ICE tables: solving for equilibrium concentrations
Master ICE tables (Initial, Change, Equilibrium): the systematic method for computing equilibrium concentrations, with a fully worked example.
Quick answer
An ICE table organizes equilibrium calculations into three rows — Initial concentrations, Change (in terms of x), and Equilibrium concentrations — which are then substituted into the K expression and solved for x.
Example: For H₂ + I₂ ⇌ 2HI (Kc = 54.3) starting at 0.150 M H₂ and 0.150 M I₂: solving 4x²/(0.150−x)² = 54.3 gives x = 0.118, so [HI] = 0.236 M and [H₂] = [I₂] = 0.0320 M at equilibrium.
What you will learn
- Set up an ICE table with correct signs from the direction of shift.
- Write the equilibrium-constant expression in terms of x.
- Solve for x and compute all equilibrium concentrations.
- Verify the answer by checking that Q equals K.
The Initial–Change–Equilibrium framework
When you know the starting concentrations and K but not the final ones, an ICE table keeps the bookkeeping straight. The three rows are:
- I — Initial: the starting concentration of each species
- C — Change: how each concentration shifts, written in terms of x (the extent of reaction)
- E — Equilibrium: initial plus change, the expressions you plug into K
The signs in the change row follow the direction of shift: species that are consumed get −x (times their coefficient); species that form get +x. When in doubt, compute Q first — Q < K means shift right, Q > K means shift left.
Setting up the table: a full example
Consider H₂(g) + I₂(g) ⇌ 2HI(g) with Kc = 54.3 at 430 °C. A flask starts with [H₂] = 0.150 M, [I₂] = 0.150 M, and [HI] = 0. Since Q = 0 < K, the reaction shifts right: H₂ and I₂ are consumed, HI forms.
| | H₂ | I₂ | 2HI |
|---|---|---|---|
| Initial (M) | 0.150 | 0.150 | 0 |
| Change (M) | −x | −x | +2x |
| Equilibrium (M) | 0.150 − x | 0.150 − x | 2x |
Substituting the equilibrium row into the K expression:
Solving for x and verifying
Taking the square root of both sides simplifies the algebra:
Now read the equilibrium concentrations from the table's last row:
- [HI] = 2x = 2(0.118) = 0.236 M
- [H₂] = [I₂] = 0.150 − 0.118 = 0.0320 M
Always sanity-check: concentrations must be positive (they are), and plugging back should recover K: (0.236)²/(0.0320)² = 0.0557/0.00102 ≈ 54.4, matching 54.3 within rounding. The check closes the loop.
The small-x approximation
When K is very small, x is tiny compared to the initial concentrations, and you can approximate (initial − x) ≈ initial. This turns a quadratic into a one-line calculation. The rule: after solving, check that x is less than 5% of the initial concentration. If it passes, the approximation stands; if not, solve the full quadratic.
Example: for a reaction with K = 1.0 × 10⁻⁵ starting at 0.20 M reactant, x ≈ √(K × 0.20) = √(2.0 × 10⁻⁶) = 1.4 × 10⁻³ M — just 0.7% of 0.20 M, so the approximation is fine.
Variables and units
| Symbol | Name | Unit | Meaning |
|---|---|---|---|
| Initial concentration | M | Initial concentration of a species before any shift | |
| Change variable | M | Extent of reaction; the unknown solved from the K expression | |
| Equilibrium concentration | M | Concentration once equilibrium is reached (initial ± change) |
Common mistakes
The signs depend on the direction of shift. If Q > K, the reaction moves left: reactants gain (+x) and products lose (−x). Always check Q versus K (or think physically) before assigning signs.
x is the change in concentration — the extent of reaction. Equilibrium concentrations are initial ± x (times coefficients). Reporting x itself as the answer is a classic mistake.
Coefficients scale the change: for 2HI forming, the change is +2x, not +x. The coefficient appears in the ICE table and again as an exponent in the K expression — both are required.
Real-world applications
Industrial chemistry: predicting equilibrium yields before building a reactor.
Environmental science: computing how pollutants partition at equilibrium.
Biochemistry: working out the final pH of buffer mixtures.
Classroom problem-solving: the standard method for every 'find the equilibrium concentrations' question.
Frequently asked questions
How do I know whether x is added or subtracted?
Use Q first. If Q < K, the reaction shifts right: reactants decrease (−x) and products increase (+x). If Q > K, flip the signs. Guessing wrong gives a negative concentration at the end — a clear signal to reverse the signs.
Do stoichiometric coefficients appear twice — in the change row and as exponents?
Yes. In the equilibrium expression, a change of +2x for 2HI becomes (2x)² — the coefficient appears both in the ICE row and as an exponent. Forgetting either one is the most common ICE error.
What if the equation has coefficients like 2 or 3?
Multiply every change entry by the same x (with its coefficient): −x, −x, +2x. There is only ever one x per reaction — it represents the single extent of reaction. Never use x and y for the same reaction.
When is the small-x approximation allowed?
Drop the tiny term: if x is less than 5% of the initial concentration, the approximation is accepted. If it fails the 5% check, solve the full quadratic — the approximation was not justified.