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Chapter 15: Problem 59

Write the equation relating \(K_{\mathrm{a}}\) for a weak acid and \(K_{\mathrm{b}}\) for its conjugate base. Use \(\mathrm{NH}_{3}\) and its conjugate acid \(\mathrm{NH}_{4}^{+}\) to derive the relationship between \(K_{\mathrm{a}}\) and \(K_{\mathrm{b}}\).

Short Answer

Expert verified
The relationship between the acid dissociation constant \(K_a\) for a weak acid and the base dissociation constant \(K_b\) for its conjugate base is given by \(K_a * K_b = K_w\), where \(K_w\) is the ion product of water. This relationship is validated by the acid and base dissociation constants of \(NH_4^+\) and \(NH_3\).

Step by step solution

01

Determine the Ion Product of Water

The ion product of water (\(K_w\)) is equal to the product of the concentrations of hydrogen ions, \([H^+]\), and hydroxide ions, \([OH^-]\), at a particular temperature. It's a constant with the value of \(1.0 x 10^{-14}\) at 25°C.\nSo, we have: \(K_w = [H^+][OH^-]\)
02

Write \(K_a\) and \(K_b\) for the Acid and Its Conjugate Base

For the weak acid \(NH_4^+\), the acid dissociation constant (\(K_a\)) is given by: \(K_a = ([H^+][NH_3])/[NH_4^+]\).\nFor the conjugate base \(NH_3\), the base dissociation constant (\(K_b\)) is: \(K_b = ([NH_4^+][OH^-])/[NH_3]\)
03

Derive the Relationship between \(K_a\), \(K_b\) and \(K_w\)

From the definitions of \(K_a\) and \(K_b\) we can solve for \([H^+]\) and \([OH^-]\) respectively: \nFor \(K_a\): \([H^+] = (K_a [NH_4^+])/[NH_3]\) and for \(K_b\): \([OH^-] = (K_b [NH_3])/[NH_4^+]\).\nSubstitute these expressions into equation for \(K_w\) from step 1: \nSo we get: \(K_w = (K_a [NH_4^+])/[NH_3] * (K_b [NH_3])/[NH_4^+]\).\nThe terms \([NH_4^+]\) and \([NH_3]\) cancel out giving the required relationship between \(K_a\), \(K_b\) and \(K_w\): \(K_a * K_b = K_w\).
04

Verify the Relationship through Base and Acid Dissociation Constants for NH3 and NH4+

By looking up the acid dissociation constant for \(NH_4^+\) and base dissociation constant for \(NH_3\), it can be verified that the product of \(K_a\) and \(K_b\) comes up to \(K_w = 1.0 x 10^{-14}\), which establishes the validity of the derived relationship.

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Most popular questions from this chapter

List the factors on which the \(K_{\mathrm{a}}\) of a weak acid depends.

Write equations for the reactions between (a) \(\mathrm{CO}_{2}\) and \(\mathrm{NaOH}(a q),\) (b) \(\mathrm{Na}_{2} \mathrm{O}\) and \(\mathrm{HNO}_{3}(a q)\).

Prove the statement that when the concentration of a weak acid HA decreases by a factor of \(10,\) its percent ionization increases by a factor of \(\sqrt{10}\). State any assumptions.

Calculate the \(\mathrm{pH}\) of a \(0.20 \mathrm{M} \mathrm{NaHCO}_{3}\) solution. (Hint: As an approximation, calculate hydrolysis and ionization separately first, followed by partial neutralization.)

The pOH of a strong base solution is 1.88 at \(25^{\circ} \mathrm{C}\). Calculate the concentration of the base if (a) the base is \(\mathrm{KOH}\) and \((\mathrm{b})\) the base is \(\mathrm{Ba}(\mathrm{OH})_{2}\).
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