What the solubility product describes
Ksp is the equilibrium constant for a solid dissolving into its ions. For AgCl(s) ⇌ Ag+(aq) + Cl−(aq) it is simply [Ag+][Cl−] at saturation. The solid does not appear, because a pure solid has activity 1 no matter how much of it is sitting at the bottom of the beaker — which is why adding more powder to a saturated solution changes nothing.
That absence is the whole point. Because Ksp is a product of ion concentrations rather than a solubility in grams, it stays constant even when the ions come from different sources. Chloride from sodium chloride counts exactly as much as chloride from the dissolving silver chloride. This is what lets one constant predict solubility in pure water, solubility in a brine, and whether two solutions will produce a precipitate when you mix them.
Under the modern IUPAC Green Book convention Ksp is dimensionless: each concentration is a ratio to the 1 mol/L standard state before the exponent is applied. Practically, that means you feed the expression concentrations in mol/L and quote the answer without units, with a temperature attached. Nearly all published values are for 25 °C, and most salts become more soluble as temperature rises, so a table value used at 60 °C will underestimate what dissolves.
From Ksp to solubility, and why the exponents dominate
Write the dissolution equation first, because the subscripts become both the coefficients and the exponents. A salt AxBy dissolving to the extent of s mol/L releases x s of the cation and y s of the anion, so
and inverting gives s = (Ksp / (xxyy))1/(x+y). Two consequences deserve attention.
You cannot rank solubility by Ksp across different stoichiometries. A 1:1 salt with Ksp = 1.0 × 10−10 has s = 1.0 × 10−5 M. A 1:2 salt with the identical Ksp has s = (10−10/4)1/3 = 2.92 × 10−4 M — nearly thirty times more soluble on the same constant. The exponent 1/(x+y) flattens differences in Ksp and the xxyy prefactor shifts them further. Comparing constants directly is only legitimate between salts of the same type.
A common ion suppresses solubility, and the exponent decides how hard. If the solution already contains C mol/L of the B ion, the anion term becomes (y s + C) and the equation is no longer a simple power. Textbooks approximate it by assuming y s is negligible against C, which is usually excellent but fails when C is small or the salt is relatively soluble. This calculator solves the full expression by bisection, so the answer stays right in both regimes.
Worked example: silver chloride in water and in 0.100 M NaCl
Silver chloride has Ksp = 1.8 × 10−10 at 25 °C and a molar mass of 143.32 g/mol.
- Write the dissolution. AgCl(s) ⇌ Ag+ + Cl−, so x = 1 and y = 1.
- Substitute the extent. [Ag+] = s and [Cl−] = s, giving Ksp = s2.
- Solve. s = √(1.8 × 10−10) = 1.3416 × 10−5 M.
- Convert to mass. 1.3416 × 10−5 × 143.32 = 1.923 × 10−3 g/L, about 1.9 mg in a litre.
Now repeat in 0.100 M sodium chloride, where chloride is already present.
- Rewrite the expression. [Ag+] = s but [Cl−] = s + 0.100, so s(s + 0.100) = 1.8 × 10−10.
- Solve the quadratic. s = [−0.100 + √(0.0100 + 7.2 × 10−10)] ÷ 2 = 1.800 × 10−9 M.
- Compare. 1.3416 × 10−5 ÷ 1.800 × 10−9 = 7,453. The common ion has cut solubility by a factor of about 7,450.
- Check the shortcut. Assuming s is negligible against 0.100 gives s = 1.8 × 10−10/0.100 = 1.8 × 10−9 M, identical to four figures — here the approximation is safe because s is nine orders of magnitude below C.
Finally, test a precipitation. Mix equal volumes of 2.0 × 10−3 M AgNO3 and 2.0 × 10−3 M NaCl. Mixing halves each concentration, so both ions arrive at 1.0 × 10−3 M. Qsp = (1.0 × 10−3)2 = 1.0 × 10−6, which exceeds Ksp by a factor of 5,556. The saturation index is log10(5,556) = +3.74, so a precipitate forms and keeps forming until the ion product falls back to 1.8 × 10−10.
Reading the saturation index
The saturation index — log10 of the ion product divided by the solubility product — is the number water chemists and scaling engineers actually watch, because it is scale-free and additive. Zero means exactly saturated. A negative value means the water can dissolve more of that mineral, so existing scale slowly disappears and bare metal is at risk of corrosion. A positive value means the water is supersaturated and will deposit scale, given a nucleation site and time.
Magnitude matters less than sign in the first instance, because the index is logarithmic: an index of +1 means the ion product is ten times the equilibrium value, and +3.74 means 5,556 times. Real waters routinely sit slightly supersaturated for calcium carbonate without depositing anything, because nucleation is slow and the barrier to forming the first crystal is high. That is a kinetic effect, not a thermodynamic one, and it is why a positive index predicts that scale can form rather than that it has formed.
The molar solubility output tells you the other half of the story: how much salt a saturated solution holds. Convert it to g/L before comparing with a specification or a solubility figure from a supplier, because product data sheets almost never quote molar values. If the calculated solubility exceeds about 0.1 M, treat the whole calculation as indicative only — at that concentration ion pairing and activity coefficients diverge from ideality and the tabulated constant will no longer reproduce a measured solubility. For genuinely soluble salts, use a measured solubility curve rather than a Ksp.
