Polyprotic Acid Equilibrium Solver
Find the exact pH and complete species distribution of any diprotic or triprotic acid — carbonic, phosphoric, oxalic, sulfuric, citric and more — from its total concentration and Ka1/Ka2/Ka3, solved with the full charge-balance equation instead of the simplified Ka1-only shortcut. Or explore the species distribution diagram at any chosen pH.
Choose an acid type and calculation, then enter the known values.
Equilibrium pH
[H+] at equilibrium: 1.189 x 10^-4 M
99.64%
H2CO3
0.36%
HCO3-
0%
CO3(2-)
6.367
pKa1
10.319
pKa2
Species Distribution Diagram
Shows what percent of the total acid exists as each species across the full 0–14 pH range, using the exact alpha-fraction formula.
Step-by-Step Polyprotic Equilibrium Calculation
Here's exactly how this answer was calculated, one step at a time.
Given: H2CO3, C = 0.033 mol/L, Ka1 = 4.300 x 10⁻⁷, Ka2 = 4.800 x 10⁻¹¹
Step 1: Write out every dissociation step
A polyprotic acid loses its protons one at a time, and each step has its own, progressively smaller, equilibrium constant.
H2CO3 <-> H+ + HCO3-, Ka1 = 4.300 x 10⁻⁷ HCO3- <-> H+ + CO3(2-), Ka2 = 4.800 x 10⁻¹¹Step 2: Set up the full charge balance
Instead of assuming only the first step matters, this equation accounts for the H+ contributed by every dissociation step at once.
[H+] = [OH-] + [HCO3-] + 2.[CO3(2-)]Step 3: Solve the charge balance numerically
Solving for [H+] (bisection on the exact equation) gives [H+] = 1.189 x 10⁻⁴ MStep 4: Convert to pH
pH = -log10[H+] = 3.925Step 5: Find the alpha (distribution) fraction of every species
alpha_i = [H+]^2.Ka-product / D, where D is the sum of every possible [H+]/Ka combination — this is exact at any pH and does not depend on concentration.
alpha(H2CO3) = 99.6397% alpha(HCO3-) = 0.3603% alpha(CO3(2-)) = 0%Step 6: Convert fractions into actual concentrations
[H2CO3] = 99.6397% x 0.033 M = 3.288 x 10⁻² M [HCO3-] = 0.3603% x 0.033 M = 1.189 x 10⁻⁴ M [CO3(2-)] = 0% x 0.033 M = 4.800 x 10⁻¹¹ M
Calculated pH:
3.925
Polyprotic Acid Equilibrium Solver: Full pH and Species Distribution, Not Just a Guess at Ka1
This free polyprotic acid equilibrium solver is built for chemistry students, lab technicians, and anyone working with acids that can hand off more than one proton — carbonic acid, phosphoric acid, oxalic acid, sulfuric acid, citric acid, and more. Most simplified calculators only look at Ka1 and quietly assume every later dissociation step is too small to matter. This tool solves the real charge-balance equation across all the dissociation steps at once, then reports the exact pH and the full breakdown of how much of the acid exists as each species at equilibrium.
Two calculation modes cover the situations you'll actually run into. The first finds the equilibrium pH and complete species distribution of a diprotic or triprotic acid from its total concentration and Ka values. The second flips the question around: given only the Ka values, it shows what percent of the acid exists as each species at any pH you choose, with a full distribution diagram plotted across the entire 0–14 pH range.
What Makes an Acid 'Polyprotic'?
A polyprotic acid is one molecule that can release more than one hydrogen ion, one at a time, in a series of separate equilibrium steps. A diprotic acid like carbonic acid (H2CO3) has two steps: H2CO3 loses a proton to become bicarbonate (HCO3-), and bicarbonate can lose a second proton to become carbonate (CO3^2-). A triprotic acid like phosphoric acid (H3PO4) goes through three steps, ending at phosphate (PO4^3-).
Each step has its own equilibrium constant — Ka1, Ka2, and (for triprotic acids) Ka3 — and each one is almost always much smaller than the last, often by four to six orders of magnitude. That's because pulling a second or third positively charged proton away from an already negatively charged ion takes noticeably more energy than removing the first proton from a neutral molecule.
