Problem 102 - Chemistry The Central Science In Si Units

Question

Which, if any, of the following statements are true? (a) The stronger the base, the smaller the \(\mathrm{p} K_{b}\). (b) The stronger the base, the larger the \(\mathrm{p} K_{b}\). (c) The stronger the base, the smaller the \(K_{b}\). (d) The stronger the base, the larger the \(K_{b}\). (e) The stronger the base, the smaller the \(\mathrm{p} K_{a}\) of its conjugate acid. (f) The stronger the base, the larger the \(\mathrm{pK}_{a}\) of its conjugate acid.

Step-by-Step Solution

Answer

The true statements are (a), (d), and (f).

Step 1: Understand the Meaning of pKb and Kb

The base dissociation constant, \(K_b\), measures the strength of a base in water. The \( \text{p}K_b \) is the negative logarithm of \(K_b\) and provides an inverse measure, where smaller \( \text{p}K_b \) values correspond to stronger bases, because the base dissociation constant \(K_b\) is larger.

Step 2: Analyze Statement (a)

This statement says "the stronger the base, the smaller the \( \text{p}K_b \)." This is true because a stronger base has a higher \(K_b\), which leads to a smaller \( \text{p}K_b\).

Step 3: Analyze Statement (b)

This statement suggests "the stronger the base, the larger the \( \text{p}K_b \)." This is false as explained in step 1 because a stronger base actually has a smaller \( \text{p}K_b\).

Step 4: Analyze Statement (c)

This statement suggests "the stronger the base, the smaller the \(K_b\)." This is false because a stronger base should have a larger \(K_b\).

Step 5: Analyze Statement (d)

This statement says "the stronger the base, the larger the \(K_b\)." This is true because a stronger base dissociates more in water, which results in a larger \(K_b\).

Step 6: Understand the Relationship Between pKa and pKb

The \(\text{p}K_a\) and \(\text{p}K_b\) are related by the equation \(\text{p}K_a + \text{p}K_b = 14\). Thus, if \(\text{p}K_b\) is smaller for stronger bases, \(\text{p}K_a\) of the conjugate acid should be larger.

Step 7: Analyze Statement (e)

This statement says "the stronger the base, the smaller the \( \text{p}K_a \) of its conjugate acid," which is false because a stronger base means a larger \( \text{p}K_a \) for its conjugate acid.

Step 8: Analyze Statement (f)

This statement suggests "the stronger the base, the larger the \(\text{p}K_a\) of its conjugate acid." This is true as explained in Step 6, because a stronger base with a smaller \(\text{p}K_b\) corresponds to a larger \(\text{p}K_a\) of its conjugate acid.

Key Concepts

pKb and Kb relationship

Understanding the relationship between \(\text{p}K_b\) and \(K_b\) can help make predictions about the strength of a base. \(K_b\) is known as the base dissociation constant. It measures how well a base dissociates into its ions in water. When a base has a large \(K_b\), it means the base dissociates well, thus indicating a stronger base. On the other hand, the \(\text{p}K_b\) value is derived from \(K_b\) and is calculated using the formula: \(\text{p}K_b = -\log(K_b)\). This formula makes it clear that a large \(K_b\) results in a small \(\text{p}K_b\) and vice-versa.
So remember:

• Smaller \(\text{p}K_b\) values ➔ Stronger base.
• Larger \(K_b\) values ➔ Stronger base.

Whether you are analyzing data from a chemical experiment or trying to solve homework problems, this inverse relationship is essential in understanding acid-base equilibrium.

Conjugate acid-base pairs

Conjugate acid-base pairs are crucial in the study of acid-base chemistry. When a base accepts a proton, it becomes its conjugate acid. Conversely, when an acid donates a proton, it forms its conjugate base. These pairs help chemists predict the behavior of substances in chemical reactions.
The pKa and pKb relationship plays a significant role in understanding these pairs. The equation \(\text{p}K_a + \text{p}K_b = 14\) at 25 °C implies that a strong base with a low \(\text{p}K_b\) will have a conjugate acid with a high \(\text{p}K_a\). Essentially, strong bases correspond to weak conjugate acids, and vice versa.
Key takeaways:

• Strong base ➔ Weak conjugate acid.
• The strength of conjugate pairs links inversely.

This dynamic is fundamental in chemical equilibrium and contributes to predicting reaction directions and product stability.

Base dissociation constant Kb

The base dissociation constant, \(K_b\), is a quantitative measure of a base's strength in water. It provides insight into how completely a base dissociates into its ions in solution. For example, a base like ammonia (NH3) dissociates partially in water, resulting in a certain \(K_b\) value.
Calculated from the concentration of the ions and molecules in water, \(K_b\) values help categorize bases:

• High \(K_b\) value ➔ Stronger base.
• Low \(K_b\) value ➔ Weaker base.

Having a firm grasp of \(K_b\) allows students and chemists alike to understand reaction tendencies, equilibrium positions, and much more. When evaluating solutions and reactions in labs, knowing \(K_b\) clarifies how well a base can neutralize an acid.