Problem 83 - Chemistry: The Central Science in SI Units, Global Edition

Question

Derive an equation similar to the Henderson-Hasselbalch equation relating the pOH of a buffer to the \(\mathrm{pK}_{b}\) of its base component.

Step-by-Step Solution

Answer

The derived equation relating pOH to the \(\mathrm{pK}_{b}\) of a buffer's base component is: \[ pOH = pK_{b} + \log{\frac{[B]}{[BH^+]}} \]

Step 1: Write the general equation for a weak base reacting with water

For a weak base B, the general equation for its reaction with water can be written as: B + H2O <=> BH+ + OH-

Step 2: Write the expression for the base ionization constant (Kb)

The base ionization constant can be described as the equilibrium constant for the ionization of B: \[ K_{b} = \frac{[BH^+][OH^-]}{[B]}\]

Step 3: Find the relationship between OH- and the weak base

The pOH is related to the concentration of OH- through the equation: \[ pOH = -\log [OH^-]\] We can rewrite this expression to find the concentration of OH- in terms of pOH: \[ [OH^-] = 10^{-pOH}\]

Step 4: Write the expression for the pKb

The relationship between the base ionization constant (Kb) and pKb is given by: \[ pK_{b} = -\log K_{b}\] Rewrite the expression to find Kb in terms of pKb: \[ K_{b} = 10^{-pK_{b}}\]

Step 5: Combine the previous expressions to derive the required equation

Substitute the expressions for [OH-] and Kb from Steps 3 and 4 into the Kb expression derived in step 2: \begin{align*} 10^{-pK_{b}} &= \frac{[BH^+][10^{-pOH}]}{[B]} \end{align*} Now, we solve for pOH: \begin{align*}pOH &= pK_{b} + \log{\frac{[B]}{[BH^+]}}\end{align*} This is the required equation, which is similar to the Henderson-Hasselbalch equation, but relates the pOH of a buffer to the \(\mathrm{pK}_{b}\) of its base component.

Key Concepts

pOH

Understanding the concept of pOH is essential in the study of bases and buffer solutions. pOH is analogous to pH, but instead of focusing on the concentration of hydrogen ions \([H^+]\), it centers on the concentration of hydroxide ions \([OH^-]\). \[\text{pOH} = -\log [OH^-] \\]Thus, pOH provides a measure of how basic or alkaline a solution is. The scale of pOH ranges from 0 to 14, where lower values indicate a more basic solution. If you know the pOH of a solution, you can easily find the pH and vice versa since the sum of pH and pOH for any aqueous solution at 25°C is always 14:

• \(\text{pH} = 14 - \text{pOH}\)
• If pOH is small, the solution is more basic.

Understanding these concepts is crucial for dealing with buffer solutions and any chemical reaction involving bases.

pKb

The term pKb is used to describe the strength of a base, similar to how pKa describes the strength of an acid. It is derived from the base ionization constant \(K_b\), which measures a base's ability to dissociate into its ions in solution. \[\text{pK}_b = -\log K_b\\]A lower pKb value indicates a stronger base because it implies a higher degree of ionization. In contrast, a higher pKb suggests a weaker base. By calculating pKb, one can compare the basic strength of different substances and use it in buffer equations. Thus, pKb is a critical aspect when applying the Henderson-Hasselbalch equation to basic buffer solutions, showing the direct relationship between base strength and solution alkalinity.

Buffer Solution

Buffer solutions are crucial in maintaining the pH balance of a system, making them invaluable in various scientific and industrial applications. They resist changes in pH when small amounts of acid or base are added, thus stabilizing the solution. A buffer solution usually consists of:

• A weak acid and its conjugate base, or
• A weak base and its conjugate acid.

In the context of bases, a buffer solution can include a weak base \(\text{B}\) and its conjugate acid \(\text{BH}^+\). The equilibrium between these two components allows the solution to neutralize added acids (by consumption of base \(\text{B}\)) or bases (by consumption of acid \(\text{BH}^+\)).
This neutralization mechanism is vital for maintaining a stable pH which is key for biochemical processes and reactions in living organisms. Understanding buffer solutions with bases and acids is essential for deriving equations, like the pOH variant of the Henderson-Hasselbalch equation.

Base Ionization Constant

The base ionization constant \(K_b\) is an equilibrium constant that reflects the degree to which a base can ionize in solution. For a general weak base \(\text{B}\), the reaction in water is expressed as:

• \(\text{B} + \text{H}_2\text{O} \rightleftharpoons \text{BH}^+ + \text{OH}^-\)

The expression for \(K_b\) is given by:\[ K_b = \frac{[BH^+][OH^-]}{[B]} \\]A larger \(K_b\) value indicates a stronger base that ionizes more completely in solution. In buffer solutions, \(K_b\) is used to relate concentrations of the conjugate acid and base, allowing derivations like the one that forms the pOH equation akin to the Henderson-Hasselbalch equation. Recognizing how \(K_b\) influences base behavior in solutions is crucial for understanding the chemical dynamics of buffer systems.