Solubility Product Constant (Ksp) Equation, Derivation & Rearrangements

Core idea

Overview

The Solubility Product Constant (Ksp) quantifies the extent to which an ionic compound dissolves in water, representing the equilibrium between the solid and its constituent ions in a saturated solution. It is a specific type of equilibrium constant, applicable only to sparingly soluble salts. The Ksp value indicates the maximum product of ion concentrations that can exist in solution before precipitation occurs, providing a crucial measure for predicting solubility and precipitation.

When to use: This equation is used to determine the Ksp value from equilibrium ion concentrations, or to calculate the concentration of an ion in a saturated solution given the Ksp. It is essential for predicting whether a precipitate will form when two solutions are mixed, by comparing the ionic product (Qsp) with Ksp. Always ensure the solution is saturated or at equilibrium for Ksp calculations.

Why it matters: Understanding Ksp is vital in various fields, including environmental chemistry for assessing water quality and pollutant solubility, and in analytical chemistry for designing precipitation reactions. In medicine, it helps in understanding the formation of kidney stones (e.g., calcium oxalate) and in pharmacology for formulating drugs with optimal solubility. Industrially, it's used in processes like water softening and mineral extraction.

Symbols

Variables

[ $A^{m +}$ ] = Concentration of Cation, [ $B^{n -}$ ] = Concentration of Anion, p = Stoichiometric Coefficient of Cation, q = Stoichiometric Coefficient of Anion, $K_{s p}$ = Solubility Product Constant

[A^{m +}]

Concentration of Cation

M

[B^{n -}]

Concentration of Anion

M

p

Stoichiometric Coefficient of Cation

Variable

q

Stoichiometric Coefficient of Anion

Variable

K_{s p}

Solubility Product Constant

Variable

Walkthrough

Derivation

Formula: Solubility Product Constant (Ksp)

The Ksp expression is derived from the general equilibrium constant for the dissolution of a sparingly soluble ionic solid.

The solution is at equilibrium, meaning the rate of dissolution equals the rate of precipitation.
The ionic compound is sparingly soluble, so the concentration of the undissolved solid remains constant and is not included in the Ksp expression.
The temperature is constant, as Ksp is temperature-dependent.
Ideal behavior of ions in solution (no significant interionic attractions).

1

Start with the Dissolution Equilibrium:

For a generic sparingly soluble ionic compound $A_{p} B_{q}$ , it dissociates into $p$ moles of cation $A^{m +}$ and $q$ moles of anion $B^{n -}$ in aqueous solution.

A_{p} B_{q} (s) ⇌ p A^{m +} (a q) + q B^{n -} (a q)

2

Write the General Equilibrium Constant Expression:

The equilibrium constant ( $K_{c}$ ) is defined as the ratio of the product of the concentrations of products raised to their stoichiometric coefficients to the product of the concentrations of reactants raised to their stoichiometric coefficients.

K_{c} = \frac{[ A ^{m +} ] ^{p} [ B ^{n -} ] ^{q}}{[ A _{p} B _{q} ( s )]}

3

Simplify for Solids:

Since $A_{p} B_{q} (s)$ is a pure solid, its concentration is constant and is incorporated into the equilibrium constant. Therefore, the new constant is called the Solubility Product Constant, $K_{s p}$ .

K_{s p} = [A^{m +}]^{p} [B^{n -}]^{q}

Result

K_{s p} = [A^{m +}]^{p} [B^{n -}]^{q}

Source: AQA A-level Chemistry — Physical Chemistry (3.1.9.2 Solubility products)

Visual intuition

Graph

The graph follows a power law curve where the solubility product constant increases exponentially from the origin as the concentration of the cation increases. For a chemistry student, this shape demonstrates that even a small increase in the concentration of the cation results in a significantly larger solubility product constant, indicating a much higher degree of ionic dissociation. The most important feature of this curve is the exponential growth, which means that the solubility product constant is highly sensitive to changes in the concentration of the cation rather than changing at a constant linear rate.

Graph type: power_law

Why it behaves this way

Intuition

Imagine a solid crystal immersed in water, with ions continuously breaking away from its surface into the solution while other ions from the solution simultaneously reattach to the crystal surface, eventually reaching a

K_{s p}

The equilibrium constant for the dissolution of a sparingly soluble ionic compound, representing the product of ion concentrations in a saturated solution.

A higher Ksp means the compound is more soluble, allowing higher concentrations of its ions to exist in solution before precipitation. It sets the upper limit for the product of ion concentrations.

[A^{m +}]

Molar concentration of the cation A at equilibrium in a saturated solution.

This value indicates how many moles of cation A are dissolved per liter of solution when no more solid can dissolve.

[B^{n -}]

Molar concentration of the anion B at equilibrium in a saturated solution.

This value indicates how many moles of anion B are dissolved per liter of solution when no more solid can dissolve.

p

Stoichiometric coefficient of the cation A from the balanced dissolution equation.

This exponent accounts for the fact that if a compound produces 'p' moles of cation A for every mole dissolved, the concentration of A effectively contributes 'p' times (as [A]^p) to the overall ion product.

q

Stoichiometric coefficient of the anion B from the balanced dissolution equation.

