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**Hint :**First calculate the value of $ E{^\circ _{cell}} $ . The value of $ E{^\circ _{cell}} $ will be used in the Nernst equation to get the value of $ Q $ which is a ratio of molarity of products to molarity of reactants. The value of $ Q $ will be used to calculate the molarity of $ SO_4^{2 - } $ . Now we can calculate the dissociation constant $ K $ .

**Complete Step By Step Answer:**

First we have to calculate the value of $ E{^\circ _{cell}} $ ,

We know that,

$ E{^\circ _{cell}} = E{^\circ _R} - E{^\circ _L} $

It is given that,

$ E{^\circ _R} = - 0.126 $ and $ E{^\circ _L} = 0.356 $

Therefore,

$ E{^\circ _{cell}} = - 0.126 + 0.356 $

$ E{^\circ _{cell}} = 0.23V $

We can now use Nernst's equation to calculate the value of $ Q $ . Nernst equation is given as:

$ {E_{cell}} = E{^\circ _{cell}} - \dfrac{{0.0592}}{n}\log Q $

Where, $ {E_{cell}} \to $ max potential which can be generated when no current is flowing.

$ E{^\circ _{cell}} \to $ cell potential

$ n \to $ number of electrons gained or lost during reaction.

If we check the reaction, we can see that the number of electrons lost is $ 2 $ .

$ Pb(s) + SO_4^{2 - } \to PbS{O_4} + 2{e^ - } $

On putting the values in Nernst equation,

$ 0.061 = 0.23 - \dfrac{{0.0592}}{2}\log Q $

On further solving,

$ \log Q = 5.709 $

$ Q = 5.122 \times {10^5} $

Since,

$ Q = \dfrac{{[products]}}{{[reac\tan ts]}} $

$ Q = \dfrac{{[PbS{O_4}]}}{{[P{b^{2 + }}][SO_4^{2 - }]}} $

Hence,

$ 5.122 \times {10^5} = \dfrac{1}{{2.5 \times {{10}^{ - 5}} \times [SO_4^{2 - }]}} $

$ [SO_4^{2 - }] = \dfrac{1}{{5.122 \times {{10}^5} \times 2.5 \times {{10}^{ - 5}}}} $

$ [SO_4^{2 - }] = 0.078M $

Hence, $ 0.078M $ of $ SO_4^{2 - } $ will make $ {E_{cell}} = 0 $ , thus, we reach equilibrium and now we can calculate the value of dissociation constant.

The dissociation constant $ K $ for $ HSO_4^ - $ can be calculated as follows,

$ K = \dfrac{{[{H^ + }][SO_4^{2 - }]}}{{[HSO_4^ - ]}} $

$ K = \dfrac{{0.078 \times 0.078}}{{0.6}} $

Hence,

$ K = 0.0106 $

Since we are asked to multiply the value of $ K $ by $ 100 $ , thus, on multiplying,

$ K \times 100 = 1.06 $

On writing the value of $ K $ to its nearest integer,

$ \to K = 1.06 \approx 1 $

Hence the value of $ K $ is $ 1 $ .

**Note :**

Neither the value of $ Q $ nor the dissociation constant $ K $ has any unit as both of them are simply a ratio of molarity of products to their reactants. $ E{^\circ _R} $ and $ E{^\circ _L} $ can be understood as potential at anode and potential at cathode respectively. The dissociation constant $ K $ can also be written as $ {K_d} $ .

The dissociation constant is called ionization constant when it is applied for salts. The inverse of dissociation constant is called association constant.

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