Question

For a conjugate acid-base pair the relation between ${{K}_{a}}$ and ${{K}_{b}}$ :
(A) ${{K}_{a}}\times {{K}_{b}}=1$
(B) ${{K}_{a}}/{{K}_{b}}={{K}_{w}}$
(C) ${{K}_{a}}\times {{K}_{b}}={{K}_{w}}$
(D) ${{K}_{a}}\times {{K}_{b}}=14$

Answer 1

We know that ${{K}_{a}}$ is dissociation constant for acid and ${{K}_{b}}$ is dissociation constant for base. Both can be found out by finding the equilibrium constant for acid and base respectively. Both constants are the measure of the strengths of acid and base respectively. The relation between these two constants gives the equation for autoionization of water or self-dissociation for water.

Complete answer:
The substance which is formed when an acid donates a proton to base in a chemical reaction is called conjugate acid. In other words, we can also say that, it is the base with which the hydrogen is added to it while the left substance after donating the proton from acid during chemical reaction is called conjugate base. A cation can be conjugate acid and an anion can be a conjugate base.
Bronsted and Martin Lowry proposed that any compound which transfers to the other compound is called acid and the compound which accepts the proton is called base. This theory is called Bronsted – Lowry theory. The strength of conjugate acid is directly proportional to its dissociation constant. If a conjugate acid is strong, its dissociation will have a higher equilibrium constant and the product of reaction will be favored. The tendency of the conjugate base to pull a hydrogen proton towards itself is called a conjugate base.
For conjugate acid-base pairs, relation between ${{K}_{a}}$ and ${{K}_{b}}$ is given by ${{K}_{a}}\times {{K}_{b}}={{K}_{w}}$ where, ${{K}_{w}}~$ is auto-ionization constant.
Therefore, the correct answer is option C.

Note:
Remember that Larger the value of ${{K}_{a}}$ , stronger is the acid meaning that the acid can easily give protons and larger the value of ${{K}_{b}}$ , stronger is the base meaning that the base can easily give hydroxyl ions. Thus, we can say that the strength of acid or base is related to the value of dissociation constant. Note that at room temperature the value of the ionic product of water, ${{K}_{w}}={{10}^{-14}}.$