Chemical reactions that can proceed in both the forward and reverse directions eventually reach a state known as chemical equilibrium. This state is not one of inactivity, but a dynamic balance where the rates of the forward and reverse reactions are exactly equal. At this point, the macroscopic properties of the system, such as the concentrations of the reactants and products, stop changing over time. The specific molarity of each substance present when this stable state is achieved is called the equilibrium concentration. Calculating these concentrations is a foundational problem in chemistry, requiring a systematic approach that links the initial amounts of substances to the final, balanced state.
The Role of the Equilibrium Constant
The relationship between the concentrations of reactants and products at equilibrium is defined by the equilibrium constant, symbolized as \(K_c\) (where ‘c’ indicates concentration in moles per liter). This constant is a numerical value that remains unchanged for a specific reaction at a fixed temperature. The expression for \(K_c\) is written as a ratio of the product concentrations raised to their stoichiometric coefficients divided by the reactant concentrations raised to their coefficients.
For a general reversible reaction, \(aA + bB \rightleftharpoons cC + dD\), the equilibrium constant expression is \(K_c = \frac{[C]^c[D]^d}{[A]^a[B]^b}\), where the square brackets denote the molar concentration of the species. Since the concentrations of pure solids and pure liquids do not change during a reaction, they are conventionally omitted from the \(K_c\) expression. The magnitude of \(K_c\) provides initial insight: a large value indicates that products are favored, while a small value means reactants are favored at equilibrium.
Structuring the Calculation: The ICE Table Method
To manage the variables involved in calculating equilibrium concentrations, the ICE table method provides a standardized framework. The acronym ICE stands for Initial, Change, and Equilibrium, representing the three rows of the table. This tool is structured beneath the balanced chemical equation, with a column for every reactant and product involved in the \(K_c\) expression.
The ‘I’ row is filled with the initial molar concentrations of all species before the reaction begins to shift toward equilibrium. The ‘C’ row represents the change in concentration that occurs as the system moves from its initial state to equilibrium. This change is defined using a single variable, typically ‘x’, multiplied by the stoichiometric coefficient of each substance. Reactants are assigned a negative change (e.g., \(-ax\)), while products are assigned a positive change (e.g., \(+cx\)).
The final ‘E’ row is calculated by algebraically summing the ‘I’ and ‘C’ rows for each substance, creating an expression for the equilibrium concentration. For instance, a reactant ‘A’ with an initial concentration \([A]_i\) and a coefficient ‘a’ would have an equilibrium concentration of \([A]_{eq} = [A]_i – ax\). These algebraic expressions are the concentrations that will be substituted into the \(K_c\) expression to solve for the unknown variable ‘x’.
Solving for Unknown Concentrations
The expressions for the equilibrium concentrations from the ‘E’ row are substituted into the \(K_c\) equation, which creates a single mathematical equation with ‘x’ as the only unknown. If the equation simplifies to a linear form or a perfect square, the value of ‘x’ can be found through simple algebra or by taking the square root of both sides.
For many problems, the substitution results in a higher-order polynomial equation, often a quadratic equation in the form \(ax^2 + bx + c = 0\). If the reaction has a very small \(K_c\) value, the “x-is-small” approximation rule can be applied. This rule assumes that ‘x’ is negligible compared to the initial concentration of a reactant, allowing terms like \((C_{initial} – x)\) to be simplified to \(C_{initial}\). This simplification avoids the need for the quadratic formula, but the calculated ‘x’ must be checked to ensure it is less than five percent of the initial concentration to validate the approximation.
If the approximation is invalid, or if \(K_c\) is large, the quadratic formula must be employed: \(x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}\). The formula yields two possible values for ‘x’, but only the root that is chemically reasonable is accepted. Since concentration cannot be negative, the solution for ‘x’ that results in a negative equilibrium concentration for any species must be discarded.
A Step-by-Step Calculation Example
Consider the dissociation of a weak acid, HA, with an initial concentration of \(0.10 \text{ M}\) and a known \(K_a\) of \(1.8 \times 10^{-5}\): \(\text{HA} \rightleftharpoons \text{H}^+ + \text{A}^-\). The equilibrium constant expression is \(K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\). The ICE table is set up with \([\text{HA}]_i = 0.10 \text{ M}\) and \([\text{H}^+]_i = [\text{A}^-]_i = 0 \text{ M}\). The change row is \(-\text{x}\) for \(\text{HA}\) and \(+\text{x}\) for \(\text{H}^+\) and \(\text{A}^-\), making the equilibrium concentrations \([\text{HA}]_{eq} = 0.10 – \text{x}\) and \([\text{H}^+]_{eq} = [\text{A}^-]_{eq} = \text{x}\).
Substituting these expressions into the \(K_a\) equation gives \(1.8 \times 10^{-5} = \frac{(\text{x})(\text{x})}{0.10 – \text{x}}\). Since the \(K_a\) value is very small, the approximation rule is applied, simplifying the denominator to \(0.10\). Solving the simplified equation \(1.8 \times 10^{-5} \approx \frac{\text{x}^2}{0.10}\) yields \(\text{x} = 1.34 \times 10^{-3} \text{ M}\).
The change ‘x’ is confirmed to be less than five percent of \(0.10 \text{ M}\) \((1.34\%)\), validating the approximation. This calculated value of ‘x’ is then substituted back into the ‘E’ row expressions to find the final equilibrium concentrations. The equilibrium concentrations are \([\text{H}^+] = 1.34 \times 10^{-3} \text{ M}\), \([\text{A}^-] = 1.34 \times 10^{-3} \text{ M}\), and \([\text{HA}] \approx 0.10 \text{ M}\).