Name: pH Scale
Uploaded: 2019-07-19
Duration: 9 min 45 s
Description: This video discusses the pH of acids, bases, and salts and also the limitations of using a pH scale for identification. The natural indicators do the verific...

Question

Briefly explain the terms—salt hydrolysis, hydrolysis constant and degree of hydrolysis. Derive expressions for the calculation of

pH

of salt of strong acid with weak base and salt of weak acid with weak base.

Accepted Answer

Salt hydrolysis is defined as the process in which a salt reacts with water to give back the acid and the base.
$\underset{Salt}{BA} + \underset{Water}{H_{2} O} \to \underset{Acid}{HA} + \underset{Base}{BOH}$

The equilibrium constant for a hydrolysis reaction is called the hydrolysis constant. It is represented by $K_{h}$ .

The degree of hydrolysis $(h)$ of a salt is defined as the fraction (or percentage) of the total salt which is hydrolysed, i.e.,
$h = \frac{Number of moles of salt hydrolysed}{Total number of moles of salt taken}$

Calculation of the $pH$ of a strong acid with a weak base : Let BA be the salt of a strong acid with a weak base.

It is a case of cationic hydrolysis.

The hydrolysis constant $K_{h}$ , will be given by $K_{h} = \frac{[BOH] [H^{+}]}{[B^{+}]}$ $. . . . . . . (1)$

For the weak base $BOH$ , the dissociation equilibrium is : $pH$
$∴$ The dissociation constant $K_{b} = \frac{[B^{+}] [{OH}^{-}]}{[BOH]}$ $. . . . . . . (2)$
The ionic product of water, $K_{w} = [H^{+}] [{OH}^{-}]$ $. . . . . . . (3)$
Multiplying equation $(1)$ with $(2)$ and dividing by $(3)$ , we get
$\frac{K_{h} K_{b}}{K_{w}} = 1$ or $K_{h} = \frac{K_{w}}{K_{b}}$ $. . . . . . . . (4)$
As the degree of hydrolysis $h = \sqrt{\frac{K_{h}}{c}}$
Substituting the value of $K_{h}$ , from equation $(4)$ ,
we get $h = \sqrt{\frac{K_{w}}{K_{b} c}}$ $. . . . . . . (5)$

$B^{+} + H_{2} O ⇌ BOH + H^{+} Initial Conc . c 0 0 At Eq . conc . c (1 - h) ch ch$
Substituting the value of $h$ from equation $(5)$ , we get
$[H^{+}] = c \cdot \sqrt{\frac{K_{w}}{K_{b} \cdot c}} = \sqrt{\frac{K_{w} \cdot c}{K_{b}}} = {(\frac{K_{w} \cdot c}{K_{b}})}^{1 / 2}$
$∴ p H = - \log [H^{+}] = - \log {(\frac{K_{w} \cdot c}{K_{b}})}^{1 / 2}$
or $pH = - \frac{1}{2} [{logK}_{w} - {logK}_{b} + logc]$
The above equation can be rewritten as: $pH = \frac{1}{2} [{pK}_{w} - {pK}_{b} - logc]$
At $298 K, {pK}_{w} = 14$
Hence, at $298 K$ , we have $pH = 7 - \frac{1}{2} [{pK}_{b} + logc]$

Calculation of the $pH$ of a weak acid with a weak base : Let BA be the salt of a weak acid with a weak base.

It involves both anionic and cationic hydrolysis. The equation for the hydrolysis constant will be: $K_{h} = \frac{[BOH] [HA]}{[B^{+}] [A^{-}]}$ $. . . . . . . . (6)$
For the weak acid, $HA ⇌ H^{+} + A^{-},$
Therefore, $K_{a} = \frac{[H^{+}] [A^{-}]}{[HA]}$ $. . . . . . . . (7)$
For the weak base, $BOH ⇌ B^{+} + {OH}^{-}$ ,
Therefore, $K_{b} = \frac{[B^{+}] [{OH}^{-}]}{[BOH]}$ $. . . . . . . . (8)$
Also, we know that $K_{w} = [H^{+}] [{OH}^{-}]$ $. . . . . . . . (9)$
Multiplying equation $(6), (7), (8)$ and dividing by equation $(9)$ , we get
$\frac{K_{h} \cdot K_{a} \cdot K_{b}}{K_{w}} = 1$ or $K_{h} = \frac{K_{w}}{K_{a} K_{b}}$ $. . . . . . . . (10)$

Therefore, the degree of hydrolysis
$K_{h} = \frac{[BOH] [HA]}{[B^{+}] [A^{-}]} = \frac{ch . ch}{c (1 - h) \cdot c (1 - h)} = \frac{h^{2}}{{(1 - h)}^{2}}$ $. . . . . . . . (11)$

For $p H$ ,
$HA ⇌ H^{+} + A^{-}$
$∴ K_{a} = \frac{[H^{+}] [A^{-}]}{[HA]}$
or $[H^{+}] = K_{a} \frac{[HA]}{[A^{-}]} = K_{a} \frac{ch}{c (1 - h)} = K_{a} \frac{h}{1 - h}$
But from equation $(11)$ , $\frac{h}{1 - h} = \sqrt{K_{h}}$
Hence, $[H^{+}] = K_{a} \sqrt{K_{h}}$
$= K_{a} \sqrt{\frac{K_{w}}{K_{a} . K_{b}}} = \sqrt{\frac{K_{a} K_{w}}{K_{b}}} [From (10)]$
Now, $pH = - \log [H^{+}]$
$= - \log \sqrt{\frac{K_{a} K_{w}}{K_{b}}} = - \log {(\frac{K_{a} K_{w}}{K_{b}})}^{\frac{1}{2}}$
Or $pH = - \frac{1}{2} [\log K_{a} + \log K_{w} - \log K_{b}]$
Or the above equation may be rewritten as: $pH = \frac{1}{2} [{pK}_{w} + {pK}_{a} - {pK}_{b}]$
At $298 K$ , ${pK}_{w} = 14$
Hence at $298 K$ , $pH = 7 + \frac{1}{2} [{pK}_{a} - {pK}_{b}]$

Briefly explain the terms—salt hydrolysis, hydrolysis constant and degree of hydrolysis. Derive expressions for the calculation of $pH$ of salt of strong acid with weak base and salt of weak acid with weak base.

Important Questions on Ionic Equilibrium in Solution

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