Ionic Product of Water ([H⁺][OH⁻]) Equation, Derivation & Rearrangements

Core idea

Overview

The ionic product of water, Kw, is the equilibrium constant for the self-ionization of water into hydronium and hydroxide ions. At a standard temperature of 25°C, this constant remains fixed at 1.0 × 10⁻¹⁴, defining the inverse relationship between acidity and alkalinity in aqueous solutions.

When to use: Use this equation whenever you need to convert between hydrogen ion concentration and hydroxide ion concentration in an aqueous environment. It is valid for acidic, basic, and neutral solutions, provided the system has reached chemical equilibrium.

Why it matters: This equation forms the mathematical basis for the pH scale, which is essential for monitoring environmental water quality and human physiology. It ensures that any change in acidity is balanced by a corresponding change in basicity, maintaining chemical stability in biological cells.

Symbols

Variables

[H^+] = [H+], [OH^-] = [OH-], K_w = Kw Constant

[H^{+}]

[H+]

M

[O H^{-}]

[OH-]

M

K_{w}

Kw Constant

V a r iab l e

Walkthrough

Derivation

Understanding Ionic Product of Water (Kw)

Kw is the equilibrium constant for water auto-ionisation and is temperature dependent.

Kw changes with temperature.

1

State the Expression:

At 298 K, Kw is approximately 1.0 $\times$ 10^{-14}.

K_{w} = [H^{+}] [O H^{-}]

Result

K_{w} = [H^{+}] [O H^{-}]

Source: AQA A-Level Chemistry — Acids and Bases

Free formulas

Rearrangements

Solve for $[H^{+}]$

Make [H^+] the subject

h = \frac{K _{w}}{o h}

To make [H⁺] the subject of the ionic product of water equation, divide both sides by [OH⁻].

Difficulty: 2/5

Solve for $[O H^{-}]$

Make [OH^-] the subject

o h = \frac{K _{w}}{h}

Start from the Ionic Product of Water equation, $K_{w}$ = [H^+][OH^-]. To make [OH^-] the subject, divide both sides by [H^+], then rewrite using shorthand symbols.

Difficulty: 2/5

Solve for $K_{w}$

Rewriting the Ionic Product of Water Equation

K_{w} = h \times o h

The equation for the ionic product of water, $K_{w}$ , is already expressed with $K_{w}$ as the subject. No algebraic rearrangement is required.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

The graph is a linear function passing through the origin where Kw increases at a constant rate as h increases. For a chemistry student, this means that a larger x-value represents a more acidic solution with a higher concentration of hydrogen ions, while a smaller x-value indicates a more basic solution. The most important feature is that the linear relationship means doubling h will always double the resulting Kw for a fixed oh value.

Graph type: linear

Why it behaves this way

Intuition

Visualize a seesaw where the concentrations of hydrogen ions ([H+]) and hydroxide ions ([OH-]) are on opposite ends, and the ionic product of water ( $K_{w}$ ) acts as the fixed pivot point

K_{w}

The equilibrium constant for the autoionization of water, representing the product of hydrogen and hydroxide ion concentrations at equilibrium.

It sets a fixed 'budget' for the product of [H+] and [OH-] in aqueous solutions at a given temperature; if one concentration increases, the other must decrease proportionally to maintain this constant product.

[H+]

The molar concentration of hydrogen ions (often represented as hydronium ions, H3O+) in an aqueous solution.

Directly quantifies the acidity of a solution; higher values mean the solution is more acidic.

[OH-]

The molar concentration of hydroxide ions in an aqueous solution.

Directly quantifies the basicity (alkalinity) of a solution; higher values mean the solution is more basic.

Free study cues

Insight

Canonical usage

Calculated using molar concentrations of hydrogen and hydroxide ions, typically expressed in moles per cubic decimetre (mol dm-3).

Common confusion

Assuming Kw is a universal constant of 1.0 x 10^-14 at all temperatures; it is endothermic and increases as temperature rises.

Dimension note

While Kw is treated as having units of (concentration)^2 in introductory chemistry, it is strictly dimensionless in IUPAC thermodynamic definitions where it is based on activities relative to a standard state of 1

Unit systems

$[H +]$ mol dm^-3 · Also written as [H3O+]; represents the molarity of hydronium ions.

$[O H -]$ mol dm^-3 · Represents the molarity of hydroxide ions.

$K w$ mol^2 dm^-6 · The units are the square of the concentration units used for the individual ions.

Ballpark figures

Quantity:

One free problem

Practice Problem

A sample of pure water at 25°C has a hydrogen ion concentration of 1.0 × 10⁻⁷ M and a hydroxide ion concentration of 1.0 × 10⁻⁷ M. Calculate the value of the ionic product of water (Kw).

[H+]1e-7 M

[OH-]1e-7 M

Solve for: $K w$

Hint: Multiply the concentration of the hydrogen ions by the concentration of the hydroxide ions.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

Calculating [H⁺] in a basic solution from its [OH⁻].

Study smarter

Tips

Kw increases with temperature because the self-ionization of water is endothermic.
In neutral solutions at 25°C, both [H⁺] and [OH⁻] are exactly 1.0 × 10⁻⁷ M.
Always ensure concentrations are in Molarity (mol/L) before calculating.

Avoid these traps

Common Mistakes

Forgetting Kw changes with temperature.
Using wrong value for Kw.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Kw is the equilibrium constant for water auto-ionisation and is temperature dependent.

Use this equation whenever you need to convert between hydrogen ion concentration and hydroxide ion concentration in an aqueous environment. It is valid for acidic, basic, and neutral solutions, provided the system has reached chemical equilibrium.

This equation forms the mathematical basis for the pH scale, which is essential for monitoring environmental water quality and human physiology. It ensures that any change in acidity is balanced by a corresponding change in basicity, maintaining chemical stability in biological cells.

Forgetting Kw changes with temperature. Using wrong value for Kw.

Calculating [H⁺] in a basic solution from its [OH⁻].

Kw increases with temperature because the self-ionization of water is endothermic. In neutral solutions at 25°C, both [H⁺] and [OH⁻] are exactly 1.0 × 10⁻⁷ M. Always ensure concentrations are in Molarity (mol/L) before calculating.

References

Sources

Atkins' Physical Chemistry
IUPAC Gold Book: Ionic product of water
Wikipedia: Ionic product of water
IUPAC Gold Book
NIST Chemistry WebBook
IUPAC Gold Book: 'autoprotolysis constant of water'
NIST Chemistry WebBook: 'Water (H2O) - Ionization Constant'
Atkins, P. W., & de Paula, J. (2018). Atkins' Physical Chemistry (11th ed.). Oxford University Press.

Ionic Product of Water ([H⁺][OH⁻])

Overview

Variables

Derivation

State the Expression:

Rearrangements

Graph

Intuition

Insight

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Ka and Kb Relationship

pH and pOH Relationship

Frequently Asked Questions

Sources