Chapter 18, Problem 18.149CP

Interpretation Introduction

Interpretation:

The solubility product of a sparingly soluble salt can be obtained from the cell potential value of the cell made by the cation and anion.

Concept introduction:

• Solubility product: For the sparingly soluble salt (AB) in the solution of the salt (AB) maximum portion of the salt remains in un-dissociated form. So in this solution there remain an equilibrium:

$AB(Un-dissolved)\rightleftharpoons AB(Dissolved)\rightleftarrows A^{+}(aq)\times B^{-}(aq)$.
AS the maximum portion of the salt remains un-dissociated state thus the equilibrium can be re-written as-

$AB(Un-dissolved)\rightleftharpoons A^{+}(aq)\times B^{-}(aq)$

So, the equilibrium constant of the reaction $K=\frac{[A^{+}]\times [B^{-}]}{[AB]}$

As the concentration of [AB] is so high so it assumed to be constant. Then, $K_{sp}=[A^{+}]\times [B^{-}]$. The $K_{sp}$ is called the solubility product of the sparingly soluble salt at a constant temperature.

• Cell potential with equilibrium constant: The equilibrium constant and formation constant of a reaction can be determined from the cell potential by the use of Nernst equation: Let assume a reaction:

$aA+bB\rightleftarrows cC+dD$.
Thus the cell potential of the reaction from Nernst equation is:

$E_{Cell}=E_{Cell}^0-\frac{RT}{nF}\times \ln \frac{{[C]}^c\times {[D]}^d}{{[A]}^a\times {[B]}^b}$

Now at equilibrium the $E_{Cell}$ =0 and the $\frac{{[C]}^c\times {[D]}^d}{{[A]}^a\times {[B]}^b}$ = Equilibrium constant, K.
Thus $\ln K=\frac{nF{E}^0}{RT}$.

• Cell reaction in shorthand notation: The cell reaction of a galvanic cell can be represented in a symbolic manner, where the left hand side is the oxidation half cell (anode) and right hand side is the reduction half cell (cathode).
The anode is represented as $Zn$ | $Z{n}^{2+}$ , where Zn is the metal which is immerged in the solution of $Z{n}^{2+}$.
Similarly the cathode represented as $C{u}^{2+}$ | $Cu$. If in any reaction salt bridge is used then the reaction is represented as $Zn$ | $Z{n}^{2+}$ || $C{u}^{2+}$ | $Cu$. The two vertical lines (|) represent the salt bridge between cathode and anode.

Given:

The following galvanic cell has a potential of 0.578V at 25 ^oC:
$Ag(s)$ | $AgCl(s)$ | $C{l}^2(1.0M)$ || $A{g}^{+}(1.0M)$ | $Ag(s)$

To determine:

The solubility product ( $K_{sp}$ ) for AgCl at 25 ^oC.