ICE Table — Chemical Equilibrium

Solve the equilibrium of aA + bB ⇌ cC + dD from K

From the initial state and the constant K, find the equilibrium.

Reaction and coefficients

\mathrm{A}_{(\text{aq})} + \mathrm{B}_{(\text{aq})} \rightleftharpoons \mathrm{C}_{(\text{aq})} + \mathrm{D}_{(\text{aq})}

Adjust each species' coefficient and phase. Gases (g) and aqueous (aq) species enter K; pure solids (s) and liquids (l) are excluded. A coefficient of 0 disables the species.

Initial state

Reactant A

Initial concentration

mol/L

Reactant B

Initial concentration

mol/L

Product C

Initial concentration

mol/L

Product D

Initial concentration

mol/L

Equilibrium constant

Value of K (dimensionless, relative to the standard state).

1.000e-61.000e+6

Results

Shift toward products (→)

Extent of reaction

0.5

mol/L

Q₀ = 0 | K = 1

Limiting-reactant conversion

Q at equilibrium = 1

Approximation invalid (change > 5%)

ICE Table

ICE Table (mol/L)

Species	A(aq)	B(aq)	C(aq)	D(aq)
Initial (I)	1	1	0	0
Change (C)	−0.5	−0.5	+0.5	+0.5
Equilibrium (E)	0.5	0.5	0.5	0.5

Fundamentals & Explanation

What is an ICE table?

When a reversible reaction $aA + bB \rightleftharpoons cC + dD$ reaches equilibrium, the concentrations stop changing but are rarely the initial ones. The ICE table (Initial, Change, Equilibrium) is the standard method to find those equilibrium concentrations: it lays out, in three rows, the Initial state, the Change imposed by stoichiometry, and the Equilibrium state.

The entire change is written with a single unknown, the extent of reaction $\;x$ . Since each species reacts in proportion to its coefficient, the equilibrium concentrations are:

[A]_{eq} = [A]_0 - a\,x \qquad [B]_{eq} = [B]_0 - b\,x

[C]_{eq} = [C]_0 + c\,x \qquad [D]_{eq} = [D]_0 + d\,x

The quotient Q sets the direction

Before solving, it helps to know which way the system will move. The reaction quotient $Q$ has the same form as $K$ , but is evaluated at the current concentrations:

Q = \frac{[C]^{c}\,[D]^{d}}{[A]^{a}\,[B]^{b}}

Comparing the initial $Q_0$ with $K$ :

$Q_0 < K$ : too few products → the reaction proceeds to the right ( $x > 0$ ).
$Q_0 > K$ : too many products → the reaction shifts to the left ( $x < 0$ ).
$Q_0 = K$ : the system is already at equilibrium ( $x = 0$ ).

Solving for x

At equilibrium $Q = K$ . Substituting the equilibrium concentrations gives an equation in $x$ :

\frac{([C]_0 + c\,x)^{c}\,([D]_0 + d\,x)^{d}}{([A]_0 - a\,x)^{a}\,([B]_0 - b\,x)^{b}} = K

For $1{:}1$ coefficients this is a quadratic you can solve by hand; in general it is a higher-degree polynomial. The calculator solves it numerically (bisection), which works for any stoichiometry without resorting to approximations.

Physically, $x$ is bounded: no concentration may go negative. Over that valid interval $Q$ increases strictly monotonically from $\;0$ to $+\infty$ , so for any $K > 0$ there is exactly one equilibrium solution.

Heterogeneous equilibria: solids and liquids

When species are in different phases, pure solids and pure liquids(and the solvent in dilute solution) do not appear in K. The reason: their activity is 1 and constant — their amount does not shift the equilibrium as long as some of that phase is present.

\text{CaCO}_3(s) \rightleftharpoons \text{CaO}(s) + \text{CO}_2(g) \qquad K_p = P_{\text{CO}_2}

That is why you pick each species' phase in the calculator: gases $(g)$ and aqueous $(aq)$ species enter $K$ ; solids $(s)$ and liquids $(l)$ are excluded (they stay in the ICE table, but not in the K expression).

The 5% rule

When $K$ is very small, $x$ is tiny compared with the initial concentrations, and many textbooks approximate $[A]_{eq} \approx [A]_0$ to simplify the algebra. The approximation is considered valid if the relative change of the limiting reactant is below 5%:

\frac{a\,x}{[A]_0} \times 100\% < 5\%

The calculator always solves exactly and, in addition, tells you whether that approximation would have been valid. It is the check textbooks require after solving — and the one that is easy to forget.

The method, step by step

Write the balanced reaction.
Tabulate the initial concentrations (row I).
Express the change with $x$ and the coefficients (row C).
Add to get the equilibrium values (row E).
Substitute into the expression for $K$ and solve for $x$ .
Compute the concentrations and check the 5% rule.

And the other way around ("Find K" mode): since $x$ is the only unknown, measuring a single species at equilibrium is enough. That value fixes $x$ , you complete the whole table and obtain $K = Q$ at equilibrium — without knowing every concentration in advance.

Kc vs Kp

The method is identical with concentrations ( $K_c$ ) or partial pressures ( $K_p$ ). By convention $K$ is dimensionless(activities are measured relative to the standard state: 1 M or 1 atm). The two are related by $\;K_p = K_c\,(RT)^{\Delta n}$ , with $\Delta n$ the change in moles of gas.

Related calculators

The dissociation of a weak acid ( $K_a$ ) is, at heart, an ICE problem — you will see it in Ionic Dissociation. Once the acid and its conjugate base already coexist, the Buffer (Henderson-Hasselbalch) calculator uses the logarithmic form of that same equilibrium.

Academic References

[1] Chang, R. & Goldsby, K. A. (2015). Chemistry (12th ed.). McGraw-Hill. (Chemical equilibrium, ICE table, 5% rule.)
[2] Atkins, P. & de Paula, J. (2018). Atkins' Physical Chemistry (11th ed.). Oxford University Press. (Equilibrium constant and standard state.)
[3] Brown, T. L., et al. (2017). Chemistry: The Central Science (14th ed.). Pearson. (Reaction quotient and the direction of change.)

ICE Table — Chemical Equilibrium

Reaction and coefficients

Initial state

Results

ICE Table

Fundamentals & Explanation

What is an ICE table?

The quotient Q sets the direction

Solving for x

Heterogeneous equilibria: solids and liquids

The 5% rule

The method, step by step

Academic References

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What is an ICE table?

The quotient Q sets the direction

Solving for x

Heterogeneous equilibria: solids and liquids

The 5% rule

The method, step by step

Academic References