ICE Tables and Equilibrium Calculations

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.

For N₂O₄ ⇌ 2 NO₂ starting with 0.500 M N₂O₄ and no NO₂:

The variable x is the concentration decrease of N₂O₄, not automatically the concentration of each product. Substitute only the last row into K_c = [NO₂]²/[N₂O₄], giving K_c = 4x²/(0.500 − x). All entries in a concentration table must use the same final vessel volume; do a rapid stoichiometric reaction or dilution first if the problem requires it.

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.

Never replace a product term 2x by zero merely because the product was initially absent. A small-x approximation applies to a sum or difference such as 0.500 − x, not to the x needed to create products. If a simplified term is C − 3x, test 3x/C, not x/C. A small K by itself is not a universal validation criterion because stoichiometric powers and starting products affect the solution.

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.

For A ⇌ 2 B with C = 0.100 M and K_c = 0.100, the equation is 0.100 = 4x²/(0.100 − x). Ignoring x in the denominator predicts x = 0.0500 M, a 50% change, so the shortcut fails. Keep all terms: 4x² + 0.100x − 0.0100 = 0. The roots are about +0.0390 and −0.0640 M. The negative root would give [B] < 0 and is rejected. The valid concentrations are [A] ≈ 0.0610 M and [B] ≈ 0.0781 M; substitution returns K_c ≈ 0.100. Testing physical concentrations and back-substituting are stronger checks than accepting whichever root the calculator shows first.

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.

When initial products are present, Q chooses a convenient sign convention but does not change equilibrium mathematics. A signed x is also possible if every entry remains consistent; the final test is that all concentrations are nonnegative and the material balance and K relation hold. Define x in words before building the row, especially when comparing a forward-oriented table with a reverse-oriented one.