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Heat and Work

Which of the ...

Question

Which of the following represents the correct relation between $K_{s p} (A g C l)$ and standard reduction potential of $A g | A g C l (saturated) in K C l$ electrode?

E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{F} l n K_{s p}

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E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{F} l n K_{s p} - E_{A g^{+} / A g}^{0}

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E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{F} l n K_{s p} + E_{A g^{+} / A g}^{0}

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E_{C l^{-} / A g C l / A g}^{0} = E_{A g^{+} / A g}^{0}

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Solution

The correct option is C $E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{F} l n K_{s p} + E_{A g^{+} / A g}^{0}$
The given electrode is a metal/metal insoluble salt electrode.
Here the silver rod is coated with a paste of saturated $A g C l$ and dipped in a solution of ionic electrolyte of $K C l$ .

The electrode reaction is,
$A g C l (s) + e^{-} \to A g (s) + C l^{-}$

Derivation:
$A g C l (s) ⇌ A g^{+} (a q) + C l^{-} (a q) . . . . . . (1)$
$Δ G_{1} = Δ G^{0} + R T l n Q$
At equilibrium,
$Δ G_{1} = 0$
$Q = K_{s p}$
$Δ G_{1}^{0} = - R T l n K_{s p}$

$A g^{+} (a q) + e^{-} \to A g (s) . . . . . . . . (2)$
$Δ G_{2}^{0} = - n F E_{A g^{+} / A g}^{0}$

Adding equation (1) and (2)
$A g C l (s) + e^{-} \to A g (s) + C l^{-} . . . . . . . (3)$
$Δ G_{3}^{0} = - n F E_{C l^{-} / A g C l / A g}^{0}$

Adding equation (1) and (2) gives (3)
$∴$
$Δ G_{3}^{0} = Δ G_{1}^{0} + Δ G_{2}^{0}$
$- n F E_{C l^{-} / A g C l / A g}^{0} = - R T l n K_{s p} - n F E_{A g^{+} / A g}^{0}$

$n F E_{C l^{-} / A g C l / A g}^{0} = R T l n K_{s p} + n F E_{A g^{+} / A g}^{0}$

$E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{n F} l n K_{s p} + n F E_{A g^{+} / A g}^{0}$

$n = 1$ for given electrode reaction,

$E_{C l^{-} / A g C l / A g}^{0} = \frac{R T}{F} l n K_{s p} + E_{A g^{+} / A g}^{0}$

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