Question

A buffer contains a weak acid, \(\mathrm{HX}\), and its conjugate base. The weak acid has a \(\mathrm{pK}_{a}\) of \(4.5\), and the buffer solution has a \(\mathrm{pH}\) of \(4.3\). Without doing a calculation, predict whether \([\mathrm{HX}]=\left[\mathrm{X}^{-}\right],[\mathrm{HX}]>\left[\mathrm{X}^{-}\right]\), or \([\mathrm{HX}]<\left[\mathrm{X}^{-}\right]\) Explain. [Section 17.2]

Short answer

Based on the Henderson-Hasselbalch equation, we can predict that in this buffer solution with a pH of 4.3 and a pKa of 4.5, the concentration of the weak acid (HX) will be greater than the concentration of its conjugate base (X⁻). This is because pH < pKa, which implies that \([\mathrm{HX}] > [\mathrm{X}^{-}]\).

Step-by-step solution

Understand the Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is used to calculate the \(\mathrm{pH}\) of a buffer solution. It relates the \(\mathrm{pH}\), \(\mathrm{pK}_{a}\), and the concentrations of a weak acid and its conjugate base. The equation is: \( \mathrm{pH} = \mathrm{pK}_{a} + \log \frac{[\mathrm{X}^{-}]}{[\mathrm{HX}]} \) In this exercise, we are asked to predict the relationship between \([\mathrm{HX}]\) and \([\mathrm{X}^{-}]\) using the given \(\mathrm{pH}\) and \(\mathrm{pK}_{a}\) values.

Analyze the relationship between \(\mathrm{pH}\) and \(\mathrm{pK}_{a}\)

We have the \(\mathrm{pH}\) and \(\mathrm{pK}_{a}\) values as follows: \( \mathrm{pH} = 4.3 \) \( \mathrm{pK}_{a} = 4.5 \) Now, look at the Henderson-Hasselbalch equation: \( \mathrm{pH} = \mathrm{pK}_{a} + \log \frac{[\mathrm{X}^{-}]}{[\mathrm{HX}]} \) Notice that if \(\mathrm{pH} > \mathrm{pK}_{a}\), then the fraction \(\frac{[\mathrm{X}^{-}]}{[\mathrm{HX}]}\) must be greater than 1, which means \([\mathrm{X}^{-}] > [\mathrm{HX}]\). Conversely, if \(\mathrm{pH} < \mathrm{pK}_{a}\), then the fraction \(\frac{[\mathrm{X}^{-}]}{[\mathrm{HX}]}\) must be less than 1, meaning \([\mathrm{X}^{-}] < [\mathrm{HX}]\).

Predict the relationship between the concentrations of the weak acid and its conjugate base

From our previous analysis, we know that if \(\mathrm{pH} < \mathrm{pK}_{a}\), then \([\mathrm{X}^{-}] < [\mathrm{HX}]\). Since the given values are: \( \mathrm{pH} = 4.3 \) \( \mathrm{pK}_{a} = 4.5 \) The condition \(\mathrm{pH} < \mathrm{pK}_{a}\) is fulfilled, and we can predict that in this buffer solution, \([\mathrm{HX}] > [\mathrm{X}^{-}]\).

Key concepts

pH and pKa Relationship

Understanding the relationship between pH (power of hydrogen) and pKa (acid dissociation constant) is essential for mastering acid-base chemistry. This relationship tells us about the concentration of hydrogen ions in a solution. The pH is a logarithmic scale used to quantify the acidity or basicity of a solution, while pKa is a specific value that indicates the strength of an acid, defining the equilibrium point at which half of the acid molecules lose their hydrogen ions.

In terms of the Henderson-Hasselbalch equation, which is \[\mathrm{pH} = \mathrm{pK}_{a} + \log \frac{[\mathrm{X}^{-}]}{[\mathrm{HX}]} \]the pH can be directly compared to the pKa to understand the predominant form of the acid-base pair in the solution. If the pH is lesser than pKa, the form that donates the proton (the acid form, HX) is predominant. Conversely, if the pH is greater than pKa, the form that accepts the proton (the conjugate base form, X⁻) is predominant. This relationship helps in predicting the buffering capacity and the form of the species present in the solution.

Acid-Base Buffer Systems

Acid-base buffer systems are crucial for maintaining stable pH levels in various biological and chemical environments, including human blood. A buffer is a mixture of a weak acid and its conjugate base, or a weak base and its conjugate acid. It resists changes in pH upon the addition of small amounts of acids or bases. This resistance is due to the presence of the two forms (the acid and its conjugate base), which can neutralize added H⁺ or OH⁻ ions.

For example, the addition of a base leads to the formation of more acid, counteracting the pH increase. Conversely, the addition of an acid converts some of the conjugate base into the acid form, mitigating the pH decrease. The effectiveness of a buffer is greatest at or near the pKa value of the weak acid or base it contains. This is where the Henderson-Hasselbalch equation becomes a practical tool for predicting buffer behavior in acid-base disturbances.

Weak Acid and Conjugate Base Concentration

In a buffer solution, the concentrations of the weak acid (HX) and its conjugate base (X⁻) are intimately related to the solution's ability to maintain its pH. The Henderson-Hasselbalch equation showcases this relationship beautifully. The equation implies that for a buffer at its optimal buffering pH (which is at the pKa), the concentrations of the weak acid and its conjugate base are equal.

However, deviations from the pKa value indicate an imbalance in the concentrations. When the pH is lower than the pKa, the concentration of the weak acid, [HX], is higher than that of the conjugate base, [X⁻], as seen in our earlier analysis. The reverse is true if the pH is higher than the pKa. Understanding these concentration dynamics is fundamental for designing effective buffers, especially in pharmaceutical and industrial processes that require strict pH control.