Question

A certain buffer solution contains equal concentration of \(X^{-}\) and \(H X\). The \(K p\) for \(X^{-}\) is \(10^{-10}\). The \(\mathrm{pH}\) of the buffer is: (a) 6 (b) 10 (c) 4 (d) 14

Short answer

The pH of the buffer is 4.

Step-by-step solution

Understand the Relationship between pH and pKa

The problem involves a buffer solution containing equal concentrations of the conjugate base, \(X^{-}\), and its conjugate acid, \(HX\). The Henderson-Hasselbalch equation describes the pH of a buffer system: \[\text{pH} = \text{pKa} + \log \left( \frac{[ \text{Base} ]}{[ \text{Acid} ]} \right)\]Since the concentrations of \([X^-]\) and \([HX]\) are equal, the log term becomes \( \log(1) = 0 \). Therefore, \( \text{pH} = \text{pKa} \).

Calculate pKa from Kb

The given problem provides the base dissociation constant, \(K_b\), for \(X^-\), which is \(10^{-10}\). To find \( \text{pKa} \), we use the relationship between \( K_a \) and \( K_b \) for a conjugate acid-base pair, given by:\[K_w = K_a \times K_b\]where \(K_w = 1 \times 10^{-14}\). Therefore,\[K_a = \frac{K_w}{K_b} = \frac{1 \times 10^{-14}}{10^{-10}} = 10^{-4}\]The \(\text{pKa}\) is calculated as:\[\text{pKa} = -\log(K_a) = -\log(10^{-4}) = 4\]

Determine the pH of the Buffer

Since \( \text{pH} = \text{pKa} \), and we previously calculated \(\text{pKa} = 4\), it follows that the \( \text{pH} \) of the buffer is also 4.

Key concepts

Henderson-Hasselbalch equation

In the world of chemistry, buffer solutions help maintain stable pH levels by resisting changes when small amounts of acid or base are added. The Henderson-Hasselbalch equation is a simple way to calculate the pH of a buffer solution. This equation states:\[ \text{pH} = \text{pKa} + \log \left( \frac{[ \text{Base} ]}{[ \text{Acid} ]} \right) \]This equation assumes a buffer system made up of a weak acid and its conjugate base or a weak base and its conjugate acid. When the concentrations of these two components are equal, the logarithmic part of the equation, \( \log(1) \), equals zero. This situation simplifies things considerably, as the pH of the solution equals the pKa of the acid. It's like finding the sweet spot of balance. Understanding how the Henderson-Hasselbalch equation functions allows you to predict how the buffer will respond to changes in pH when acids or bases are introduced.

pKa and pKb relationship

In acid-base chemistry, knowing the dissociation constants provides a lot of insight into a reaction. The pKa and pKb (base dissociation constant) are essential in describing the strength of acids and bases respectively. They are interconnected through the following equation:\[ K_w = K_a \times K_b \]where \(K_w\) is the ion-product constant of water, which is \(1 \times 10^{-14}\) at 25°C. By rearranging this formula, you can find either \(K_a\) or \(K_b\) provided you know one and \(K_w\). This relationship shows that the stronger the acid, the weaker its conjugate base and vice versa. In practical terms, finding \(K_a\) from \(K_b\) involves a bit of arithmetic manipulation, but it ultimately tells you a great deal about how acids and bases will behave in solution. From \(K_a\), the pKa can be deduced using the formula:\[ \text{pKa} = -\log K_a \]This measure tells you just how easily a proton can be given off, guiding predictions on buffer behavior.

pH calculation

Calculating pH is crucial in many chemical solutions, especially buffers that must maintain certain pH levels. The pH scale ranges from 0 to 14 and indicates how acidic or basic a solution is, with lower numbers being more acidic and higher numbers more basic. The formula to find pH is:\[ \text{pH} = -\log[H^+] \]where \([H^+]\) is the concentration of hydrogen ions in the solution. For buffers, the process includes using the previously mentioned Henderson-Hasselbalch equation when dealing with its components. In our example, since the concentrations of the acid and base parts are equal, this simplifies calculations: the pH is directly equal to the pKa of the acid in the buffer. Given the simplicity and power of these tools, once you grasp how to apply them, predicting the pH of various solutions becomes straightforward and reliable. Understanding pH doesn't just help in theoretical scenarios, it also aids in practical applications like calibrating buffer solutions for experiments.