Why the 'Only Ka1 Matters' Shortcut Falls Short
A common textbook shortcut treats a polyprotic acid as if it were a simple monoprotic acid, solving only the first dissociation step with the usual ICE-table quadratic and ignoring everything after it. For the pH itself, this is often a reasonable approximation, since Ka2 and Ka3 are usually far too small to add meaningfully more H+ to the solution.
But 'close enough for pH' is not the same as 'a full picture of what's actually in the beaker.' At a given pH, real solutions contain a mix of every species simultaneously — some fully protonated acid, some partially deprotonated, and (in small amounts) fully deprotonated. Questions about buffer behavior, titration curves, and which form of an acid dominates at a particular pH all require knowing the full species distribution, not just the final pH number.
The Exact Charge-Balance Approach This Calculator Uses
Instead of assuming only the first step counts, this solver sets up the full charge balance for the solution: [H+] = [OH-] + [HA(n-1)-] + 2.[A(n-2)-] + 3.[A(n-3)-] and so on, where each later species contributes proportionally more H+ per molecule because it required giving up more protons to form. It then finds the exact value of [H+] that satisfies this equation using a numerical root-finding method (bisection), rather than a hand-wavy shortcut.
Once [H+] is known, the fraction of the total acid present as each species — called the alpha (distribution) fraction — is calculated with alpha_i = (product of the first i Ka values x [H+]^(n-i)) divided by the sum of every such term across all species. These alpha fractions are a clean, well-established result from equilibrium chemistry and, notably, don't depend on the total concentration at all — only on [H+] and the Ka values.
Reading the Species Distribution Diagram
The distribution diagram plots the percent of the total acid present as each species across the full pH range from 0 to 14. At very low pH, essentially all of the acid sits in its fully protonated form. As pH rises, each species rises to a peak, then falls away as the next species in the sequence takes over — and every crossover point where two neighboring species are present in exactly equal amounts happens precisely at that step's pKa.
This makes the diagram genuinely useful beyond just answering 'what's the pH': it shows exactly which species dominates at any pH of interest, which is the key question behind choosing a buffer system, interpreting a titration curve, or understanding how a molecule's charge state (and therefore its chemistry) shifts across a pH range.
Common Polyprotic Acids and Their Ka Values
Carbonic acid (H2CO3), formed when CO2 dissolves in water, has Ka1 around 4.3 x 10^-7 and Ka2 around 4.8 x 10^-11 — the carbonic acid/bicarbonate pair is the primary buffer that keeps human blood pH stable near 7.4. Phosphoric acid (H3PO4) is triprotic with Ka1 = 7.5 x 10^-3, Ka2 = 6.2 x 10^-8, and Ka3 = 4.2 x 10^-13, and its second step (H2PO4-/HPO4^2-) is the backbone of the phosphate buffer widely used in biochemistry and cell biology labs.
Oxalic acid (H2C2O4) and sulfurous acid (H2SO3) are both meaningfully stronger diprotic acids, with Ka1 values around 10^-2, so their first dissociation step approaches the behavior of a strong acid even though the second step is still weak. Sulfuric acid (H2SO4) is unusual: its first proton comes off essentially completely (Ka1 is enormous), while its second dissociation, to sulfate (SO4^2-), behaves like an ordinary weak acid with Ka2 around 1.2 x 10^-2. Citric acid, a triprotic acid found in citrus fruit, has all three Ka values fall within a fairly narrow range, which is part of why it makes such an effective buffer across a wide pH window.
Where Polyprotic Acid Equilibrium Actually Gets Used
In human physiology, the carbonic acid/bicarbonate equilibrium is the body's main defense against pH swings in blood, constantly shifting between H2CO3, HCO3-, and dissolved CO2 to buffer against acid or base loads. In biochemistry and molecular biology labs, phosphate buffers built around the H2PO4-/HPO4^2- pair are a standard choice for keeping reaction mixtures, cell culture media, and protein purification steps at a stable, biologically relevant pH.
