Question

The \(\mathrm{pH}\) of a \(0.30-\mathrm{M}\) solution of a weak base is 10.66 at \(25^{\circ} \mathrm{C}\). What is the \(K_{\mathrm{b}}\) of the base?

Short answer

The base ionization constant, \(K_{\mathrm{b}}\), is approximately \(6.96 \times 10^{-7}\).

Step-by-step solution

Calculate the pOH

Starting with the pH given as 10.66, we can find the pOH using the relationship: \[ \text{pOH} = 14 - \text{pH} \]Substitute the value for pH: \[ \text{pOH} = 14 - 10.66 = 3.34 \]

Calculate the hydroxide ion concentration

Using the pOH, we can calculate the concentration of hydroxide ions \([\text{OH}^-]\):\[ \text{pOH} = -\log [\text{OH}^-] \]Rearrange to find \([\text{OH}^-]\):\[ [\text{OH}^-] = 10^{-\text{pOH}} = 10^{-3.34} \approx 4.57 \times 10^{-4} \text{ M} \]

Use the hydroxide concentration to find Kb

For a weak base \(B\), the equilibrium can be expressed as:\[ B + H_2O \rightleftharpoons BH^+ + OH^- \]The equilibrium expression for \(K_b\) is:\[ K_b = \frac{[BH^+][OH^-]}{[B]} \]Assuming \([BH^+] \approx [OH^-] = 4.57 \times 10^{-4} \text{ M}\) for a weak base starting at initial concentration 0.30 M:\[ K_b = \frac{(4.57 \times 10^{-4})^2}{0.30 - 4.57 \times 10^{-4}} \]Since \(4.57 \times 10^{-4}\) is much smaller than 0.30, approximate to:\[ K_b \approx \frac{(4.57 \times 10^{-4})^2}{0.30} \approx 6.96 \times 10^{-7} \]

Key concepts

Understanding Weak Bases

A weak base is a chemical compound that does not completely dissociate in an aqueous solution. Unlike strong bases, weak bases partially accept protons or donate hydroxide ions, thereby establishing an equilibrium in the solution.
Weak bases are important in chemistry because they help maintain pH balance in systems, and their behavior is predictable using specific calculations. In a solution, only a fraction of the weak base will react with water.

• As weak bases do not dissociate completely, they result in an equilibrium state where both the weak base and its ions are present in the solution.
• This incomplete dissociation affects both the solution's pH value and its corresponding hydroxide ion concentration.

The ability to only partially dissociate gives weak bases unique properties distinct from those of strong bases, which dissociate fully. This property is critical when calculating pH, pOH, and associated equilibrium constants like the dissociation constant \( K_b \).

Equilibrium Expression for Weak Bases

Equilibrium expressions are essential for understanding how weak bases behave in a solution. When a weak base \( B \) is dissolved in water, an equilibrium is established between the base and its ions.

• The reaction can be represented as: \( B + H_2O \rightleftharpoons BH^+ + OH^- \), where \( BH^+ \) and \( OH^- \) are the products.
• Given this reaction, the equilibrium expression defines the dissociation constant \( K_b \).
• It is described by the equation: \( K_b = \frac{[BH^+][OH^-]}{[B]} \).

This expression lays the foundation for calculating the dissociation constant for weak bases, which informs us about the extent of dissociation and the strength of the base in solution.

Calculating Hydroxide Ion Concentration

The hydroxide ion concentration \([\text{OH}^-]\) in a solution is pivotal for understanding the basicity of that solution.
For weak bases, knowing the pOH helps ascertain \([\text{OH}^-]\), which is crucial for further calculations like finding the \( K_b \).

• The relationship between the pOH and \([\text{OH}^-]\) is given by: \[ \text{pOH} = -\log [\text{OH}^-] \]
• To find \([\text{OH}^-]\), rearrange this equation: \[ [\text{OH}^-] = 10^{-\text{pOH}} \]
• In practice, once the pOH is known, calculating \([\text{OH}^-]\) becomes a straightforward process.

Understanding the hydroxide ion concentration is not only vital for determining \( K_b \) but also for predicting how the solution will interact with acids and other chemicals.

Dissociation Constant \( K_b \) of Weak Bases

The dissociation constant, \( K_b \), is a crucial parameter that helps quantify the strength of a weak base in solution.
\( K_b \) indicates how well a weak base ionizes in water.

• A higher \( K_b \) value suggests a stronger base that dissociates more in water.
• The calculation of \( K_b \) often involves using the equilibrium concentrations of ions in a solution.
• For example, a typical calculation can be done using: \[ K_b = \frac{[BH^+][OH^-]}{[B]} \] where \( [BH^+] \) and \( [OH^-] \) are approximated to be equal in weak bases, making the formula easier to apply.

The concept of \( K_b \) not only helps in determining the behavior of weak bases but also plays a key role in various chemical calculations, including pH determination.