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pH Calculator

Free pH calculator — solve pH from [H+], [H+] from pH, pOH, and [OH-] with temperature-dependent Kw (CRC handbook). Worked examples.

pH Calculator

Solve for
Concentration of H+ (or H3O+) in mol/L. Used when solving for pH. Must be strictly positive — log of zero is undefined.
mol/L
Dimensionless pH value, typically 0 to 14 in dilute aqueous solutions at 25 °C. Used when solving for [H+] or pOH from pH.
Concentration of OH- in mol/L. Used when solving for pOH from [OH-]. Must be strictly positive.
mol/L
Temperature in Celsius. Defaults to 25 °C (pKw = 14.000). Affects the [H+]/[OH-]/pOH relationship via CRC handbook Kw table (0–60 °C).
°C
pH
3
Dimensionless pH value, defined as pH = −log10[H+]. At 25 °C, pH 7 is neutral; below 7 is acidic, above 7 is basic.
pOH
11
[H+]
0.001 mol/L
[OH-]
0 mol/L
Acidity category
1

Background.

This pH calculator computes the pH of an aqueous solution from its hydrogen-ion concentration, the inverse mapping from pH back to [H+], and the corresponding pOH and hydroxide-ion concentration — with a CRC-Handbook-backed correction for the temperature dependence of the water ion product Kw, because pH is one of those quantities that is silently temperature-dependent and the textbook answer of 'neutral water is pH 7' only holds at exactly 25 °C.

The pH scale was introduced by the Danish biochemist Søren Peder Lauritz Sørensen in his 1909 paper 'Über die Messung und die Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen' (Biochemische Zeitschrift 21, 131–304), originally as a pragmatic shorthand for the very small concentrations of H+ that arise in enzymatic and physiological work. Sørensen defined the quantity by pH = −log10[H+], turning a concentration that ranges over fourteen orders of magnitude (from roughly 1 mol/L in concentrated strong acid down to 10^−14 mol/L in concentrated strong base) into a single-digit number, which is why the scale is logarithmic.

The factor of ten between adjacent pH units is not a stylistic choice but a mathematical consequence of the definition: lemon juice at pH 2 is one hundred times more acidic in [H+] terms than orange juice at pH 4, and ten thousand times more acidic than tap water at pH 6. Pure water self-ionises into H+ and OH-, and at 25 °C the equilibrium constant for that reaction is Kw = [H+][OH-] = 1.0 × 10^−14, which forces [H+] = [OH-] = 10^−7 mol/L and therefore pH = 7 exactly at neutrality.

The standard '0 to 14' range that students memorise is the consequence of those bounds being where almost all dilute aqueous chemistry happens — but it is not a hard limit. Concentrated strong acids and bases can push pH outside the [0, 14] window, and at higher temperatures Kw rises significantly (pKw drops from 14.000 at 25 °C to about 13.617 at body temperature 37 °C and 13.262 at 50 °C), which means neutral water at 37 °C is not pH 7 but pH 6.81 — a small but measurable shift that matters for clinical chemistry, brewing, hot-spring geochemistry, and any thermostatted reactor.

This calculator implements all of that with a CRC-Handbook lookup of Kw between 0 °C and 60 °C, linearly interpolated between tabulated knots; outside that range it extrapolates conservatively. The four solve-for modes — pH from [H+], [H+] from pH, pOH from pH, pOH from [OH-] — cover every elementary acid-base bookkeeping question, and every result is internally consistent because the engine always computes the full quartet (pH, pOH, [H+], [OH-]) before returning.

What is ph calculator?

