Question

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

Short answer

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

Step-by-step solution

Write the base ionization expression

To derive the desired equation, we first need to write the base ionization expression. Let B be a weak base, and let BH+ be its conjugate acid. The ionization of the base in water can be represented as follows: \(B + H_{2}O \rightleftharpoons BH^{+} + OH^{-}\)

Write the base ionization constant expression

We can now write the base ionization constant expression, as follows: \(K_{b} = \frac{ [BH^{+}][OH^{-}]}{[B]}\) Here, \(K_{b}\) is the base ionization constant, and [BH+], [OH-], and [B] represent the equilibrium concentrations of the conjugate acid, hydroxide ions, and the base, respectively.

Express pOH and \(pK_{b}\)

Next, we need to express the pOH and the \(pK_{b}\) in terms of their respective logarithms. By definition, pOH is the negative logarithm of the hydroxide ion concentration, and \(pK_{b}\) is the negative logarithm of the base ionization constant. Therefore, we can write: \(pOH = - \log [OH^{-}]\) and \(pK_{b} = - \log K_{b}\)

Take the logarithm of the base ionization constant expression

Now, we will take the logarithm of both sides of the base ionization constant (Step 2) expression: \(\log K_{b} = \log \frac{ [BH^{+}][OH^{-}]}{[B]}\)

Simplify using logarithmic properties

Using logarithmic properties, we can simplify the equation as follows: \(\log K_{b} = \log [BH^{+}] + \log [OH^{-}] - \log [B] \)

Insert pOH and \(pK_{b}\) expressions

Now, replace the \(\log K_{b}\) with \(-pK_{b}\) and \(\log [OH-]\) with \(-pOH\): \(-pK_{b} = \log [BH^{+}] - pOH - \log [B] \)

Rearrange the equation

Finally, let's rearrange the equation to relate pOH to the \(pK_{b}\) of the base component: \(pOH = pK_{b} + \log \frac{[B]}{[BH^{+}]}\) This is the derived equation that relates the pOH of a buffer solution to the \(pK_{b}\) of its base component, similar to the Henderson-Hasselbalch equation.

Key concepts

pOH

When we talk about pOH, we are discussing a measure related to the concentration of hydroxide ions \(OH^-\). Essentially, pOH provides insight into how basic a solution is.
It's the counterpart to pH, which measures acidity. Calculating pOH is straightforward; it is the negative logarithm of the hydroxide ion concentration:

• \(pOH = - \log [OH^-]\)

The scale for pOH is similar to the pH scale, ranging from 0 to 14. A low pOH corresponds to a high concentration of hydroxide ions and therefore a strong basic or alkaline solution. Conversely, a high pOH is indicative of a lower base concentration and thus a weaker base solution.
Remember, pOH and pH are related:

• \(pOH + pH = 14\)

This relationship helps in understanding the full picture of a solution's acidity or basicity.

pKb

In chemical terms, the pKb is a measure of how easily a base ionizes in water. It tells us about the strength of the base in a similar way that pKa does for acids.
The definition of pKb is the negative logarithm of the base ionization constant (Kb):

• \(pK_b = - \log K_b\)

A lower pKb indicates a stronger base, which means that it ionizes more readily in water. Conversely, a higher pKb signifies a weaker base. The ionization is described by the base ionization equation, where a base B reacts with water to form the conjugate acid BH+ and hydroxide ions OH-.The derived equation that connects pOH with pKb shows us that the relation of base components also factors in the logarithm of the base and its conjugate acid ratio:

• \(pOH = pK_b + \log \frac{[B]}{[BH^+]}\)

Understanding pKb is vital when working with buffer solutions, and it's comparable in meaning and importance to pKa.

buffer solution

A buffer solution plays an essential role in maintaining the pH of a solution despite additions of acids or bases. These solutions are composed of a weak acid and its conjugate base or a weak base and its conjugate acid.
This unique mix helps the solution resist significant changes in pH. When an acid or base is added to a buffer, the components of the buffer will react with the addition:

• If an acid is added, the conjugate base in the buffer will neutralize it.
• Similarly, if a base is added, the weak acid of the buffer will neutralize it.

Buffer solutions are crucial in various scientific fields, ranging from biochemistry to environmental science. The closeness in concentration of the base to its conjugate acid influences how well the buffer can resist pH changes.
The Henderson-Hasselbalch equation, and its analogous forms like the one including pOH, help us calculate the expected pH or pOH for given buffer compositions.

base ionization constant

The base ionization constant, labeled as Kb, provides information about the extent to which a base can ionize in solution. This constant is fundamental in specifying the strength of a base.
For the general base ionization reaction: \(B + H_2O \rightleftharpoons BH^+ + OH^-\),
the expression for the base ionization constant is:

• \(K_b = \frac{[BH^+][OH^-]}{[B]}\)

In this formula:

• \([BH^+]\) represents the concentration of the conjugate acid.
• \([OH^-]\) denotes the concentration of hydroxide ions.
• \([B]\) is the concentration of the base.

A greater Kb value means the base is stronger and ionizes more completely in solution, while a smaller Kb value implies a weaker base. Calculating and understanding Kb is essential when applying the Henderson-Hasselbalch-type equations to buffer systems.
It helps determine how a base will behave in a buffer solution and how it will affect the equilibrium of the system.