An ICE (Initial, Change, Equilibrium) table is simple matrix formalism that used to simplify the calculations in reversible equilibrium reactions (e.g., weak acids and weak bases or complex ion formation). ICE tables are composed of the concentrations of molecules in solution in different stages of a reaction, and are usually used to calculate the K, or equilibrium constant expression, of a reaction (in some instances, K may be given, and one or more of the concentrations in the table will be the unknown to be solved for). ICE tables automatically set up and organize the variables and constants needed when calculating the unknown. ICE is a simple acronym for the titles of the first
column of the table. The procedure for filling out an ICE table is best illustrated
through example. Example 1 Use an ICE table to determine \(K_c\) for the following balanced general reaction: \[ \ce{ 2X(g) <=> 3Y(g) + 4Z(g)} \nonumber\] where the capital letters represent the products and reactants. A sample consisting of 0.500 mol of x is placed into a system with a volume of 0.750
liters. At equilibrium, the amount
of sample x is known to be 0.350 mol. Desired Unknown \[ K_c = ? \nonumber \] Solution The equilibrium constant expression is expressed as products over reactants, each raised to the power of their respective stoichiometric coefficients: \[ K_c = \dfrac{[Y]^3[Z]^4}{[X]^2} \nonumber \] The equilibrium concentrations of Y and Z are unknown, but they can be
calculated using the ICE table. STEP 1: Fill in the given amounts
This is the first step in setting up the ICE table. As mentioned above, the ICE mnemonic is vertical and the equation heads the table horizontally, giving the rows and columns of the table, respectively. The numerical amounts were given. Any amount not directly given is unknown. STEP 2: Fill in the amount of change for each compound
Notice that the equilibrium in this equation is shifted to the right, meaning that some amount of reactant will be taken away and some amount of product will be added (for the Change row). The change in amount (\(x\)) can be calculated using algebra: \[ Equilibrium \; Amount = Initial \; Amount + Change \; in \; Amount \nonumber \] Solving for the Change in the amount of \(2x\) gives: \[ 0.350 \; mol - 0.500 \; mol = -0.150 \; mol \nonumber \] The change in reactants and the balanced equation of the reaction is known, so the change in products can be calculated. The stoichiometric coefficients indicate that for every 2 mol of x reacted, 3 mol of Y and 4 mol of Z are produced. The relationship is as follows: \[ \begin{eqnarray} Change \; in \; Product &=& -\left(\dfrac{\text{Stoichiometric Coefficient of Product}}{\text{Stoichiometric Coefficient of Reactant}}\right)(\text{Change in Reactant}) \\ Change \; in \; Y &=& -\left(\dfrac{3}{2}\right)(-0.150 \; mol) \\ &= +0.225 \; mol \end{eqnarray} \nonumber \] Try obtaining the change in Z with this method (the answer is already in the ICE table). STEP 3: Solve for the equilibrium amounts
If the initial amounts of Y and/or Z were nonzero, then they would be added together with the change in amounts to determine equilibrium amounts. However, because there was no initial amount for the two products, the equilibrium amount is simply equal to the change: \[\begin{eqnarray} Equilibrium \; Amount &=& Initial \; Amount + Change \; in \; Amount \\ Equilibrium \; Amount \; of \; Y &=& 0.000 \; mol\; + 0.225 \; mol \\ &=& +0.225 \; mol \end{eqnarray} \nonumber \] Use the same method to find the equilibrium amount of Z. Convert the equilibrium amounts to concentrations. Recall that the volume of the system is 0.750 liters. \[[Equilibrium \; Concentration \; of \; Substance] = \dfrac{Amount \; of \; Substance}{Volume \; of \; System}\nonumber \] \[ [X] = \dfrac{0.350 \; mol}{0.750 \; L} = 0.467 \; M \nonumber \] \[ [Y] = \dfrac{0.225 \; mol}{0.750 \; L} = 0.300 \; M \nonumber \] \[ [Z] = \dfrac{0.300 \; mol}{0.750 \; L} = 0.400 \; M \nonumber \] Use the concentration values to solve the \(K_c\) equation: \[ \begin{eqnarray} K_c &=& \dfrac{[Y]^3[Z]^4}{[X]^2} \\ &=& \dfrac{[0.300]^3[0.400]^4}{[0.467]^2} \\ K_c &=& 3.17 \times 10^{-3} \end{eqnarray}\nonumber \] Example 2: Using an ICE Table with Concentrations n this example an ICE table is used to find the equilibrium concentration of the reactants and products. (This example will be less in depth than the previous example, but the same concepts are applied.) These calculations are often carried out for weak acid titrations. Find the concentration of A- for the generic acid dissociation reaction: \[ \ce{HA(aq) + H_2O(l) <=> A^{-}(aq) + H_3O^{+}(aq)} \nonumber \] with \([HA (aq)]_{initial} = 0.150 M\) and \(K_a = 1.6 \times 10^{-2}\) Solution This equation describes a weak acid reaction in solution with water. The acid (HA) dissociates into its conjugate base (\(A^-\)) and protons (H3O+). Notice that water is a liquid, so its concentration is not relevant to these calculations. STEP 1: Fill in the given concentrations
STEP 2: Calculate the change concentrations by using a variable 'x'
STEP 3: Calculate the concentrations at equilibrium
STEP 4: Use the ICE table to calculate concentrations with \(K_a\) The expression for Ka is written by dividing the concentrations of the products by the concentrations of the reactants. Plugging in the values at equilibrium into the equation for Ka gives the following: \[K_a = \dfrac{x^2}{0.150-x} = 1.6 \times 10^{-2} \nonumber\] To find the concentration x, rearrange this equation to its quadratic form, and then use the quadratic formula to find x: \[\begin{align*} (1.6 \times 10^{-2})({0.150-x}) &= {x^2} \\[4pt] x^2+(1.6 \times 10^{-2})x-(0.150)(1.6 \times 10^{-2}) &= 0 \end{align*}\] This is the typical form for a quadratic equation: \[Ax^{2}+Bx+C=0\nonumber \] where, in this case:
The quadratic formula gives two solutions (but only one physical solution) for x: \[x = \dfrac{-B+\sqrt{B^2-4AC}}{2A}\nonumber \] and \[x = \dfrac{-B-\sqrt{B^2-4AC}}{2A}\nonumber \] Intuition must be used in determining which solution is correct. If one gives a negative concentration, it can be eliminated, because negative concentrations are unphysical. The x value can be used to calculate the equilibrium concentrations of each product and reactant by plugging it into the elements in the E row of the ice table. [Solution: x = 0.0416, -0.0576. x = 0.0416 makes chemical sense and is therefore the correct answer.] For some problems like example 2, if x is significantly less than the value for Ka, then the x of the reactants (in the denominator) can be omitted and the concentration for x should not be greatly affected. This will make calculations faster by eliminating the necessity of the quadratic formula. Checklist for ICE tables
Partial pressures may also be substituted for concentrations in the ICE table, if desired (i.e., if the concentrations are not known, \(K_p\) instead of \(K_c\) is desired, etc.). "Amount" is also acceptable (the ICE table may be done in amounts until the equilibrium amounts are found, after which they will be converted to concentrations). For simplicity, assume that the word "concentration" can be replaced with "partial pressure" or "amounts" when formulating ICE tables. Exercise \(\PageIndex{1}\) 0.200 M acetic acid is added to water. What is the concentration of H3O+ in solution if \(K_c = 1.8 \times 10^{-6}\)? Answer5.99×10-4 Exercise \(\PageIndex{2}\) If the initial concentration of NH3 is 0.350 M and the concentration at equilibrium is 0.325 M, what is \(K_c\) for this reaction? Answer1.92×10-3 Exercise \(\PageIndex{3}\) How is \(K_c\) derived from \(K_p\)? Answer\(K_p = K_c(RT)^{\Delta n}\) then solve for \(K_c\) Exercise \(\PageIndex{4}\) Complete this ICE table:
Contributors and Attributions
What is the formula for calculating equilibrium?Here is how to find the equilibrium price of a product:. Use the supply function for quantity. You use the supply formula, Qs = x + yP, to find the supply line algebraically or on a graph. ... . Use the demand function for quantity. ... . Set the two quantities equal in terms of price. ... . Solve for the equilibrium price.. Is ICE an equilibrium?ICE stands for Initial, Change, Equilibrium.
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