Problem 75 - Chemistry The Molecular Nature of Matter

Question

The \(K_{\mathrm{a}}\) for \(\mathrm{HF}\) is \(3.5 \times 10^{-4}\). What is the \(K_{\mathrm{b}}\) for \(\mathrm{F}^{-}\) ?

Step-by-Step Solution

Answer

Kb for F- is approximately 2.86 x 10^(-11)

Step 1: Understanding the Relationship between Ka and Kb

For a conjugate acid-base pair, the ionization constant for the acid (Ka) and the base (Kb) are related by the equation Kw = Ka * Kb, where Kw is the ion-product constant for water at a given temperature. At 25 degrees Celsius, Kw is typically 1.0 x 10^(-14).

Step 2: Calculate the Kb value

Use the relationship Kw = Ka * Kb to calculate Kb for F-. To find Kb, divide Kw by the given Ka value for HF. Kb = Kw / Ka.

Step 3: Perform the Calculation

Plug the known values into the equation from the previous step. Therefore, Kb = (1.0 x 10^(-14)) / (3.5 x 10^(-4)).

Step 4: Solve for Kb

Calculate the result of the division to find the Kb value for F-.

Key Concepts

Ka and Kb Relationship

Understanding the relationship between Ka and Kb is crucial when studying acid-base chemistry. Ka, or the acid dissociation constant, measures how completely an acid releases its protons (H+) in the solution. On the other hand, Kb, the base dissociation constant, gauges the ability of a base to attract and bind protons.

For every acid, there exists a conjugate base formed when the acid donates a proton. The strength of an acid is inversely related to the strength of its conjugate base, meaning that a strong acid has a weak conjugate base, and vice versa. This inverse relationship is mathematically represented by the equation:
\[K_{\text{w}} = K_{\text{a}} \times K_{\text{b}}\] where Kw is the ion-product constant for water, a constant value at a given temperature. In practical terms, this means if you know the value of Ka for an acid, you can calculate Kb for its conjugate base using the formula:
\[K_{\text{b}} = \frac{K_{\text{w}}}{K_{\text{a}}}\]

Ion-Product Constant for Water

The ion-product constant for water (Kw) is a fundamental principle in acid-base chemistry. It refers to the equilibrium constant for the self-ionization of water and is calculated as follows:
\[K_{\text{w}} = [\text{H}^+] \times [\text{OH}^-]\] At 25°C, the value of Kw is always 1.0 x 10^-14. This constant value is the product of the concentrations of hydrogen ions (H+) and hydroxide ions (OH-) in pure water. Since water dissociates to a very small extent into these ions, the concentration of each in pure water is 1.0 x 10^-7 M.

In an acidic solution, the concentration of H+ exceeds that of OH-, shifting the equilibrium towards more free protons. Conversely, in a basic solution, the concentration of OH- is higher. The constant nature of Kw means that if the concentration of one goes up, the other must go down in order to maintain the value of Kw.

Conjugate Acid-Base Pair

A conjugate acid-base pair consists of two species that transform into each other by the gain or loss of a proton. Understanding this concept is essential to grasp the nuances of acid-base reactions. In the case of the acid HF (hydrofluoric acid), when it donates a proton, it becomes its conjugate base F− (fluoride ion). Similarly, if F− were to accept a proton, it would become HF.

In a conjugate acid-base pair, the acid and base differ by the presence or absence of a proton. The formula can be represented as:
\[\text{Acid} \rightleftharpoons \text{Base} + \text{H}^+\] where the arrow indicates the reversible nature of the reaction. This relationship plays a key role in determining the pH of a solution and in buffer systems that resist changes in pH upon addition of small amounts of acid or base.