Question

Derive an equation analogous to the Henderson-Hasselbalch equation but relating pOH and \(\mathrm{p} K_{\mathrm{b}}\) of a buffered solution composed of a weak base and its conjugate acid, such as \(\mathrm{NH}_{3}\) and \(\mathrm{NH}_{4}^{+}\)

Short answer

The short answer based on the step-by-step solution is: To derive an equation analogous to the Henderson-Hasselbalch equation relating pOH and pKb of a buffered solution composed of a weak base (B) and its conjugate acid (HB+), we first write the expression for the base dissociation constant (Kb): \[K_{b} = \frac{[HB^{+}][OH^{-}]}{[B]}\] After assuming that the dissociation is small compared to the initial concentrations and isolating the concentration of hydroxide ions, we obtain the relationship between pOH and pKb: \[pOH = pK_{b} - \log{\frac{[B]_{0}}{[HB^{+}]_{0}}}\]

Step-by-step solution

Write the dissociation expression for the weak base

Write the expression for the dissociation of the weak base (B) in water: \[B + H_{2}O \rightleftharpoons HB^{+} + OH^{-}\]

Write the expression for the base dissociation constant, Kb

Write the expression for the base dissociation constant (Kb): \[K_{b} = \frac{[HB^{+}][OH^{-}]}{[B]}\]

Express the concentrations of the species in terms of the initial concentrations and dissociation amount

Let x be the amount dissociated, we can write the following: \[[HB^{+}] = [HB^{+}]_{0} + x\] \[[B] = [B]_{0} - x\] \[[OH^{-}] = x\]

Substitute the expressions in Step 3 in the Kb expression

Substituting the concentrations of the species from step 3 into the Kb expression from step 2, we get: \[K_{b} = \frac{([HB^{+}]_{0} + x)(x)}{([B]_{0} - x)}\]

Assume that the dissociation is small compared to the initial concentrations

For a weak base, we can assume that x (dissociation) is much smaller than the initial concentrations. Therefore, we can simplify the equation in step 4: \[K_{b} \approx \frac{([HB^{+}]_{0})(x)}{([B]_{0})}\]

Isolate x (concentration of OH- ions)

Isolate x in the equation from step 5 to find the concentration of hydroxide ions: \[[OH^{-}] = x = \frac{K_{b}[B]_{0}}{[HB^{+}]_{0}}\]

Write the expressions for pOH and pKb

Write the expressions for pOH and pKb: \[pOH = -\log{[OH^{-}]}\] \[pK_{b} = -\log{K_{b}}\]

Substitute the expression for the concentration of OH- ions from step 6 in the pOH expression

Substituting the expression for [OH-] from step 6 into the expression for pOH, we get: \[pOH = -\log{\left(\frac{K_{b}[B]_{0}}{[HB^{+}]_{0}}\right)}\]

Manipulate the expression to obtain a relationship between pOH and pKb

Use the properties of logarithms to simplify the expression from step 8: \[pOH = - \left(\log{K_{b}} + \log{\frac{[B]_{0}}{[HB^{+}]_{0}}}\right) = pK_{b} -\log{\frac{[B]_{0}}{[HB^{+}]_{0}}}\] And our final relation between pOH and pKb is: \[pOH = pK_{b} - \log{\frac{[B]_{0}}{[HB^{+}]_{0}}}\]

Key concepts

Buffered Solution

A buffered solution is a special type of solution that resists changes in pH when small amounts of an acid or base are added. These solutions are typically composed of a weak acid and its conjugate base, or a weak base and its conjugate acid.

When a strong acid or base is added, the buffer neutralizes it, preventing significant changes in the pH level.

• Buffers are crucial in maintaining the pH of various environments, such as biological systems and chemical reactions.
• They ensure stability and balance by modulating acidity and basicity in solutions.

Understanding buffered solutions is key in chemistry because it helps control reactions and processes that depend on specific pH ranges.

Weak Base

A weak base is a substance that partially dissociates into ions in an aqueous solution, unlike strong bases, which dissociate completely. This partial dissociation makes weak bases less efficient at deprotonating water.

The equation for a weak base (B) in water is: \[ B + H_{2}O \rightleftharpoons HB^{+} + OH^{-} \]
In this reaction, the weak base B accepts a proton from water, creating its conjugate acid \(HB^{+}\) and hydroxide ions \(OH^{-}\).

• Weak bases include compounds like ammonia \(NH_3\) and other amines.
• They play a vital role in forming buffers with their conjugate acids.

Understanding weak bases is vital for buffering capacity and calculating pOH in solutions.

Conjugate Acid

A conjugate acid is the species that forms when a base accepts a proton during a chemical reaction. In the reaction of a weak base, the conjugate acid is represented as \(HB^{+}\).

When \(B\) accepts a proton, \(HB^{+}\) is formed, creating a critical component of a buffered solution. The relationship between bases and their conjugate acids is foundational in understanding buffering actions.

• Conjugate acids stabilize the solution by reacting with any added base, which helps control pH changes.
• They are part of the equilibrium that defines how buffers work.

This equilibrium allows conjugate acids to prevent drastic pH changes by providing a repository for protons.

pOH

pOH is a measure of the hydroxide ion concentration in a solution and is pivotal in understanding basicity. It is interrelated with pH through the equation:

\[ pH + pOH = 14 \] in pure water at 25°C.The pOH can be calculated using the formula:\[ pOH = -\log{[OH^{-}]} \]Here, \([OH^{-}]\) is the concentration of hydroxide ions in the solution. A low pOH indicates a high concentration of hydroxide ions, which means the solution is more basic.

• pOH is used alongside pKb to determine the strength and behavior of basic solutions.
• It's essential for understanding and calculating buffered solutions and reactions involving weak bases.

By mastering these calculations, students can predict and control the basicity of various solutions.

pKb

pKb is the negative logarithm of the base dissociation constant \(K_b\) and helps quantify the strength of a base. The calculated pKb value provides insight into the basicity of a substance.

Mathematically, it's expressed as:\[ pK_b = -\log{K_b} \]
A lower pKb value indicates a stronger base—a base that more readily accepts protons. This concept ties directly into the Henderson-Hasselbalch equation for bases, relating pOH and pKb:\[ pOH = pK_b - \log{\frac{[B]_{0}}{[HB^{+}]_{0}}} \]Understanding pKb is crucial when working with buffers, as it provides a parameter to balance reactions and achieve desired pH levels. It reflects a base's ability to maintain equilibrium with its conjugate acid and stabilize pH fluctuations.