ICE Table Calculator

What is an ICE table?

An ICE table is the grid used to work out an equilibrium: a row of Initial amounts, a row for the Change as the reaction moves by an unknown x, and a row of Equilibrium amounts, each the initial amount plus its change. The changes follow the coefficients, so for aA + bB ⇌ cC + dD running forward the reactants fall by ax and bx and the products rise by cx and dx. Putting the bottom row into the equilibrium constant, Kc = [C]c[D]d / ([A]a[B]b), gives one equation in x, and solving it gives every equilibrium amount.

This calculator builds the table for a reaction you type and solves for x exactly, then shows what the small-x shortcut would have given and whether it passes the 5 percent check. It also runs the other way, finding K from amounts measured at equilibrium.

Using the ICE table calculator

Type the balanced reaction with = between the sides, such as N2O4(g) = 2NO2(g). A number in front of a species is its coefficient, and up to two species on each side can appear in K. Mark a pure solid or liquid with (s) or (l), as in CaCO3(s) = CaO(s) + CO2(g), and it is left out of K and assumed to be in excess. Write a charge as NH4+ or Fe^3+. When every species is a real formula the calculator also checks that the equation balances, and warns if it does not.

Then choose what to find:

• Equilibrium amounts from K. Give K and the starting amounts. The calculator works out Q at the start to decide the direction, fills in the table and solves for x.
• K from one equilibrium amount. Give the starting amounts and one amount measured at equilibrium. Its change, divided by its coefficient, is x, and the rest of the table follows.
• K from every equilibrium amount. Give the equilibrium row and the calculator evaluates the expression.

Amounts are concentrations in mol/L for Kc, or partial pressures in atm or bar for Kp. Type a power of ten as 4.63e-3 or 4.63x10^-3. The readouts give x, the direction and the shortcut’s verdict, the table shows each change both as algebra and as a number, and the working sets out every step, including the quadratic when there is one.

Worked example: hydrogen iodide at 430 °C

The calculator opens on a standard problem from Chang’s Chemistry: 0.500 mol/L each of H₂ and I₂ in a flask at 430 °C, where Kc = 54.3 for H₂(g) + I₂(g) ⇌ 2HI(g).

• No HI is present yet, so Q = 0, which is below K, and the reaction runs forward.
• The rows read Initial 0.500, 0.500 and 0; Change −x, −x and +2x; Equilibrium 0.500 − x, 0.500 − x and 2x.
• Substituting gives 54.3 = (2x)² / (0.500 − x)², which rearranges to 50.3x² − 54.3x + 13.575 = 0.
• The quadratic formula gives x = (54.3 ± √217.2) / 100.6, so x is 0.3933 or 0.6863.
• Only the first is physical. The second would leave 0.500 − 0.6863 = −0.1863 mol/L of H₂, which is impossible, so x = 0.3933 mol/L.
• That gives [H₂] = [I₂] = 0.500 − 0.3933 = 0.1067 mol/L and [HI] = 2x = 0.7865 mol/L, the figures the calculator opens with.

Both sides of this equation are perfect squares, so Chang takes the square root instead: √54.3 = 7.369 = 2x / (0.500 − x) gives the same x with no quadratic. The book prints [HI] as 0.786 mol/L because it doubles x after rounding it to 0.393; unrounded, the answer is 0.7865.

The small-x approximation and the 5 percent rule

When K is small the reaction barely moves, and a term such as 0.100 − x is close to 0.100. Dropping x from every term that starts above zero leaves x only in the terms that start at zero, and the equation solves without a quadratic. Whether that was safe is decided by the 5 percent rule: the change the shortcut predicts must be under 5 percent of each starting amount it simplified.

For 0.100 mol/L ethanoic acid, with Ka = 1.8 × 10⁻⁵ from OpenStax Chemistry 2e, the shortcut gives x ≈ √(1.8 × 10⁻⁵ × 0.100) = 1.342 × 10⁻³ mol/L, which is 1.34 percent of 0.100, so it passes. The exact answer is 1.333 × 10⁻³ mol/L, and the shortcut is 0.67 percent above it. For hydrogen iodide the shortcut gives x ≈ √(54.3 × 0.500 × 0.500 / 4) = 1.842 mol/L, 368 percent of the 0.500 mol/L there is, so it fails outright. With several species the check uses each one’s own coefficient times x, and reports the worst.

Finding K from equilibrium amounts

Switch to the second mode with the same start and 0.786 mol/L of HI measured at equilibrium. HI rose by 0.786, so x = 0.786 / 2 = 0.393 mol/L, both reactants fall to 0.500 − 0.393 = 0.107 mol/L, and Kc = 0.786² / (0.107 × 0.107) = 53.96. That matches 54.3 only to two figures, because amounts rounded to three figures lose precision once they are squared and divided, so keep every figure you have. If you already know every equilibrium amount, the third mode evaluates the expression directly.

Kc, Kp and partial pressures

For gases the same table works in partial pressures, with Kp in place of Kc, as long as the volume and temperature stay fixed, since each partial pressure is then proportional to the amount of that gas. The two constants are linked by Kp = Kc(RT)^Δn, where Δn is the moles of gas on the right minus the left, T is in kelvin, and R is 0.08206 L·atm/(mol·K) for pressures in atm or 0.08314 L·bar/(mol·K) for bar. Give a temperature and the calculator converts. For N₂O₄ ⇌ 2NO₂, with Kc = 4.63 × 10⁻³ at 25 °C as Chang gives it, Δn = 1 and Kp = 4.63 × 10⁻³ × 0.08206 × 298.15 = 0.1133 in atm. Hydrogen iodide has Δn = 0, so its Kp equals its Kc at every temperature. A reaction with ions or dissolved species has no Kp, and the calculator says so rather than converting.

What this model leaves out

• Activities. Concentrations and pressures stand in for activities. That is accurate for dilute solutions and for gases at modest pressure, and drifts at high ionic strength or high pressure.
• How K depends on temperature. K holds at one temperature, and the temperature field only converts between Kc and Kp. K is tied to thermodynamics by ΔG° = −RT ln K, which the Gibbs free energy calculator works with.
• Other equilibria. One reaction at a time: the self-ionisation of water, a second dissociation step and side reactions are ignored. For the pH of a weak acid with water’s own ions included, use the pH calculator, and for a buffer made from a weak acid and its salt, the Henderson-Hasselbalch calculator.
• Solids and liquids running out. A species marked (s) or (l) is taken to be present in excess. How far a salt dissolves depends partly on its lattice enthalpy, which the Born-Haber cycle calculator finds from a cycle of enthalpy changes.
• A changing volume. An ICE table in pressures assumes a fixed volume and temperature. Compressing the mixture changes every partial pressure at once.

Common mistakes

• Putting moles into K. Divide by the volume first. Moles give the right answer only when the volume is 1 L or there are equal moles on each side.
• Changing every species by x. The change follows the coefficients, so HI rises by 2x while H₂ falls by x.
• Squaring in the wrong place. A coefficient of 2 makes [HI] = 2x and its term (2x)², which is 4x², not 2x².
• Keeping the wrong root. A quadratic has two, and the one that makes any amount negative is not the answer.
• Trusting the shortcut without the check. With a large K or a dilute solution, the small-x approximation can be far off.
• Guessing the direction. Work out Q first. With products present at the start, the reaction may run in reverse.
• Putting solids, liquids or the solvent into K. They are left out, so the water in an aqueous reaction does not appear.