In food chemistry, citric, oxalic, and other polyprotic organic acids determine the tartness, preservation, and buffering capacity of citrus juices, wine, and many processed foods. In environmental and geochemistry, carbonic acid equilibrium controls the pH of rainwater, rivers, and the ocean, and shifts in that equilibrium (from rising atmospheric CO2) are the direct chemical mechanism behind ocean acidification.
Common Mistakes to Avoid
The most frequent error is treating Ka2 (or Ka3) as if it contributes the same order of magnitude of H+ as Ka1 — in reality these later constants are almost always far smaller, and lumping them together in the wrong step of a calculation produces a badly wrong pH. Another common mistake is looking up a species' concentration using only the total concentration C, without applying the correct alpha fraction for the pH in question — a species' alpha fraction changes constantly as pH shifts, even though C stays fixed.
It's also easy to mix up Ka and pKa when reading a reference table for a polyprotic acid with three separate values to track — always double check which of Ka1/Ka2/Ka3 (or pKa1/pKa2/pKa3) a source is actually reporting, since a factor-of-ten mistake anywhere in the chain compounds through the whole calculation.
Polyprotic Acid Equilibrium Solver: Quick Reference Summary
For an acid with n dissociation steps and constants Ka1 through Kan, the alpha fraction of species i is alpha_i = (Ka1.Ka2...Ka_i x [H+]^(n-i)) / D, where D is the sum of that same term across every species (i = 0 to n). The true equilibrium [H+] for a given total concentration C solves the charge balance [H+] = [OH-] + sum of (i x alpha_i x C) across all deprotonated species — this calculator solves that equation exactly rather than assuming only the first step matters.
This free polyprotic acid equilibrium solver is intended to support learning, buffer planning, and everyday chemistry questions. Always confirm the acid you're working with is genuinely polyprotic with well-established Ka values before relying on these formulas, and for safety-critical, regulated, clinical, or industrial work, always confirm results with validated lab instruments and your organization's approved procedures.
Frequently Asked Questions
What is a polyprotic acid?
A polyprotic acid is one that can donate more than one hydrogen ion (proton), one at a time, through a series of separate equilibrium steps — each with its own Ka value. Diprotic acids like H2CO3 have two steps; triprotic acids like H3PO4 have three.
Why is Ka2 always so much smaller than Ka1?
Removing a proton from an already negatively charged ion takes more energy than removing it from a neutral molecule, because the remaining negative charge attracts the departing (positive) proton more strongly. This makes each successive Ka value smaller, typically by several orders of magnitude.
Does the pH of a polyprotic acid depend on Ka2 and Ka3?
Usually only very slightly. Because later Ka values are so much smaller than Ka1, the first dissociation step almost always sets the pH. This calculator still solves the exact charge balance including every step, so any small contribution from later steps is captured too.
What is an alpha fraction?
The alpha (distribution) fraction is the percent of the total dissolved acid present as one particular species at a given pH. Alpha fractions always sum to 100% and, unlike concentration, depend only on pH and the Ka values, not on the total amount of acid dissolved.
Why does each species' fraction peak and then fall as pH rises?
At low pH, the fully protonated form dominates. As pH rises past each pKa, that step's equilibrium shifts toward the next, more deprotonated species, so each species rises to a peak near its own pKa range and then declines as the next species takes over.
What's the connection between pKa and the crossover points on the distribution diagram?
At pH = pKa for any given step, the two species on either side of that step are present in exactly equal amounts (50% each). This is the same relationship used in the Henderson–Hasselbalch equation for buffers.
Can this calculator handle sulfuric acid, where the first proton is essentially fully dissociated?
Yes — enter a very large Ka1 (sulfuric acid's first dissociation behaves like a strong acid) alongside the normal Ka2, and the solver will correctly treat the first step as essentially complete while still solving the second step as a genuine weak-acid equilibrium.
How is this different from the Weak Acid pH & Ka Calculator?
The Weak Acid pH & Ka Calculator models a single dissociation step (one Ka). This solver is built specifically for acids with two or three sequential dissociation steps, and additionally reports the full species distribution and distribution diagram across every step, not just the final pH.