This exponent accounts for the fact that if a compound produces 'q' moles of anion B for every mole dissolved, the concentration of B effectively contributes 'q' times (as [B]^q) to the overall ion product.

Signs and relationships

p, q (exponents): The exponents 'p' and 'q' are positive because they represent the stoichiometric coefficients of the product ions in the dissolution reaction.

Free study cues

Insight

Canonical usage

Ksp is technically dimensionless when defined via activities, though in many educational contexts, units are derived from the product of molar concentrations based on the stoichiometry of the salt.

Common confusion

Assuming Ksp has a universal unit; the 'units' change depending on the number of ions in the balanced dissolution equation.

Dimension note

In rigorous chemical thermodynamics, Ksp is a ratio of activities. Since activity is the ratio of a concentration to the standard state concentration (c°), the units cancel out.

Unit systems

$[A^{m} +]$ mol dm^-3 · Molar concentration of the cation at equilibrium.

$[B^{n} -]$ mol dm^-3 · Molar concentration of the anion at equilibrium.

One free problem

Practice Problem

A sparingly soluble salt, $P b C l_{2}$ , dissolves according to the equilibrium $P b C l_{2} (s) ⇌ P b^{2 +} (a q) + 2 C l^{-} (a q)$ . If the equilibrium concentration of $P b^{2 +}$ ions in a saturated solution is $1.6 \times 1 0^{- 2} M$ , calculate the solubility product constant ( $K_{s p}$ ) for $P b C l_{2}$ .

Concentration of Cation0.016 M

Concentration of Anion0.032 M

Stoichiometric Coefficient of Cation1

Stoichiometric Coefficient of Anion2

Solve for: Ksp

Hint: Remember to account for the stoichiometry of the chloride ions when calculating their concentration and raising it to the correct power.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

When predicting the formation of scale (e.g, Solubility Product Constant (Ksp) is used to calculate Solubility Product Constant from Concentration of Cation, Concentration of Anion, and Stoichiometric Coefficient of Cation. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Always write the balanced dissolution equilibrium equation first to determine the stoichiometric coefficients (p and q).
Remember that Ksp is temperature-dependent; its value changes with temperature.
Pure solids and liquids are not included in the Ksp expression.
Pay close attention to the units of concentration (Molarity, M) and ensure they are consistent.
The common ion effect will decrease the solubility of a sparingly soluble salt, but Ksp remains constant at a given temperature.

Avoid these traps

Common Mistakes

Forgetting to raise ion concentrations to their stoichiometric powers (p and q).
Incorrectly determining the ion concentrations from the molar solubility, especially when stoichiometry is not 1:1.
Confusing Ksp with molar solubility (s); Ksp is a constant, while s is a concentration.
Not considering the common ion effect when calculating solubility in solutions already containing one of the ions.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

The Ksp expression is derived from the general equilibrium constant for the dissolution of a sparingly soluble ionic solid.

This equation is used to determine the Ksp value from equilibrium ion concentrations, or to calculate the concentration of an ion in a saturated solution given the Ksp. It is essential for predicting whether a precipitate will form when two solutions are mixed, by comparing the ionic product (Qsp) with Ksp. Always ensure the solution is saturated or at equilibrium for Ksp calculations.

Understanding Ksp is vital in various fields, including environmental chemistry for assessing water quality and pollutant solubility, and in analytical chemistry for designing precipitation reactions. In medicine, it helps in understanding the formation of kidney stones (e.g., calcium oxalate) and in pharmacology for formulating drugs with optimal solubility. Industrially, it's used in processes like water softening and mineral extraction.

Forgetting to raise ion concentrations to their stoichiometric powers (p and q). Incorrectly determining the ion concentrations from the molar solubility, especially when stoichiometry is not 1:1. Confusing Ksp with molar solubility (s); Ksp is a constant, while s is a concentration. Not considering the common ion effect when calculating solubility in solutions already containing one of the ions.

When predicting the formation of scale (e.g, Solubility Product Constant (Ksp) is used to calculate Solubility Product Constant from Concentration of Cation, Concentration of Anion, and Stoichiometric Coefficient of Cation. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Always write the balanced dissolution equilibrium equation first to determine the stoichiometric coefficients (p and q). Remember that Ksp is temperature-dependent; its value changes with temperature. Pure solids and liquids are not included in the Ksp expression. Pay close attention to the units of concentration (Molarity, M) and ensure they are consistent. The common ion effect will decrease the solubility of a sparingly soluble salt, but Ksp remains constant at a given temperature.

References

Sources

Atkins, P. W., de Paula, J., & Keeler, J. (2018). Atkins' Physical Chemistry (11th ed.). Oxford University Press.
International Union of Pure and Applied Chemistry. "Solubility product." Compendium of Chemical Terminology, Gold Book.
Wikipedia: Solubility product constant
IUPAC Gold Book
Atkins' Physical Chemistry
Cambridge International AS and A Level Chemistry Coursebook
AQA A-level Chemistry — Physical Chemistry (3.1.9.2 Solubility products)

Solubility Product Constant (Ksp)

Overview

Variables

Derivation

Start with the Dissolution Equilibrium:

Write the General Equilibrium Constant Expression:

Simplify for Solids:

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Ionic Product (Qsp)

Common Ion Effect

Frequently Asked Questions

Sources