Worked example 1
For H₂(g) + I₂(g) ⇌ 2 HI(g), K = 50.0. Starting with 1.00 M H₂ and 1.00 M I₂, find the equilibrium concentrations.
Try it first: Write the change row using the coefficients 1, 1 and 2.
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Chemical Equilibrium & SolubilityEquilibrium CalculationsContent level: Core 26 min
ICE Tables and Equilibrium Calculations
What you'll be able to do: Set up an ICE table for any equilibrium and solve for the unknown concentrations, using the small-x approximation when it is valid.
Introduction
Once you can predict direction, the natural next question is how far. An ICE table is the bookkeeping that turns a starting mixture and a value of K into actual equilibrium concentrations.
These are recommended, not required. You can start this lesson at any time.
Learning objectives
- Build an ICE table with correct stoichiometric changes
- Substitute the equilibrium row into the K expression
- Apply and test the small-x approximation with the 5% rule
- Solve the quadratic when the approximation fails
Lesson
The three rows
Initial holds the concentrations before any reaction. Change holds the amounts consumed and formed, written in terms of x. Equilibrium is simply the sum of the two rows above. Everything you need for the K expression lives in the bottom row.
Getting the change row right
The changes follow the coefficients exactly. For N₂ + 3 H₂ ⇌ 2 NH₃ the changes are -x, -3x and +2x. Reactants that are consumed get a minus sign and products formed get a plus sign, and the multipliers are never optional.
Coefficients multiply x in the change row and become exponents in the K expression. Both jobs, every time.
Choosing the direction
If the mixture starts with products present, calculate Q first. When Q > K the reaction runs in reverse and the signs in the change row flip. Assigning the direction before writing the change row prevents a negative concentration later.
The small-x approximation
When an initial concentration is much larger than the expected change, replacing C − x with C may simplify the equilibrium equation. Whether this is valid depends on the reaction stoichiometry, K, and starting mixture; no single initial-concentration-to-K ratio works for every ICE table. Use the approximation to estimate x, then check the percent change in each concentration whose C ± change term you simplified.
Try the approximation when the initial concentration divided by K is greater than about 400.
Checking with the 5% rule
After solving with an approximation, compare the magnitude of the concentration change with the nonzero initial concentration it was subtracted from or added to. If the change is 5% or less for each approximated concentration, the shortcut is acceptable; otherwise solve the full equation and keep only roots that give physically possible concentrations.
percent change = 100 × |concentration change| / initial concentration
Use Q to choose the sign of x
An ICE table does not always start with products increasing. Calculate Q from initial concentrations when both sides are present: Q<K means forward change, Q>K means reverse change. Then use one variable x with stoichiometric coefficients and reject roots that imply negative concentrations.
Determine direction before filling the change row.
Key ideas
Definition
ICE table
A three-row layout of Initial, Change and Equilibrium concentrations used to solve equilibrium problems.
Rule
Change row follows coefficients
Every change is a coefficient multiple of x, negative for consumption and positive for formation.
Rule
5% rule
The small-x approximation is acceptable when x is no more than 5% of the initial concentration.
Rule
Reject impossible roots
A quadratic root that gives a negative concentration is discarded.
Key concept
Direction first
Compare Q with K before assigning signs in the change row.
Equation
Quadratic formula
x = (-b +/- sqrt(b² - 4ac)) / (2a)
- a, b, c = coefficients of ax² + bx + c = 0
Equation
Approximation check
(x / [A]0) × 100 <= 5%
- [A]0 = initial concentration
Worked examples
Worked example 2
For N₂O₄(g) ⇌ 2 NO₂(g), Kc = 4.6 × 10⁻³. Starting with 0.50 M N₂O₄, find [NO₂] at equilibrium.
Try it first: Set up the equilibrium expression, find a candidate x, and then check whether the concentration approximation is within 5%.
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Worked example 3
For A(g) ⇌ B(g), K = 2.0, starting with 1.0 M A and 0.0 M B. Solve exactly.
Try it first: Would the small-x approximation be valid here?
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Worked example 4
For A ⇌ B, K=4.0. Initially [A]=2.0 M and [B]=4.0 M. Find equilibrium concentrations.
Try it first: Qinitial = 4.0/2.0 = 2.0, below K.
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Common mistakes
Leaving the coefficient off the change row, for example writing +x instead of +2x.
Why it's wrong: The change must be in the stoichiometric ratio.
Check instead: Copy each coefficient into the change row before substituting.
Using the change row instead of the equilibrium row in the K expression.
Why it's wrong: K is defined by equilibrium concentrations only.
Check instead: Always substitute the bottom row.
Applying the small-x approximation when K is large.
Why it's wrong: A large K means x is comparable to the initial concentration.
Check instead: Run the 5% check, or solve the quadratic from the start.
Keeping a quadratic root that gives a negative concentration.
Why it's wrong: Concentrations cannot be negative.
Check instead: Discard that root and keep the physically possible one.
Reporting x when the question asks for a product concentration of 2x.
Why it's wrong: x is the extent of reaction, not necessarily the answer.
Check instead: Read the equilibrium row for the species asked about.
After the lesson · Question bank
Independent practice
Work through new problems on this topic on your own. The lesson's quick checks and worked examples stay above if you need to review a step.
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What you should now know
An ICE table tracks Initial, Change and Equilibrium concentrations for every species. Changes follow the balanced equation’s stoichiometric ratios. Substituting the equilibrium row into K gives an equation in x. A small-x approximation may simplify that equation, but its validity depends on the reaction and starting mixture; estimate x and check the percent change in each concentration being approximated. If any change exceeds 5%, solve the full equation and keep only roots that give physically possible concentrations.
- ICE tracks Initial, Change and Equilibrium concentrations
- Changes follow the stoichiometric coefficients
- Substitute the equilibrium row into K
- Neglect x only when K is small, then check with the 5% rule
- Discard roots that give negative concentrations
Sources and further reading
This lesson is original Chem Help content. No external sources were adapted.