pH is defined as the negative base-10 logarithm of the hydrogen-ion activity in a solution, written pH = −log10(a_H+), or to a very good approximation in dilute solutions pH = −log10[H+] where [H+] is the molar concentration of hydrogen ions (more precisely the hydronium ion H3O+) in mol/L. The IUPAC operational definition (Compendium of Chemical Terminology, 'pH' entry, doi:10.1351/goldbook.P04524) ties the laboratory measurement to a chain of pH-buffer reference materials with assigned values, so 'pH' as you read it off an electrochemical meter is the operationally defined IUPAC pH, not literally −log10[H+]. For dilute aqueous work the two definitions agree to within the precision a typical meter can deliver. The companion quantity pOH = −log10[OH-] handles the basic end of the scale symmetrically. Because water self-ionises with equilibrium constant Kw = [H+][OH-], taking the negative log of both sides gives the fundamental relationship pH + pOH = pKw. At 25 °C, pKw = 14.000 exactly (NIST / IUPAC reference value), which is the origin of the familiar '0 to 14' pH scale and the rule that neutral water is pH 7. Kw is temperature-dependent — the self-ionisation reaction is endothermic, so heating water shifts the equilibrium toward more H+ and OH-, raising Kw and lowering pKw. The CRC Handbook of Chemistry and Physics (100th ed., Table 5-71) tabulates pKw from 0 °C (14.938) through 25 °C (14.000), 37 °C (13.617), 50 °C (13.262), to 60 °C (13.017). 'Neutral pH' — the pH at which [H+] = [OH-] — therefore drifts: 7.47 at 0 °C, 7.00 at 25 °C, 6.81 at body temperature 37 °C, 6.63 at 50 °C. Anything outside the [0, 14] band — concentrated 6 M HCl is around pH −0.8, and 1 M NaOH is around pH 14 — is still well-defined by the definition itself, but the simple [H+] = molarity-of-acid approximation breaks down at high concentrations because activity coefficients diverge from unity.

How to use this calculator.

  1. Pick a solve-for mode from the dropdown. 'pH from [H+]' is the usual lab calculation; 'pOH from [OH-]' is its base-side mirror; the two 'from pH' modes are useful when you have a meter reading and need the underlying concentrations.
  2. Enter the known quantity in SI units: concentrations in mol/L (molarity), not mmol/L or g/L. Convert if necessary: 1 mM = 0.001 mol/L.
  3. Adjust the temperature field if your solution is not at 25 °C. The calculator interpolates the CRC Handbook Kw table between 0 °C and 60 °C; outside that range it linearly extrapolates and should be treated as an estimate.
  4. Read the primary result (your solved-for quantity) and the three companion outputs — pH, pOH, [H+], [OH-] — which are always returned together and are internally consistent through pH + pOH = pKw.
  5. Check the acidity category code: 0 strong acid, 1 weak acid, 2 neutral (within ±0.05 of 7.0 at 25 °C), 3 weak base, 4 strong base. Use it to sanity-check your inputs.
  6. For a strong monoprotic acid (HCl, HNO3, HClO4) at concentrations above about 10^−6 mol/L, [H+] equals the acid molarity to excellent precision. For weak acids (acetic, citric, carbonic) you must first solve the equilibrium Ka expression and only then feed [H+] into this calculator.
  7. For very dilute solutions (below 10^−6 mol/L of strong acid), the contribution of water's own self-ionisation becomes significant and a simple −log10[acid concentration] underestimates the pH. Use the full charge-balance equation in that regime.

The formula.

pH = −log₁₀[H⁺]
pH · acidity value[H⁺] · hydrogen-ion concentration (mol/L)

The core relationships are:

pH = −log10[H+] (Sørensen definition) pOH = −log10[OH-] Kw = [H+] × [OH-] (water self-ionisation equilibrium) pKw = −log10(Kw) = pH + pOH (taking −log10 of the Kw expression) [H+] = 10^(−pH) [OH-] = 10^(−pOH) = Kw / [H+]

At 25 °C, Kw = 1.0 × 10^−14 mol²/L² exactly (the IUPAC/NIST reference value), so pKw = 14.000 and the familiar formulas pH + pOH = 14 and [H+][OH-] = 10^−14 are special cases. Outside 25 °C they are wrong by a measurable amount.

The calculator handles the temperature dependence by linearly interpolating between CRC Handbook (100th ed., Table 5-71) tabulated knots:

T (°C) | pKw -------|------- 0 | 14.938 10 | 14.535 15 | 14.346 20 | 14.167 25 | 14.000 30 | 13.833 37 | 13.617 40 | 13.535 50 | 13.262 60 | 13.017

For temperatures between two knots the engine uses straight linear interpolation; below 0 °C or above 60 °C it linearly extrapolates from the nearest interval, which is a deliberately conservative choice (the true Kw(T) curve is slightly nonlinear, captured better by the Marshall-Franck or Bandura-Lvov correlations, but the linear knot interpolation is accurate to better than 0.01 pH units across the tabulated range).

