Problem 36 - Chemistry and Chemical Reactivity

Question

The chromium(III) ion in water, \(\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{6}\right]^{3+},\) is a weak acid with \(\mathrm{p} K_{\mathrm{a}}=3.95 .\) What is the value of \(K_{\mathrm{b}}\) for its conjugate base, \(\left[\mathrm{Cr}\left(\mathrm{H}_{2} \mathrm{O}\right)_{5} \mathrm{OH}\right]^{2+} ?\)

Step-by-Step Solution

Answer

The value of \(K_b\) is approximately \(8.93 \times 10^{-11}\).

Step 1: Understand the Relationship between Ka and Kb

For a conjugate acid-base pair, the product of the acid dissociation constant \(K_a\) and the base dissociation constant \(K_b\) is equal to the ion-product constant for water, \(K_w\). This can be written as: \( K_a \times K_b = K_w \). At 25°C, \( K_w = 1.0 \times 10^{-14} \).

Step 2: Convert pKa to Ka

The \( pK_a \) is given as 3.95. We first need to convert this to \( K_a \) using the formula: \( K_a = 10^{-pK_a} \). Thus, \( K_a = 10^{-3.95} \).

Step 3: Calculate Ka Value

Calculate \( K_a \) using the formula: \( K_a = 10^{-3.95} \). By computing this, we find \( K_a \approx 1.12 \times 10^{-4} \).

Step 4: Calculate Kb using Ka and Kw

With \( K_a \) known, use the relationship \( K_a \times K_b = K_w \) to solve for \( K_b \). Rearrange the equation to find \( K_b = \frac{K_w}{K_a} = \frac{1.0 \times 10^{-14}}{1.12 \times 10^{-4}} \).

Step 5: Solve for Kb Value

Perform the division \( \frac{1.0 \times 10^{-14}}{1.12 \times 10^{-4}} \) to find \( K_b \). This results in \( K_b \approx 8.93 \times 10^{-11} \).

Key Concepts

Acid-Base Equilibrium

Acid-base equilibrium is a central concept in chemistry that explains how acids and bases interact in a solution. When an acid donates a proton (H⁺), it forms its conjugate base. Conversely, when a base accepts a proton, it forms its conjugate acid.

The equilibrium is represented by the constant values, known as dissociation constants, for both acids and bases. This equilibrium helps understand how changes in conditions like concentration or temperature can shift the balance between acid and base forms. The principle of Le Chatelier’s can be applied to predict these shifts.

• Acids increase the concentration of H⁺ ions in the solution.
• Bases reduce the concentration of H⁺ ions by forming hydroxide ions (OH⁻).
• The balance between these ions determines the pH of the solution.

Understanding acid-base equilibrium allows chemists to predict the behavior of solutions in reactions, essential for fields ranging from pharmacology to environmental science.

Dissociation Constant

The dissociation constant ( K_a for acids and K_b for bases) quantifies the strength of an acid or base in a solution. It measures how completely an acid or base dissociates into ions.

The larger the dissociation constant, the stronger the acid or base, indicating more complete dissociation into ions. For an acid AB, the dissociation can be written as: AB ↔ A⁻ + H⁺ with K_a = [A⁻][H⁺]/[AB] . Similarly, for a base B, it dissociates as B + H₂O ↔ BH⁺ + OH⁻ with K_b = [BH⁺][OH⁻]/[B] .

• A higher K_a or K_b indicates a stronger acid or base.
• Weak acids and bases have low dissociation constants.
• Dissociation is incomplete for weak acids and bases.

Understanding and calculating these constants helps predict how substances behave in different chemical reactions.

pKa and pKb

The terms pK_a and pK_b are the negative logarithms of the acid and base dissociation constants, respectively. They provide a more manageable way to express dissociation constants.

Using logarithms simplifies the expression of very large or very small numbers. The pK_a of an acid is calculated using the formula pK_a = - log (K_a) , and similarly for a base with pK_b = - log (K_b) . These values help to easily compare the strengths of different acids and bases.

• Lower pK_a or pK_b values indicate stronger acids or bases.
• Provides a straightforward way to rank acidity or basicity.
• Useful in calculations involving equilibrium and titrations.

In practice, chemists use pK_a and pK_b to predict the direction of acid-base reactions and to design buffers effectively.

Ion-Product Constant for Water

The ion-product constant for water ( K_w ) is a crucial concept in understanding acid-base chemistry. Defined at a specific temperature (usually 25°C), it is the product of the concentrations of hydrogen ions ( [H⁺] ) and hydroxide ions ( [OH⁻] ) in pure water. Mathematically, it is written as: K_w = [H⁺][OH⁻] .

Under standard conditions, K_w is equal to 1.0 × 10^{-14} . This constant reflects the self-ionization of water, where water molecules dissociate into hydrogen and hydroxide ions at a very low extent.

• Maintains the balance between H⁺ and OH⁻ in water.
• At neutral pH, [H⁺] = [OH⁻] , thus pH = 7.
• Used to relate the strengths of acids and bases through the equation K_a × K_b = K_w .

Recognizing the significance of K_w is essential for comprehensive understanding of aqueous equilibria and pH calculations.