictly monoprotic — say so in your working.
• Panicking at pH −0.30 for 2.0 M HCl. Negative pH is legitimate arithmetic — any strong monoprotic acid above 1 M produces it — though up there activity effects make the formula an approximation, which is why the calculator warns instead of refusing.
• Copying every digit off the display. Decimal places of pH = significant figures of concentration: from [H+] = 2.5e-3 M, report pH 2.60, not 2.602059991.
• Check the whole-number part against the exponent: [H+] = 4.2e-6 M must give a pH between 5 and 6, because 4.2e-6 sits between 1e-6 and 1e-5. If it doesn't, a sign slipped somewhere.
• Check the product: however you got there, [H+] × [OH−] must multiply back to 1.0e-14 at 25 °C, and pH + pOH must add to 14.00. The four-value readout above makes this a five-second audit.
• Check the direction: at 25 °C an acid must land below 7 and a base above it. An "acid" coming out basic means a swapped subtraction — or a dilute case that needed the charge balance, not the shortcut.