Question

Two solutions are made containing the same concentrations of solutes. One solution is composed of \(\mathrm{H}_{3} \mathrm{PO}_{4}\) and \(\mathrm{Na}_{3} \mathrm{PO}_{4}\), while the other is composed of HCN and \(\mathrm{NaCN}\). Which solution should have the larger capacity as a buffer?

Short answer

The solution with \( \mathrm{H}_3 \mathrm{PO}_4 \) and \( \mathrm{Na}_3 \mathrm{PO}_4 \) has a larger buffer capacity.

Step-by-step solution

Understanding Buffer Solutions

A buffer solution consists 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.

Identify the Buffer Components

Both solutions contain a weak acid and its conjugate base. The first solution consists of \( \mathrm{H}_3 \mathrm{PO}_4 \) (weak acid) and \( \mathrm{Na}_3 \mathrm{PO}_4 \) (its conjugate base). The second solution contains HCN (weak acid) and \( \mathrm{NaCN} \) (its conjugate base).

Compare Acid-Base Strengths

The buffering capacity is influenced by the strength of the weak acid. \( \mathrm{H}_3 \mathrm{PO}_4 \) is a stronger weak acid than HCN because it has a higher dissociation constant \( K_a \).

Analyze Buffer Capacity

A solution with a stronger weak acid in terms of higher \( K_a \) will typically have a larger buffer capacity because it is more effective at resisting pH changes. Since \( \mathrm{H}_3 \mathrm{PO}_4 \) has a higher \( K_a \) than HCN, the first solution (\( \mathrm{H}_3 \mathrm{PO}_4 \) and \( \mathrm{Na}_3 \mathrm{PO}_4 \)) should have the larger buffer capacity.

Key concepts

Weak Acid and Conjugate Base

A buffer solution typically contains a mixture of a weak acid and its conjugate base. This unique combination is what gives buffer solutions their ability to resist significant changes in pH. Imagine a weak acid as a molecule that doesn't completely release its hydrogen ions in a solution. Instead, it partially dissociates, leaving most of its acid molecules intact. Similarly, its conjugate base is formed by removing a hydrogen ion from the weak acid.

• For example, in one of the solutions given in the exercise, the weak acid is \( \mathrm{H}_3 \mathrm{PO}_4 \) and its conjugate base is part of \( \mathrm{Na}_3 \mathrm{PO}_4 \).
• In the other solution, the weak acid is HCN, and the conjugate base is \( \mathrm{NaCN} \).

By having both a weak acid and its conjugate base, a buffer solution can neutralize small added amounts of either acid or base, preventing drastic pH changes. This is crucial in many natural systems, such as human blood, where a stable pH is vital for proper function.

Buffer Capacity

Buffer capacity refers to the effectiveness of a buffer solution to resist pH changes. It's like the strength of a wall holding back a flood. The higher the buffer capacity, the more resistant the solution is to changes. Two main factors affect buffer capacity: the absolute concentrations of the weak acid and its conjugate base, and the inherent strength of the weak acid, often measured by its dissociation constant, \( K_a \).
A solution with a larger \( K_a \) indicates the weak acid is stronger within the buffer system, providing greater buffer capacity. This is because a stronger weak acid can better donate and accept protons.

• In our case, since \( \mathrm{H}_3 \mathrm{PO}_4 \) has a higher \( K_a \) value than HCN, it's anticipated to provide a larger buffer capacity for that particular solution.

Thus, when comparing two buffer solutions, like the ones in the exercise, the solution with the weak acid having the higher \( K_a \) usually wins in terms of buffer capacity.

Dissociation Constant

The dissociation constant, symbolized as \( K_a \), is a measure that indicates the extent to which a weak acid dissociates into its ions in solution. It's a critical factor in understanding the strength of a weak acid. A higher \( K_a \) means that more molecules of the acid are dissociating, contributing more hydrogen ions to the solution.

• For example, the dissociation of \( \mathrm{H}_3 \mathrm{PO}_4 \) can be represented by the equation: \[ \mathrm{H}_3 \mathrm{PO}_4 \rightleftharpoons \mathrm{H}^+ + \mathrm{H}_2 \mathrm{PO}_4^- \]
• The equation for HCN would be: \[ \mathrm{HCN} \rightleftharpoons \mathrm{H}^+ + \mathrm{CN}^- \]

The \( K_a \) value helps in determining how readily an acid can donate protons and thus influence the buffer capacity. Thus, in our solutions, since \( \mathrm{H}_3 \mathrm{PO}_4 \) has a larger \( K_a \) than HCN, we can conclude that it is a stronger weak acid and should form a buffer system with greater capacity. Understanding the dissociation constant aids in predicting the behavior of buffer solutions and their efficacy in resisting pH changes.