Chemical reactions often settle into a state of balance known as chemical equilibrium. This is a dynamic condition where the rates of the forward and reverse reactions are equal, resulting in constant concentrations of all chemical species. Understanding the position of this equilibrium is necessary for predicting the outcome of a reaction, especially in acid-base chemistry. Acid strength quantifies a substance’s tendency to donate a proton. Quantitatively linking the strength of the participating acids to the final equilibrium position provides a powerful tool for chemical analysis. This article explains how to calculate the overall reaction equilibrium constant ($K_{eq}$) based on the known strengths ($\text{p}K_a$) of the acids involved.
Defining Keq and pKa
The equilibrium constant, symbolized as $K_{eq}$, measures the extent to which a reversible reaction proceeds. It is mathematically defined as the ratio of product concentrations to reactant concentrations at equilibrium. A large $K_{eq}$ indicates that products are heavily favored, meaning the reaction proceeds far to the right. Conversely, a small value suggests the reactants are favored.
The $\text{p}K_a$ value measures the strength of an acid in solution. It is derived from the acid dissociation constant ($K_a$) using a logarithmic scale, where $\text{p}K_a$ is the negative logarithm of $K_a$. $K_a$ represents the equilibrium constant for the specific reaction where the acid donates a proton to water.
A low $\text{p}K_a$ corresponds to a strong acid that readily releases its proton. Conversely, a high $\text{p}K_a$ means the acid holds onto its proton more tightly and is considered weaker. While $K_{eq}$ describes the overall equilibrium of any reversible reaction, $\text{p}K_a$ focuses only on the proton-donating ability of a single compound.
The Thermodynamic Bridge
The ability to connect the acid strength of individual components to the overall equilibrium of an acid-base reaction stems from Gibbs Free Energy ($\Delta G^\circ$). This thermodynamic quantity determines the spontaneity and direction of a chemical change under standard conditions. A negative $\Delta G^\circ$ means the forward reaction is spontaneous, while a positive $\Delta G^\circ$ indicates the reverse reaction is spontaneous. When $\Delta G^\circ$ is zero, the system is at equilibrium.
Both the equilibrium constant ($K_{eq}$) and the acid dissociation constant ($K_a$, from which $\text{p}K_a$ is derived) are mathematically related to this same standard Gibbs Free Energy change. Specifically, $\Delta G^\circ$ is directly proportional to the negative logarithm of the equilibrium constant for the corresponding reaction. This shared relationship establishes the theoretical foundation that allows chemists to calculate one constant from the other. By determining the difference in the energy change between the reactant acid and the product acid, the overall equilibrium position of the combined reaction can be precisely determined.
The Core Calculation Method
The calculation method simplifies the thermodynamic relationship into a direct comparison between the two acids participating in the equilibrium. Any acid-base reaction involves two conjugate acid-base pairs, meaning there is a reactant acid and a product acid present in the overall equation. The first step is identifying these two acids and obtaining their respective $\text{p}K_a$ values, which are typically found in standardized chemical tables.
Once the $\text{p}K_a$ values are known, the next step is to calculate the difference in acid strength, symbolized as $\Delta \text{p}K_a$. This difference is found by subtracting the $\text{p}K_a$ of the product acid (the acid on the right side of the equation) from the $\text{p}K_a$ of the reactant acid (the acid on the left side). This specific order ensures consistency with the overall equilibrium constant definition.
This difference in acid strength directly relates to the logarithm of the overall reaction’s equilibrium constant using the equation: $\log K_{eq} = \text{p}K_{a(\text{reactant acid})} – \text{p}K_{a(\text{product acid})}$. A positive $\Delta \text{p}K_a$ means the reactant acid is stronger, resulting in a positive $\log K_{eq}$. This translates to a $K_{eq}$ greater than 1, indicating that the products are favored at equilibrium.
Conversely, a negative $\Delta \text{p}K_a$ signifies that the product acid is stronger, resulting in a negative $\log K_{eq}$ and an overall $K_{eq}$ less than 1. This confirms that the reactants are favored, and the equilibrium lies predominantly to the left. The final step is to calculate the actual $K_{eq}$ value by taking the antilog of the $\log K_{eq}$ result ($K_{eq} = 10^{\log K_{eq}}$).
Applying the Formula: A Worked Example
Consider the reaction between acetic acid ($\text{CH}_3\text{COOH}$) and the ethoxide ion ($\text{CH}_3\text{CH}_2\text{O}^-$). This results in the acetate ion ($\text{CH}_3\text{COO}^-$) and ethanol ($\text{CH}_3\text{CH}_2\text{OH}$). The two acids in this equilibrium are the reactant acid (acetic acid) and the product acid (ethanol).
The standard $\text{p}K_a$ for acetic acid is approximately 4.76, and the $\text{p}K_a$ for ethanol is around 15.9. Following the core method, the difference in acid strength is calculated: $4.76 – 15.9 = -11.14$. This result means that $\log K_{eq}$ equals $-11.14$.
The resulting negative $\log K_{eq}$ value immediately predicts that the equilibrium strongly favors the left side, meaning the reactants are preferred. To find the exact equilibrium constant, the antilog is calculated: $K_{eq} = 10^{-(-11.14)}$. This yields a $K_{eq}$ of approximately $1.38 \times 10^{11}$. The large magnitude confirms the reaction proceeds almost entirely to the right, moving from the stronger acid (acetic acid) to the weaker acid (ethanol).