Question

Given, $HF+H_{2}O\stackrel{K_a}{\to }{H}_3{O}^{+}+F^{-}$
$F^{-}+H2O\stackrel{K_b}{\to }HF+O{H}^{-}$
Which of the following relation is correct?

• A. $K_b=K_w$
• B. $K_b=\frac{1}{K_w}$
• C. $K_a\times K_b=K_w$
• D. $\frac{K_a}{K_b}=K_w$

Answer 1

Answer: (C)

$K_a\times K_b=K_w$

The reaction for the dissociation of acid is as shown below.
$HF+H_{2}O\stackrel{K_a}{\to }{H}_3{O}^{+}+F^{-}$
The expression for the acid dissociation constant $K_a$ is as shown below.
$K_a=\frac{\left[H_3{O}^{+}\right]\left[F^{-}\right]}{\left[HF\right]}$ ......(1)
The reaction of fluoride ion with water is as shown below
$F^{-}+H2O\stackrel{K_b}{\to }HF+O{H}^{-}$.
The expression for the constant $K_b$ is as shown below.
$K_b=\frac{\left[HF\right]\left[O{H}^{-}\right]}{\left[F^{-}\right]}$ ......(2)
Multiply equation (1) with equation (2).
$K_a\times K_b=\left(\frac{\left[H_3{O}^{+}\right]\left[F^{-}\right]}{\left[HF\right]}\right)\times \left(\frac{\left[HF\right]\left[O{H}^{-}\right]}{\left[F^{-}\right]}\right)=\left[H^{+}\right]\left[O{H}^{-}\right]=K_w$.