Four solve-for modes are dispatched on the 'solveFor' input:

• 'ph' — given [H+], compute pH = −log10[H+]. • 'hydrogenIonMolarity'— given pH, compute [H+] = 10^(−pH). • 'pohFromPh' — given pH, compute pOH = pKw(T) − pH. • 'pohFromMolarity' — given [OH-], compute pOH = −log10[OH-], then pH = pKw(T) − pOH.

In all four modes the engine returns the full quartet (pH, pOH, [H+], [OH-]) — every result is computed from a single self-consistent pH value, so pH + pOH always equals pKw at the requested temperature to floating-point precision. The engine refuses non-physical inputs: [H+] ≤ 0 and [OH-] ≤ 0 raise InvalidInputError (log of zero is −∞, log of a negative number is undefined), and pH outside [−2, 16] is rejected as outside any realistic aqueous range.

A worked example.

Example

Classic textbook problem: what is the pH of a 0.001 mol/L (1 mM) solution of HCl at 25 °C? HCl is a strong monoprotic acid — it dissociates essentially completely, so [H+] equals the acid molarity, 0.001 mol/L. Applying the Sørensen definition: pH = −log10(0.001) = −log10(10^−3) = 3.000 exactly. The calculator returns pH = 3, pOH = pKw − pH = 14.000 − 3.000 = 11.000, [H+] = 10^−3 = 0.001 mol/L (the input echoed back, as expected), and [OH-] = 10^−11 = 1.0 × 10^−11 mol/L. The acidity category is 1 (weak acid by the calculator's pH-band classification — note that 'weak acid' here is a label for the pH range 3 ≤ pH ≤ 6.95 in the displayed output, not a statement about HCl itself, which is by any chemistry definition a strong acid; the dilute solution simply produces a moderate, not extreme, pH). Two sanity checks. (1) Multiply [H+] × [OH-] = 10^−3 × 10^−11 = 10^−14 = Kw at 25 °C. (2) Confirm against the rule of thumb that −log10 of a power of ten is just the magnitude of the exponent: −log10(10^−3) = 3. Now change temperature to 37 °C (body temperature) with the same 0.001 M HCl: pH is still 3.000 (the definition uses only [H+]), but pOH = pKw(37 °C) − pH = 13.617 − 3.000 = 10.617, and [OH-] = 10^−10.617 ≈ 2.42 × 10^−11 mol/L — slightly higher than at 25 °C because Kw is larger. The acid solution is still as acidic in absolute [H+] terms, but the corresponding [OH-] companion shifts with temperature.

hydrogen Ion Molarity0.001
temperature25
solve Forph

Frequently asked questions.

