91Ó°ÊÓ

Benzoic acid, represented by the chemical formula \( \text{C}_6\text{H}_5\text{COOH} \), is an organic compound known for its role as a weak acid. This means it does not fully dissociate in water. Instead, it exists in equilibrium with its ions. The dissociation of benzoic acid in water is given by the equation:

• \( \text{C}_6\text{H}_5\text{COOH} \rightleftharpoons \text{C}_6\text{H}_5\text{COO}^- + \text{H}_3\text{O}^+ \)

In this reaction, a single molecule of benzoic acid dissociates to form one molecule each of benzoate ion \( \text{C}_6\text{H}_5\text{COO}^- \) and hydronium ion \( \text{H}_3\text{O}^+ \). Knowing this, the benzoic acid behaves according to its acid-dissociation constant \( K_a \), which is a measure of the strength of an acid in solution.

• A low \( K_a \) value, like the one for benzoic acid (\( 6.3 \times 10^{-5} \)), indicates a weak acid.

This low value explains why only a small fraction of benzoic acid molecules dissociate to produce ions.

To determine equilibrium concentrations in any chemical reaction, it's crucial to understand the initial state and the changes that occur. For the benzoic acid system, equilibrium is achieved when the rate of the forward reaction equals the rate of the reverse reaction. This means the concentrations of all species involved remain constant over time.Initially, the concentration of benzoic acid \([\text{C}_6\text{H}_5\text{COOH}]\) is 0.100 M, while the products \([\text{C}_6\text{H}_5\text{COO}^- ]\) and \([\text{H}_3\text{O}^+ ]\) start at 0 M.At equilibrium, the concentration of benzoic acid decreases by \( x \), the change due to dissociation:

• \([\text{C}_6\text{H}_5\text{COOH}] = 0.100 - x\)
• \([\text{C}_6\text{H}_5\text{COO}^- ] = x\)
• \([\text{H}_3\text{O}^+ ] = x\)

These equations allow us to express the point at which these concentrations stabilize, marking the system's chemical equilibrium.

An ICE table is a handy tool for tracking changes in the concentration of species in a chemical equilibrium. ICE stands for Initial, Change, and Equilibrium, reflecting the information you need and how you calculate it. Here's how you set it up for benzoic acid:Initially, before any dissociation, the concentrations are as follows:

• \([\text{C}_6\text{H}_5\text{COOH}] = 0.100\)
• \([\text{C}_6\text{H}_5\text{COO}^-] = 0\)
• \([\text{H}_3\text{O}^+] = 0\)

During the reaction, \( x \) amount of \( \text{C}_6\text{H}_5\text{COOH} \) dissociates to form \( x \) amounts each of \( \text{C}_6\text{H}_5\text{COO}^- \) and \( \text{H}_3\text{O}^+ \). Thus, the equilibrium concentrations become:

• \([\text{C}_6\text{H}_5\text{COOH}] = 0.100 - x\)
• \([\text{C}_6\text{H}_5\text{COO}^-] = x\)
• \([\text{H}_3\text{O}^+] = x\)

The ICE table helps visualize this dynamic, simplifying the problem by structuring how the concentration shifts impact the equilibrium.

The acid-dissociation constant, \( K_a \), is a key concept describing the extent of dissociation of an acid in water. For benzoic acid, the \( K_a \) expression is key in resolving equilibrium concentrations.The expression for \( K_a \) is derived from the equilibrium concentrations:\[K_a = \frac{[\text{C}_6\text{H}_5\text{COO}^-][\text{H}_3\text{O}^+]}{[\text{C}_6\text{H}_5\text{COOH}]} = \frac{x^2}{0.100 - x}\]For weak acids, a simplifying assumption can often be made: if \( x \) is much smaller than the initial concentration (0.100 M), then \( 0.100 - x \approx 0.100 \).Solving this simplified form yields:\[6.3 \times 10^{-5} = \frac{x^2}{0.100}\]From which \( x^2 = 6.3 \times 10^{-6} \), providing \( x \approx 2.51 \times 10^{-3} \text{ M} \). This \( x \) allows calculation of equilibrium concentrations of all species, illustrating how \( K_a \) guides understanding of acid behavior in equilibrium conditions.