Question

Consider a buffer solution where [weak acid] \(>\) [conjugate base]. How is the pH of the solution related to the \(\mathrm{p} K_{\mathrm{a}}\) value of the weak acid? If [conjugate base \(]>[\text { weak acid }]\) , how is pH related to \(\mathrm{p} K_{\mathrm{a}} ?\)

Short answer

In a buffer solution, the pH is related to the pKa value of the weak acid following the Henderson-Hasselbalch equation: \( pH = pKa + \log \left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right) \). If [weak acid] > [conjugate base], the fraction inside the log is less than 1, resulting in a negative log value and a pH lower than the pKa. Conversely, if [conjugate base] > [weak acid], the fraction inside the log is greater than 1, leading to a positive log value and a pH higher than the pKa.

Step-by-step solution

Understand the Buffer Solution

A buffer solution is an aqueous solution that consists of a weak acid and its conjugate base, or a weak base and its conjugate acid. The purpose of a buffer solution is to maintain a relatively constant pH when a small amount of acid or base is added to it.

Recall the Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a way to determine the pH of a buffer solution. The equation is given by: \[ pH = pKa + \log \left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right) \]

Find the relationship between pH and pKa for [weak acid] > [conjugate base]

Using the Henderson-Hasselbalch equation: \[ pH = pKa + \log \left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right) \] Since [weak acid] > [conjugate base], the fraction inside the log will be less than 1. The log of a number smaller than 1 is negative, so the pH will be lower than the pKa in this scenario.

Find the relationship between pH and pKa for [conjugate base] > [weak acid]

Again, using the Henderson-Hasselbalch equation: \[ pH = pKa + \log \left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right) \] In this case, since [conjugate base] > [weak acid], the fraction inside the log will be greater than 1. The log of a number greater than 1 is positive, so the pH will be higher than the pKa in this situation. In conclusion, if [weak acid] > [conjugate base], the pH of the buffer solution will be lower than the pKa value of the weak acid, while if [conjugate base] > [weak acid], the pH of the buffer solution will be higher than the pKa value of the weak acid.

Key concepts

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a fundamental tool in understanding the pH of buffer solutions. It provides a mathematical relationship between the pH, the pKa, and the concentrations of the acid and its conjugate base. The equation is expressed as: \[ pH = pKa + \log \left(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\right) \]This formula allows us to predict how the pH will adjust if either the acid or base concentration changes. Here,

• pH represents the acidity level of the solution.
• pKa is a constant that indicates the strength of the weak acid. It represents the pH at which half of the acid is dissociated.
• The fraction \(\frac{[\text{conjugate base}]}{[\text{weak acid}]}\) shows the ratio of the concentrations of the base form to the acid form.

Using this equation, one can determine how strong the buffer solution is at maintaining a steady pH, even when small amounts of acids or bases are added.

Weak acid and conjugate base

In chemistry, buffers are solutions that resist changes in pH. They are composed of a weak acid and its conjugate base. What are weak acids?
Weak acids partially dissociate in solution. This means they do not completely ionize into ions in water. Their dissociation is reversible and the equilibrium lies to the left, meaning most of the acid remains undissociated. Examples include acetic acid and citric acid.Understanding conjugate bases
A conjugate base is what remains of the acid after it donates a proton (H\(^{+}\)). For example, when acetic acid (CH\(_3\)COOH) loses a proton, it forms its conjugate base, acetate (CH\(_3\)COO\(^{-}\)).

• The weak acid equilibrates with its conjugate base, forming a buffer.
• This buffer solution can absorb excessive H\(^{+}\) or OH\(^{-}\) ions, preventing significant changes in pH.

The balance between the weak acid and its conjugate base is crucial. If more conjugate base exists, it can neutralize added acids. Conversely, more weak acid can neutralize added bases.

pKa and pH relationship

The concepts of pKa and pH are intimately related and pivotal in understanding buffer systems. The pKa is the negative logarithm of the acid dissociation constant (Ka) and gives insight into the strength of an acid. Understanding the relationship

• The pKa value tells us the pH at which an acid is half dissociated, meaning half of the acid molecules have donated a proton.
• In a buffer, comparing the pH with the pKa allows us to predict the behavior of the solution. A pH lower than the pKa indicates a higher concentration of the weak acid than its conjugate base, while a pH higher than the pKa suggests more conjugate base than weak acid.

This relationship is critical for designing buffers for specific pH levels. For example, to prepare a buffer with a pH close to 7.4 (the physiological pH), one would select an acid with a pKa near this value. This ensures the buffer can effectively resist changes and maintain the desired pH even with minor additions of acids or bases.