What is the formula for pH?
pH = −log10[H+], where [H+] is the molar concentration of hydrogen (hydronium) ions in mol/L. Strictly, the IUPAC operational definition uses hydrogen-ion activity a_H+ rather than concentration, with pH = −log10(a_H+); in dilute aqueous solutions the two are equal to better than typical instrument precision. The companion definition for the base side is pOH = −log10[OH-], and the two are linked by pH + pOH = pKw, with pKw = 14.000 at 25 °C.
Why does pH range from 0 to 14?
The 0-to-14 range is not a hard mathematical limit; it is the range that covers almost all dilute aqueous chemistry. The lower end comes from the fact that 1 mol/L of a strong monoprotic acid gives [H+] ≈ 1 and therefore pH ≈ 0; the upper end comes from the fact that pH + pOH = pKw = 14.000 at 25 °C, so 1 mol/L of strong base gives pOH ≈ 0 and pH ≈ 14. Concentrated solutions push outside this window: 6 M HCl is around pH −0.8, and saturated NaOH around pH 15. At elevated temperatures the upper end shrinks because pKw falls — at 60 °C the equivalent range is roughly 0 to 13.
Why is the pH scale logarithmic?
Because [H+] in aqueous solutions ranges over fourteen-plus orders of magnitude — roughly 10^0 mol/L in concentrated strong acid to 10^−14 mol/L in concentrated strong base — and a logarithmic scale compresses that span into a single-digit number that is far easier to read, plot, and reason about. Sørensen introduced the convention in 1909 explicitly for that reason. The factor-of-ten consequence is that pH 4 is ten times more acidic than pH 5, and a hundred times more acidic than pH 6, which is the practical reason small pH differences matter so much in biology, brewing, and water treatment.
How does temperature affect pH?
Temperature affects Kw, the equilibrium constant for water's self-ionisation, which in turn affects what 'neutral' means. The self-ionisation reaction is endothermic, so heating water pushes the equilibrium toward more H+ and OH-, raising Kw and lowering pKw. CRC Handbook (Table 5-71) gives pKw = 14.938 at 0 °C, 14.000 at 25 °C, 13.617 at 37 °C, and 13.017 at 60 °C. Because neutrality is defined by [H+] = [OH-] and therefore pH = pKw/2, neutral water is pH 7.47 at 0 °C, pH 7.00 at 25 °C, pH 6.81 at body temperature, and pH 6.51 at 60 °C. The pH of an acid solution itself changes much less with temperature — the −log10[H+] definition does not have Kw in it — but the implicit reference point 'neutral = 7' moves.
What is the difference between a strong and weak acid in terms of pH?
A strong acid is one that dissociates essentially completely in water: HCl, HBr, HI, HNO3, HClO4, H2SO4 (first proton). For a strong monoprotic acid at concentrations above about 10^−6 mol/L, [H+] equals the acid molarity, so the pH is just −log10 of that molarity. A weak acid (acetic, citric, formic, lactic, carbonic, all amino acids) only partially dissociates; you need its acid dissociation constant Ka, and the pH comes from solving the equilibrium expression Ka = [H+][A-]/[HA]. The Henderson-Hasselbalch equation, pH = pKa + log10([A-]/[HA]), is the buffer-friendly form of that solution. This calculator does not solve weak-acid equilibria — it takes [H+] as an input. For a weak acid you must first determine [H+] from its Ka, then feed that into the pH calculator.
Why does my pH meter sometimes read above 14 or below 0?
Because the [0, 14] interval is a feature of dilute aqueous chemistry at 25 °C, not a property of the pH scale itself. Very concentrated strong acids and bases can produce pH values outside that window — 12 M HCl, for instance, reads around pH −1.1 — and a properly calibrated meter with an electrode rated for that range will report exactly that. The caveat is that activity coefficients diverge significantly from 1 at these concentrations, so the meter is reporting operational IUPAC pH (defined against the buffer chain) which is not numerically equal to −log10 of the analytical concentration. In ordinary educational chemistry you will almost never see this; in industrial process chemistry, geochemistry, or pickling and etching baths it is routine.
How is pH measured in the lab?
Two main methods. (1) A glass-electrode pH meter measures the potential difference between a hydrogen-ion-selective glass membrane and a reference electrode, calibrated against standard IUPAC pH-buffer reference materials (typically pH 4.01, 7.00, and 10.01 phosphate/phthalate/borate buffers at 25 °C); this is the operational IUPAC definition of pH. (2) Indicator dyes — phenolphthalein, methyl orange, universal indicator, litmus — give a colour change over a known pH range and are accurate to roughly half a pH unit by eye, much better with a spectrophotometer. Both methods are temperature-sensitive: a quality meter has automatic temperature compensation (ATC) so the reading reported is corrected back to the calibration temperature.
What is the relationship between [H+] and pH numerically?
[H+] = 10^(−pH), and conversely pH = −log10[H+]. Memorising a small table helps: pH 0 → [H+] = 1 mol/L, pH 1 → 0.1, pH 2 → 0.01, pH 3 → 10^−3, pH 7 → 10^−7, pH 10 → 10^−10, pH 14 → 10^−14. For non-integer pH values, e.g. pH 5.5, the concentration is 10^−5.5 ≈ 3.16 × 10^−6 mol/L — note that the antilog of half a pH unit is √10 ≈ 3.16, not 5. This is the most common arithmetic slip in introductory chemistry: people linearly interpolate pH between integer values when in fact each tenth of a pH unit is a factor of 10^0.1 ≈ 1.259 in concentration.
What is the difference between pH and pOH?
pH and pOH are mirror-image quantities for the acid and base sides of the same solution. pH = −log10[H+] tells you how acidic; pOH = −log10[OH-] tells you how basic. They are not independent — they are linked by the water self-ionisation constant Kw = [H+][OH-], which gives pH + pOH = pKw. At 25 °C, pKw = 14.000, so pOH = 14 − pH; at 37 °C, pOH = 13.617 − pH; at 0 °C, pOH = 14.938 − pH. Either quantity fully determines the other if you know the temperature. By convention chemists report pH (acidity-centric), but pOH is occasionally more natural for problems involving strong bases — and it is the quantity that determines the rate of base-catalysed reactions, just as pH determines acid-catalysed rates.
Can pH be negative or greater than 14?
Yes, both. pH = −log10[H+] is well-defined for any positive concentration, so [H+] > 1 mol/L gives negative pH and [OH-] > 1 mol/L gives pH > 14 at 25 °C. Concentrated 12 M HCl (about 37% by weight, the standard reagent bottle) has been measured at pH ≈ −1.1, and concentrated NaOH solutions reach pH well above 14. The familiar 0–14 'limits' are pedagogical, not physical. The complication at these extremes is that the simple [H+] = analytical-concentration approximation fails because activity coefficients move sharply away from 1 and ion-pairing becomes important; you need to use activities, not concentrations, to get the operational IUPAC pH right. Within dilute aqueous chemistry (say, [H+] between 10^−14 and 1 mol/L) the simple definition this calculator uses agrees with rigorous treatments to better than 0.05 pH units.

