ICE Table Calculator

Calculate equilibrium concentrations using ICE table method for simple equilibria.

Reaction type

Initial concentration C₀ (M) (M)

Equilibrium constant Ka, Kb, or K

Please check your inputs and try again.

How an ICE Table Works

An ICE table is bookkeeping for equilibrium problems. The three rows are Initial concentrations, the Change as the system moves toward equilibrium, and the Equilibrium values. Almost every weak acid, weak base and gas-phase equilibrium problem reduces to filling one in and solving for x.

Row	What it holds	Note
Initial	Concentrations before any reaction	Products usually start at 0
Change	±x multiplied by the stoichiometric coefficient	Reactants lose, products gain
Equilibrium	Initial + Change	These go into the K expression

For a weak acid HA dissociating to H⁺ and A⁻, the equilibrium expression becomes:

K_a = x² / (C₀ − x)

The 5% Approximation

Solving that exactly requires the quadratic formula. The standard shortcut assumes x is small compared with C₀, so the denominator becomes just C₀ and the answer simplifies to x = √(K_aC₀).

The rule for when this is acceptable: the approximation holds if x is less than 5% of C₀. Equivalently, it works when C₀/K_a exceeds about 400. Below that the neglected term matters and the quadratic is required.

C₀/K_a	x as % of C₀	Approximation?
> 1000	< 3%	Safe
~400	~5%	Borderline — check
~100	~10%	Use the quadratic
< 20	> 20%	Quadratic essential

Notice the pattern: dilute solutions of weak acids are precisely where the approximation fails. Diluting reduces C₀ while K_a stays fixed, so the ratio falls and the fraction dissociated rises. This is Ostwald’s dilution law — a weak acid dissociates more completely as it becomes more dilute.

Worked Examples

Example 1: 0.1M acetic acid Ka=1.8e-5

x=√(1.8e-5×0.1)=1.34e-3

Result: pH=2.87, 1.34% dissociation

<5% so approximation valid

Example 2: 0.001M weak acid Ka=1.8e-5

5% approximation check: x=√(1.8e-5×0.001)=1.34e-4

Result: 13.4% — need quadratic!

Approximation breaks down at low concentration

Example 3: Quadratic required

0.001 M acid, K_a = 1.8 × 10⁻⁵

Result: x = 1.25 × 10⁻⁴, pH = 3.90

The approximation gives 1.34 × 10⁻⁴, which is 13.4% of C₀ — well past the 5% limit. Solving the quadratic gives a value 7% lower.

Example 4: Common ion effect

0.1 M acetic acid plus 0.1 M sodium acetate

Result: pH = pK_a = 4.76

The acetate ion starts at 0.1 M in the Initial row rather than zero. Dissociation is suppressed and the system becomes a buffer at its pK_a.

Example 5: Ostwald dilution

0.1 M gives 1.3% dissociated; 0.001 M gives 12.5%

Result: More dilute means more dissociated

Counter-intuitive but correct. Diluting shifts the equilibrium toward the ions, which is why the 5% approximation fails exactly where students most often apply it.

Common Mistakes

⚠️

Applying the approximation without checking

Always verify that x is under 5% of C₀ afterwards. For dilute solutions or moderately strong weak acids the shortcut can be off by 10% or more.

⚠️

Forgetting stoichiometric coefficients in the Change row

For a reaction producing 2 moles of product per mole of reactant, the change is +2x, and that coefficient becomes an exponent in the K expression.

⚠️

Ignoring water autoionisation in very dilute solutions

Below about 10⁻⁶ M, the H⁺ from water is comparable to that from the acid. The simple ICE treatment gives a pH above 7 for a dilute acid, which is impossible.

⚠️

Using the wrong K for the direction written

K_a applies to acid dissociation and K_b to base hydrolysis. For the conjugate of a weak acid, use K_b = K_w/K_a rather than K_a directly.

Frequently Asked Questions

5% rule for approximation?

If K×C0 is very small: x≈√(KC0) (neglect x vs C0). Valid if x/C0 < 5%. At higher ratios, use full quadratic formula. Strong acid (K→∞): x=C0 (complete dissociation). ICE tables work for any equilibrium type.

What if problem has common ion?

Common ion effect: if A⁻ is already present at concentration y, ICE changes: E: C0-x, y+x, x. K=x(y+x)/(C0-x). Approximation: x≈K×C0/y when y>>x (suppressed dissociation).

What does ICE stand for?

Initial, Change, Equilibrium — the three rows of a table used to organise concentrations when solving equilibrium problems.

When can I use the 5% approximation?

When x turns out to be less than 5% of the initial concentration, which generally holds when C₀/K_a exceeds about 400. Always check afterwards rather than assuming.

Why does a weak acid dissociate more when diluted?

Because dilution reduces the concentration term while K_a is fixed, shifting the equilibrium toward the ions. This is Ostwald’s dilution law.

How do I handle a common ion?

Enter its starting concentration in the Initial row instead of zero. The added ion suppresses dissociation, which is the basis of buffer behaviour.

When must I use the quadratic formula?

Whenever x exceeds 5% of the initial concentration — typically for dilute solutions, or acids with K_a large enough that appreciable dissociation occurs.

Formula Explorer connections

Interpretation: This formula describes how reactants, products, ions or phases distribute when opposing processes reach equilibrium. Assumption: Use equilibrium rather than initial concentrations, correct stoichiometric exponents, and the specified temperature; activities may replace concentrations in nonideal systems.

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