A chemical reaction does not always proceed until one of the reactants is completely used up. Instead, many reactions reach a state known as chemical equilibrium, where the transformation of reactants into products and the reverse process occurs simultaneously. This is a state of dynamic equilibrium, meaning the reactions continue at exactly the same rate in both the forward and reverse directions. Because the opposing rates are balanced, the concentrations of all species—reactants and products—appear constant over time.
Calculating the final, unchanging amounts of these substances is necessary to understand the final composition of the reaction mixture. Determining these concentrations requires a structured, multi-step approach that combines chemical principles with algebraic problem-solving.
Defining the Equilibrium Constant
The foundation for calculating equilibrium concentrations is the equilibrium constant, symbolized as \(K\). This constant is a numerical value that describes the ratio of product concentrations to reactant concentrations once a reaction has reached equilibrium. For any general reversible reaction, the mathematical expression for \(K\) is a fraction where the products are in the numerator and the reactants are in the denominator. Each concentration term is raised to the power of its stoichiometric coefficient from the balanced chemical equation.
The value of \(K\) is constant only at a specific temperature, and its magnitude offers insight into the reaction’s final composition. A large value for \(K\) indicates that products are favored at equilibrium, while a small value suggests the reaction mixture contains mostly unreacted starting materials. \(K_c\) is used for concentrations in moles per liter, while \(K_p\) is used for reactions involving gases where partial pressures are used.
Pure solids and pure liquids, including the solvent in a dilute solution, are excluded when constructing the \(K\) expression. These substances are omitted because their concentrations, defined by their density, do not change significantly during the reaction. Their effective concentration is considered constant and is incorporated into the value of \(K\) itself, simplifying the expression to include only gases and solutes.
Organizing Data with the ICE Table Method
The ICE table method is used to organize the initial conditions and the changes that occur as the reaction approaches equilibrium. The acronym ICE stands for Initial, Change, and Equilibrium, representing the three rows of data tracked for every species. The first step involves converting all known starting amounts into molar concentrations (\(K_c\)) or partial pressures (\(K_p\)) and entering these values into the ‘Initial’ row.
The ‘Change’ row introduces the reaction stoichiometry using the variable ‘x’, which represents the change in concentration. Reactants have a negative change, and products have a positive change. The coefficient of ‘x’ for each species must correspond to its stoichiometric coefficient in the balanced equation. For example, in the reaction \(A \rightleftharpoons 2B\), the change for \(A\) would be \(-x\), and the change for \(B\) would be \(+2x\).
This relationship ensures the changes are chemically consistent with the reaction’s proportions. The final row, ‘Equilibrium,’ is generated by algebraically adding the ‘Initial’ and ‘Change’ rows for each species. Using the example \(A \rightleftharpoons 2B\), if the initial concentration of \(A\) was \([A]_0\) and \(B\) was zero, the equilibrium concentrations would be \([A]_0 – x\) and \(2x\).
The expressions in the ‘Equilibrium’ row are the algebraic representations of the final concentrations. These expressions are substituted into the equilibrium constant expression, translating the chemical problem into solvable algebraic terms.
Solving for Equilibrium Concentrations
Once the equilibrium expressions from the ICE table are established, the next step is to substitute them into the \(K\) expression and solve for the variable ‘x’. In the simplest scenarios, the resulting algebraic equation can be solved directly. This often occurs when the expression simplifies to a perfect square, allowing the square root of both sides to be taken.
For many equilibrium problems, especially those involving weak acids or bases, the value of \(K\) is very small (typically \(10^{-4}\) or less). In these cases, the change ‘x’ is often negligible compared to the initial concentration of the reactants. This allows for the small ‘x’ approximation, where terms like \((C – x)\) are approximated as \(C\), simplifying the algebra. This approximation is considered valid if the calculated value of ‘x’ is less than five percent of the initial concentration.
When the approximation fails, the equation must be rearranged into the standard quadratic form: \(ax^2 + bx + c = 0\). The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\), must then be used to find the two possible values for ‘x’.
Since the formula yields two roots, a chemical constraint must be applied to choose the physically meaningful answer. Because ‘x’ represents a concentration change, and concentrations must be non-negative, any calculated root that results in a negative equilibrium concentration must be discarded. The correct value of ‘x’ is the one that yields positive concentrations for all species, which is then used to calculate the final numerical concentrations.