References& sources.

  1. [1]Sørensen, S. P. L. (1909). Über die Messung und die Bedeutung der Wasserstoffionenkonzentration bei enzymatischen Prozessen. Biochemische Zeitschrift, 21, 131–304. Original introduction of the pH notation and the pH = −log10[H+] definition.
  2. [2]IUPAC Compendium of Chemical Terminology (Gold Book), 2nd ed. — entry 'pH' (doi:10.1351/goldbook.P04524). International Union of Pure and Applied Chemistry. The operational definition of pH used by all modern pH-meter calibration standards.
  3. [3]Atkins, P. W. & de Paula, J. (2010). Physical Chemistry, 9th ed., §7B 'The response of equilibria to temperature' and §9C 'Acid–base equilibria'. Oxford University Press. ISBN 978-1-4292-1812-0. Derivation of Kw(T), the pH + pOH = pKw identity, and worked treatments of strong and weak acid pH.
  4. [4]Lide, D. R. (ed.) (2019). CRC Handbook of Chemistry and Physics, 100th ed., Table 5-71 'Ionization Constant of Water' — tabulated pKw values from 0 °C to 60 °C. CRC Press / Taylor & Francis. ISBN 978-1-138-36729-6. Source of the temperature-knot table used for Kw(T) interpolation in this calculator.
  5. [5]Bates, R. G. (1973). Determination of pH: Theory and Practice, 2nd ed. John Wiley & Sons. The standard reference on operational pH measurement, electrode calibration, and the link between activity-based and concentration-based pH definitions.
  6. [6]Bandura, A. V. & Lvov, S. N. (2006). The Ionization Constant of Water over Wide Ranges of Temperature and Density. Journal of Physical and Chemical Reference Data, 35(1), 15–30. Peer-reviewed NIST-affiliated reference for Kw(T) at higher temperatures and pressures than the CRC table covers.
  7. [7]Buck, R. P., Rondinini, S., Covington, A. K., Baucke, F. G. K., Brett, C. M. A., Camões, M. F., Milton, M. J. T., Mussini, T., Naumann, R., Pratt, K. W., Spitzer, P. & Wilson, G. S. (2002). Measurement of pH. Definition, standards, and procedures (IUPAC Recommendations 2002). Pure and Applied Chemistry, 74(11), 2169–2200. The current authoritative IUPAC recommendations on pH measurement that supersede the original Sørensen definition for laboratory practice.
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