Salt type, the Ksp expression, and solubility at Ksp = 1.0 × 10⁻¹⁰
| Salt type | Example | Ions released | Ksp in terms of s | s at Ksp = 1.0 × 10⁻¹⁰ |
|---|---|---|---|---|
| AB | AgCl, BaSO₄ | 1 + 1 | s² | 1.000 × 10⁻⁵ M |
| AB₂ or A₂B | CaF₂, Ag₂CrO₄ | 1 + 2 | 4s³ | 2.924 × 10⁻⁴ M |
| AB₃ or A₃B | Fe(OH)₃ | 1 + 3 | 27s⁴ | 1.387 × 10⁻³ M |
| A₃B₂ or A₂B₃ | Ca₃(PO₄)₂, Bi₂S₃ | 2 + 3 | 108s⁵ | 3.922 × 10⁻³ M |
| AB₄ or A₄B | Ag₄[Fe(CN)₆] type | 1 + 4 | 256s⁵ | 3.299 × 10⁻³ M |
Read across: five salts sharing one solubility product differ by a factor of 390 in molar solubility. Comparing Ksp values across salt types is meaningless without this correction.
Published Ksp values disagree with each other
Different compilations list noticeably different solubility products for the same salt, because they come from different measurement methods, different ionic strengths, and different extrapolations to zero ionic strength. Discrepancies of a factor of two or three between a general chemistry appendix and a critically evaluated thermodynamic table are ordinary, and for hydroxides and sulfides they can be far larger.
So: use one source consistently, record which one you used, and never mix a Ksp from one table with a Ka or complexation constant from another when the two feed the same calculation. If a computed solubility has to be defensible — in a pharmaceutical formulation or a discharge consent — measure it rather than compute it.
Mistakes that produce a wrong solubility
- Forgetting the coefficient inside the bracket. For Ag2CrO4 the silver term is (2s)2 = 4s2, not s2. Dropping the 2 changes the answer by a factor of 41/3.
- Forgetting to allow for dilution on mixing. Combining equal volumes halves both concentrations before the ion product is formed. This single omission quadruples Qsp for a 1:1 salt.
- Including the solid in the expression. A pure solid has activity 1. Its amount never appears, which is why adding more powder does not shift the equilibrium.
- Comparing Ksp values across salt types. As the table above shows, identical constants can mean solubilities that differ by more than two orders of magnitude.
- Ignoring acid-base or complexation side reactions. Carbonates, phosphates, hydroxides and sulfides all dissolve far more readily in acid because the anion is protonated and removed; silver halides dissolve in ammonia because the cation is complexed. A bare Ksp models neither.
- Using a 25 °C table value at another temperature. Most salts become more soluble on heating, and a few — calcium sulfate and calcium carbonate among them — become less soluble, which is exactly why they scale hot surfaces.
- Applying Ksp at high ionic strength. Activity coefficients fall well below 1 in concentrated brine, so real solubility exceeds the ideal prediction. Analytical work uses conditional constants measured at a fixed ionic strength instead.
What this calculator does and does not model
It models one salt dissolving into two ion types, with optional additional anion from another source, treating concentrations as activities. It solves the common-ion case exactly rather than assuming the dissolved contribution is negligible, which matters when the added ion is at a similar level to the intrinsic solubility.
It does not model protonation of the anion, hydrolysis of the cation, complex formation, ion pairing, or incongruent dissolution. Those are the reasons a laboratory solubility often exceeds the Ksp prediction, sometimes by orders of magnitude. If your anion is a weak base — carbonate, phosphate, sulfide, hydroxide — the pH of the solution belongs in the calculation, and you should pair this tool with the pH calculator and a stepwise treatment of the acid equilibrium.
The precipitation test assumes the trial concentrations you enter are the concentrations after mixing. It does not divide by two for you, because it has no way to know your volumes. Work out the diluted concentrations first with the solution dilution calculator if the mixing ratio is not 1:1.
Where Ksp sits among the other constants
Ksp is an ordinary equilibrium constant with the solid omitted, so everything you know about K applies. The comparison of Qsp against Ksp is the same test as Q against K in the reaction quotient calculator, and the general machinery for solving a composition from a constant is in the equilibrium constant calculator.
In practice Ksp is most often used alongside another equilibrium rather than alone. Gravimetric analysis picks a precipitating agent whose Ksp is small enough that losses to the filtrate are negligible, and then checks that competing complexation does not redissolve the product. Water treatment combines carbonate solubility with the carbonic acid equilibria to decide whether a supply will scale or corrode. Pharmaceutical formulation uses the common ion effect deliberately, adding a counter-ion to keep a salt from crystallising out of an injection during storage.
For gases dissolving rather than solids, the governing relation is Henry's law rather than a solubility product, and for a solute partitioning between two liquids it is a distribution coefficient. Both are equilibrium constants too; only the standard states differ. Once you can read one equilibrium expression you can read them all.