Question

If the \(\mathrm{p} K_{\mathrm{a}}\) of a weak acid is 5 , the \(\mathrm{pH}\) will be 6 : 1\. when the concentration of dissociated acid is one-tenth the concentration of undissociated acid. 2\. when half the acid is dissociated. 3\. when the concentration of dissociated acid is ten times the concentration of undissociated acid. 4\. only after a base has been added.

Short answer

The pH of the weak acid with a pKa of 5 will be 6 when the concentration of dissociated acid is ten times the concentration of undissociated acid (Condition 3).

Step-by-step solution

Write down the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation relates the pH, pKa, and concentrations of dissociated ([A-]) and undissociated acid ([HA]) as follows: \(pH = pKa + log \frac{[A^{-}]}{[HA]}\)

Plug in the given pKa value

Since the pKa of the weak acid is given as 5, the equation becomes: \(pH = 5 + log \frac{[A^{-}]}{[HA]}\) Now we will evaluate this equation in each of the given conditions.

Evaluate for Condition 1

In condition 1, the concentration of dissociated acid is one-tenth the concentration of undissociated acid. In other words, \(\frac{[A^{-}]}{[HA]} = \frac{1}{10}\). Plug in this value to the equation: \(pH = 5 + log(\frac{1}{10})\) \(pH = 5 - 1 = 4\) The pH under condition 1 is 4.

Evaluate for Condition 2

In condition 2, half the acid is dissociated. In other words, \(\frac{[A^{-}]}{[HA]} = 1\). Plug in this value to the equation: \(pH = 5 + log(1)\) Since the log of 1 is 0, \(pH = 5\) The pH under condition 2 is 5.

Evaluate for Condition 3

In condition 3, the concentration of dissociated acid is ten times the concentration of undissociated acid. In other words, \(\frac{[A^{-}]}{[HA]} = 10\). Plug in this value to the equation: \(pH = 5 + log(10)\) Since the log of 10 is 1, \(pH = 5 + 1 = 6\) The pH under condition 3 is 6. Thus, statement 3 is correct.

Evaluate Condition 4

In condition 4, we are asked if the pH will be 6 only after a base has been added. However, we have already determined that the pH is 6 under condition 3, even without adding a base. Therefore, statement 4 is not correct. In conclusion, the pH of the weak acid with a pKa of 5 will be 6 when the concentration of dissociated acid is ten times the concentration of undissociated acid.

Key concepts

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity, represented on a logarithmic scale where 7 is considered neutral. pH values below 7 indicate acidity, while values above 7 imply alkalinity. To calculate the pH of a weak acid solution, we typically use the Henderson-Hasselbalch equation, particularly when dealing with a buffer solution or when the acid is not fully dissociated.

Understanding how the pH corresponds to the concentration of hydrogen ions ( H^+ }) in the solution can be complex, but the Henderson-Hasselbalch equation simplifies this by relating the pH to the pKa, which is the acid's dissociation constant, and the ratio of the concentrations of the dissociated form ([A-]) to the undissociated form ([HA]).

In practical applications, by knowing either the pH or the pKa, and the ratio of the concentrations of the acid and its conjugate base, one can solve for the unknown. For instance, if the pKa is known, and the concentrations of [A-] and [HA] are measured, the pH can be easily determined with the Henderson-Hasselbalch equation.

Weak Acid Dissociation

Weak acids do not fully dissociate in solution, meaning that at equilibrium, both the acid (HA) and its conjugate base (A-) coexist. The extent to which a weak acid dissociates is important in many chemical and biological processes, affecting buffer capacity, solubility, and reactivity. Weak acid dissociation is explained by the equilibrium constant, Ka, which provides a ratio of the concentrations of the dissociated and undissociated species.

As a crucial concept in understanding acid-base chemistry, the degree of dissociation of a weak acid helps us predict the pH of the solution. When interpreting the Henderson-Hasselbalch equation, we can manipulate the ratio of [A-] to [HA] to predict how the pH will change under varying concentrations. For example, an increase in the concentration of [A-] relative to [HA] will result in an increased pH, meaning the solution becomes less acidic.

Acid-Base Equilibrium

Acid-base equilibrium refers to the state in which the rates of the forward and reverse reactions of an acid dissociation are equal, resulting in no net change in concentration of the reactants and products over time. This concept is central to the Henderson-Hasselbalch equation as it assumes that the solution is at equilibrium to relate the [A-] and [HA] concentrations with the pKa and pH.

It's important to remember that the equilibrium does not mean that the amounts of reactants and products are equal, but rather that their ratios do not change. Various factors such as temperature, pressure, and the presence of other substances can shift this equilibrium, leading to changes in pH. The ability to predict and control the outcomes of these shifts is a fundamental aspect of chemistry, particularly in the fields of pharmacology, environmental science, and industrial processes.

By mastering the principles of acid-base equilibrium, students can better understand the behavior of acids and bases in various contexts, such as in biological systems or industrial applications, and anticipate how changes in conditions may impact the pH and overall reactivity of a system.