Question

If pKb for fluoride ion at \(25^{\circ} \mathrm{C}\) is \(10.83\), the ionization constant of hydrofluoric acid at this temperature is: (a) \(1.74 \times 10^{-5}\) (b) \(3.52 \times 10^{-3}\) (c) \(6.75 \times 10^{-4}\) (d) \(5.38 \times 10^{-2}\)

Short answer

The ionization constant of hydrofluoric acid is approximately \(6.75 \times 10^{-4}\), matching option (c).

Step-by-step solution

Understanding the Relation Between pKa and pKb

The relationship between the ionization constant of an acid, pKa, and that of its conjugate base, pKb, at a given temperature is given by the equation: \[ pKa + pKb = 14 \]. This relationship is derived from the fact that the product of the acid dissociation constant (Ka) and the base dissociation constant (Kb) is equal to the ion-product constant for water, Kw, which is \(1.0 \times 10^{-14}\) at \(25^{\circ} \mathrm{C}\).

Using the Relationship to Find pKa

Using the formula \( pKa + pKb = 14 \), substitute the given pKb for the fluoride ion, which is 10.83, into the equation: \[ pKa + 10.83 = 14 \]. To solve for pKa, subtract 10.83 from 14 to find \( pKa = 14 - 10.83 \).

Calculation of pKa

Perform the subtraction: \( 14 - 10.83 = 3.17 \). Therefore, \( pKa = 3.17 \).

Converting pKa to Ka

pKa is the negative base-10 logarithm of the acid dissociation constant (Ka). To find Ka, use the formula \[ Ka = 10^{-pKa} \]. Substitute pKa = 3.17 into the equation: \[ Ka = 10^{-3.17} \].

Calculate Ka

Calculate the power of ten to find \( Ka = 10^{-3.17} \approx 6.76 \times 10^{-4} \).

Compare with Given Options

The result \( 6.76 \times 10^{-4} \) closely matches option (c) \( 6.75 \times 10^{-4} \), so this is the correct answer.

Key concepts

pKa and pKb Relationship

Understanding the relationship between pKa and pKb is crucial because it connects the properties of conjugate acids and bases. This relationship is defined by the equation:

• \( pKa + pKb = 14 \)

At a temperature of \(25^{\circ} \mathrm{C}\), the ion-product constant for water, \(Kw\), is \(1.0 \times 10^{-14}\). This value establishes the link between the dissociation constants for water, acids, and bases.
Since \(Kw = Ka \times Kb\), taking the logarithm and rearranging gives us the equation linking pKa and pKb. For acids and their conjugate bases, the sum of their pKa and pKb must always equal 14 under these temperature conditions.
This concept is essential for predicting the behavior of acid-base reactions and for calculating unknown dissociation constants when one constant is known.

Acid Dissociation Constant

The acid dissociation constant, \(Ka\), measures the strength of an acid in solution. It quantifies how much an acid dissociates into its ions.
When an acid dissolves in water, it donates a proton and forms its conjugate base:

• \( HA \rightarrow H^+ + A^- \)

The equilibrium constant for this reaction is \(Ka\):

• \( Ka = \frac{[H^+][A^-]}{[HA]} \)

A larger \(Ka\) indicates a stronger acid, as it means more acid molecules ionize.
The pKa value, which is the negative logarithm of \(Ka\), makes it easier to work with small constants like these in equations:

• \( pKa = -\log(Ka) \)

Remember, smaller pKa values correspond to stronger acids because they signify higher Ka values.

Conjugate Base

The term "conjugate base" refers to what remains of an acid molecule once it donates a proton during a chemical reaction.
In the general reaction of an acid \(HA\), the conjugate base is represented as \(A^-\) following the reaction:

• \( HA \rightarrow H^+ + A^- \)

The strength of a conjugate base is inversely related to the strength of its parent acid. This means if an acid is strong, its conjugate base will be weak.
This is because strong acids dissociate completely, leaving very few ions to recombine back into the original acid. Students often find the relationship between conjugate acids and bases critical when predicting the outcomes of buffer solutions and various acid-base equilibrium reactions.
Understanding these concepts can help anticipate how substances will behave, aiding in both laboratory and theoretical chemistry.