Question

Mathematically prove that the pH at the halfway point of a titration of a weak acid with a strong base (where the volume of added base is half of that needed to reach the equivalence point) is equal to \(\mathrm{p}_{u}\) for the acid.

Short answer

At the halfway point of a titration of a weak acid with a strong base, the concentrations of the weak acid and its conjugate base are equal. Using the Henderson-Hasselbalch equation, \( pH = pKa + \log{\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}} \), and substituting the equal concentrations, we get \( pH = pKa + \log{\frac{[\mathrm{HA}]}{[\mathrm{HA}]}} \). Since the logarithm of 1 is 0, the equation simplifies to \( pH = pKa \). Thus, at the halfway point, the pH of the solution is equal to the pKa of the weak acid.

Step-by-step solution

Write down the Henderson-Hasselbalch equation.

The Henderson-Hasselbalch equation helps us relate the pH, pKa, and the ratio of the concentrations of the conjugate base and the weak acid. The equation is: \[ pH = pKa + \log{\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}} \] Here, \(pH\) is the pH of the solution, \(pKa\) is the negative logarithm of the acid dissociation constant Ka, \([\mathrm{A}^-]\) represents the molar concentration of the conjugate base, and \([\mathrm{HA}]\) represents the molar concentration of the weak acid.

Determine the relationship between the weak acid and conjugate base concentrations.

At the halfway point of the titration, exactly half of the weak acid has been neutralized by the strong base and converted into its conjugate base. This means that the concentrations of the weak acid and its conjugate base should be equal at the halfway point: \[[\mathrm{A}^-] = [\mathrm{HA}]\]

Substitute the relationship into the Henderson-Hasselbalch equation.

Now, we will substitute the relationship we found in Step 2 into the Henderson-Hasselbalch equation from Step 1: \[ pH = pKa + \log{\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}} \] Since \([\mathrm{A}^-] = [\mathrm{HA}]\), we can rewrite the equation as: \[ pH = pKa + \log{\frac{[\mathrm{HA}]}{[\mathrm{HA}]}} \]

Simplify the equation and prove that pH equals pKa.

In this step, we will simplify the equation from Step 3: \[ pH = pKa + \log{\frac{[\mathrm{HA}]}{[\mathrm{HA}]}} = pKa + \log{1} \] Since the logarithm of 1 is 0, the equation simplifies further: \[ pH = pKa + 0 = pKa \] Thus, at the halfway point of a titration of a weak acid with a strong base, the pH of the solution is equal to the pKa of the weak acid. This completes the proof as required by the exercise.

Key concepts

Henderson-Hasselbalch Equation

Understanding the Henderson-Hasselbalch equation is crucial for students studying chemistry, particularly when exploring the titration of weak acids with strong bases. Simply put, this equation offers a direct relationship between the pH of a solution and the pKa (acid dissociation constant) of the acid in question. The equation reads:

\[ pH = pKa + \log{\frac{[\mathrm{A}^-]}{[\mathrm{HA}]}} \]

This formula comes into play during titration, where it acts as a bridge connecting the pH levels to the ratio of the concentrations of the conjugate base (\(\mathrm{A}^-\)) to the weak acid (\(\mathrm{HA}\)). The equation shines at the halfway point of a titration process, where it can be demonstrated that the pH equals the pKa of the weak acid due to the equal concentrations of the weak acid and its conjugate base.

Acid Dissociation Constant (pKa)

Delving into the acid dissociation constant, denoted as pKa, is vital to grasp the behavior of acids in solution. The pKa value is a quantitative measure of the strength of an acid in solution, and it is the negative logarithm of the acid dissociation constant (Ka). It provides a clear indication of the extent to which an acid can donate a proton in an aqueous solution. From the exercise, when we discuss the titration of a weak acid, the pKa takes center stage as it equals the pH at the midpoint of the titration. It isn't just a number; it's a significant piece of the puzzle for predicting the pH at various points during a titration. The lower the pKa, the stronger the acid, and this becomes a handy reference during titration analysis.

\[ pKa = -\log(Ka) \]

It's also the point of inflection on a titration curve, where 50% of the acid has been neutralized and thus represents a crucial checkpoint in the process.

pH Calculation

The pH calculation is a cornerstone in chemistry that quantifies the acidity or alkalinity of an aqueous solution. pH is calculated as the negative logarithm of the hydrogen ion concentration in moles per liter (\(M\)). The pH scale runs from 0 to 14, where 7 is neutral. A pH less than 7 indicates an acidic solution, while a pH greater than 7 indicates a basic one.

\[ pH = -\log([H^+]) \]

Calculating pH is a daily task for chemists, especially when conducting titrations. As seen in the textbook exercise, by employing the Henderson-Hasselbalch equation at the halfway point of a titration, the pH is found to be equal to the pKa. This moment is called the buffering region, where the solution's pH remains relatively stable, despite the addition of small amounts of acid or base.

Chemical Equilibrium

Chemical equilibrium is the point in a reaction at which the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of reactants and products remain constant over time. This balance is essential for reactions involving weak acids and their conjugate bases, like in a titration scenario.

\[ aA + bB \rightleftharpoons cC + dD \]

Here, in a state of equilibrium, the ratio of the products over reactants raised to the power of their stoichiometric coefficients is constant (the equilibrium constant, \(K_eq\)). Equilibrium plays an indispensable role during titration, as we continually shift the equilibrium as we add base to acid. It's the reason why the concentrations of the acid and its conjugate base at the halfway point of a titration are equal, creating a perfect case study for understanding equilibrium in action.

Buffer Solution

Finally, the concept of a buffer solution warrants exploration, as it relates directly to the behavior observed at the halfway point of the titration exercise. A buffer solution is one that can resist changes in pH upon the addition of small amounts of acid (H+) or base (OH−). It typically consists of a weak acid and its conjugate base and is most effective at the pH near its acid's pKa.
During the titration process, at the halfway point, we essentially have a buffer system because the amounts of the weak acid and its conjugate base are equal. As the strong base is added to the weak acid, the acid neutralizes the base, and its conjugate form accumulates, thus mitigating vast swings in pH. This property is crucial in many biological and chemical applications where maintaining a stable pH is imperative. The understanding of buffers, and how they are established during titration processes, highlights the practical application of concepts like the Henderson-Hasselbalch equation and acid-base equilibrium in real-world scenarios.