Problem 115 - Chemistry: The Central Science in SI Units, Global Edition

Question

The \(\mathrm{p} K_{\mathrm{b}}\) of water is (a) 1 (b) 7 (c) 14 (d) not defined (e) none of the above

Step-by-Step Solution

Answer

The pKb of water is 14. This is because water acts as both an acid and a base, and its pKa equals its pKb. Since the pKa of water is 14, the pKb is also 14.

Step 1: Review Definition of pKb

pKb is the negative logarithm of the base dissociation constant (Kb). pKb allows us to quantify the strength of a base. A low pKb value indicates a strong base, and a high pKb value denotes a weak base.

Step 2: Relationship between pKa and pKb

For a given compound, the sum of its acidity constant (pKa) and its basicity constant (pKb) equals 14, which represents the ion product of water (Kw) at 25°C: \(pK_a + pK_b = 14\)

Step 3: Calculate pKb of Water

As water acts as both an acid and a base, we know that its pKa equals its pKb. First, we will calculate the pKa value of water. The Ka value for water is 1 x 10^(-14) (at 25°C). To find the pKa, we will take the negative logarithm of the Ka value: \(pK_a = -\log_{10}(1 \times 10^{-14})\) \(pK_a = 14\) As pKa = pKb for water, the pKb value is also 14. Therefore, the correct answer is: (c) 14

Key Concepts

pKb

The term "pKb" represents the negative log of the base dissociation constant (Kb), which is a measure of how fully a base dissociates in a solution. The formula to calculate pKb is \[ \text{pKb} = -\log_{10}(K_b)\]Using logarithms helps us simplify the comparison of base strengths by transforming potentially tiny numbers like dissociation constants into manageable, more comparable values.

• A lower pKb value indicates a strong base because it implies greater dissociation in solution.
• Conversely, a higher pKb value signals a weaker base, as it dissociates less in solution.
This concept enables chemists and students alike to predict the behavior of different bases in chemical reactions, ensuring that they understand the potency of a base in a solution.

Base Dissociation Constant

The Base Dissociation Constant, symbolized as Kb, quantifies the strength of a base in solution. This constant gives insight into how well a base "accepts" protons from water, thus dissociating into various ions.

To better understand this concept, consider ammonia in water. The dissociation equation is:\[ \text{NH}_3 + \text{H}_2\text{O} \rightleftharpoons \text{NH}_4^+ + \text{OH}^- \]Here, Kb is calculated using the equilibrium concentrations:\[ K_b = \frac{[\text{NH}_4^+][\text{OH}^-]}{[\text{NH}_3]}\]This calculation helps determine the position of equilibrium in the dissociation reaction.

• A high Kb value signifies a strong base, meaning the base dissociates significantly, producing more hydroxide ions (OH^-).
• If Kb is low, the base is weaker, releasing fewer hydroxide ions in solution.

Understanding Kb provides a window into the molecular dynamics in solution and can be instrumental in applications ranging from industrial manufacturing to biochemical analyses.

pKa

"pKa" stands for the negative logarithm of the acid dissociation constant (Ka). It reflects how easily an acidic molecule releases a proton. This measure allows us to gauge acid strength efficiently using the formula:\[ \text{pK}a = -\log_{10}(K_a)\]Just like with bases, using logarithmic values for acids makes comparisons straightforward:

• A small pKa value indicates a strong acid, with a high tendency to dissociate and donate protons.
• A large pKa value points to a weak acid, with minimal tendency to donate protons.

The relationship between pKa and pKb is especially noteworthy. For any given acid-base pair, the sum of their pKa and pKb values equals 14, due to the ion product of water (Kw), given by the expression \[ \text{pKa} + \text{pK}b = 14\]This interconnection enables chemists to assess the equilibrium properties and strengths of acids and bases rapidly. Thus, understanding pKa helps predict an acid's behavior in further chemical processes, aiding in making precise chemical formulations and reactions.