Question

From the relationship \(K_{\mathrm{a}} K_{\mathrm{b}}=K_{\mathrm{w}},\) what can you deduce about the relative strengths of a weak acid and its conjugate base?

Short answer

A weak acid has a relatively stronger conjugate base than a strong acid.

Step-by-step solution

Understand the Relationship

The equation \(K_a K_b = K_w\) relates the acid dissociation constant \(K_a\), the base dissociation constant \(K_b\), and the ionic product of water \(K_w\). \(K_w\) is approximately \(1.0 \times 10^{-14}\) at 25 degrees Celsius.

Consider the Implications for a Strong Acid

If an acid is strong, \(K_a\) is very large, because the acid fully dissociates in water. For its conjugate base, \(K_b\) would need to be extremely small for their product to equal \(K_w\). This indicates that a strong acid has a very weak conjugate base.

Consider the Implications for a Weak Acid

For a weak acid, \(K_a\) is small, as the acid does not fully dissociate in solution. Therefore, its conjugate base will have a larger \(K_b\) compared to the conjugate base of a strong acid. However, \(K_b\) will still be small overall, as the product of \(K_a\) and \(K_b\) must equal \(K_w\).

Relate the Acid and Base Strengths

In conclusion, a weak acid has a conjugate base that is stronger than that of a strong acid's conjugate base, but still a weak base in general. The relative strengths are inversely related based on the equation \(K_a \times K_b = K_w\).

Key concepts

Weak Acid

A weak acid is a type of acid that does not fully dissociate in water. This means that, in a solution, most of the acid molecules remain intact while only a small fraction releases hydrogen ions (H⁺). The extent of dissociation is represented by the acid dissociation constant, denoted as \( K_a \).

If \( K_a \) is less than 1, the acid is considered weak since it indicates a low tendency to lose protons and form its conjugate base. Weak acids are very common, with examples including acetic acid (found in vinegar) and citric acid (found in citrus fruits).

Remember:

• A smaller \( K_a \) means weaker acid strength.
• Weak acids partially dissociate in solution.
• They are in a chemical equilibrium between the acid and its ions.

This incomplete dissociation leads to interesting behaviors in solution chemistry, where the interaction between the weak acid and its conjugate base plays a critical role.

Conjugate Base

When an acid donates a proton (H⁺), it forms what is known as a conjugate base. This conjugate base is the substance that results from the loss of a hydrogen ion and is capable of gaining a proton back. The strength of a conjugate base is linked directly to the strength of its parent acid.

For weak acids, the conjugate base tends to be stronger than its counterpart from a strong acid. This means that the base is more likely to re-associate with a proton. However, even the conjugate bases of weak acids are still weak compared to strong bases like hydroxides.

Key Points:

• The conjugate base of a weak acid is moderately strong.
• It is in equilibrium with the weak acid in the solution.
• The relative strength is determined by the base dissociation constant, \( K_b \).

The interplay between an acid and its conjugate base can determine the pH and properties of a solution.

Dissociation Constant

The dissociation constant is a crucial term in understanding the behavior of acids and bases in water. For an acid, the dissociation constant is represented by \( K_a \), while for a base, it is \( K_b \). These constants quantify the extent to which an acid or base dissociates in a solution.

For a weak acid:
\[K_a = \frac{[H^+][A^-]}{[HA]}\]
Here, \( [H^+] \) is the concentration of hydrogen ions, \( [A^-] \) is the concentration of the conjugate base, and \( [HA] \) is the concentration of the undissociated acid. A low \( K_a \) signifies a weak acid, while a low \( K_b \) signifies a weak conjugate base.

Essential Aspects:

• \( K_a \) and \( K_b \) are inversely related as they multiply to give \( K_w \), the ionic product of water.
• These constants provide insight into the acid and base strength and their behavior.

Understanding dissociation constants helps predict the resulting chemical species in a solution and their concentrations.

Ionic Product of Water

The ionic product of water, represented as \( K_w \), is a fundamental constant in the study of acid-base equilibrium in aqueous solutions. At 25 °C, \( K_w \) has a value of approximately \( 1.0 \times 10^{-14} \).

This constant arises from the self-ionization of water, which can be represented as:
\[H_2O(l) \rightleftharpoons H^+(aq) + OH^-(aq)\]
For pure water, the concentrations of hydrogen ions \([H^+]\) and hydroxide ions \([OH^-]\) are equal, leading to:
\[K_w = [H^+][OH^-] = 1.0 \times 10^{-14} \]
Important Takeaways:

• \( K_w \) helps in calculating the pH and pOH of solutions.
• It is the product of respective dissociation constants \( K_a \) for acids and \( K_b \) for bases, such that \( K_a \times K_b = K_w \).

Understanding \( K_w \) aids in comprehending the intricate balance of ions in water, which impacts the acidity or basicity of solutions.