We have taken a saturated solution of \(\mathrm{AgBr}, K_{\mathrm{sp}}\) is \(12 \times 10^{-14}\). If \(10^{-7}\) mole of \(\mathrm{AgNO}_3\) are added to 1 litre of this solution, find conductivity (specific conductance) of this solution in terms of \(10^{-7} \mathrm{Sm}^{-1}\) units.
\( \text {Given, } \lambda_{\left(\mathrm{Ag}^{+}\right)}^{\circ} =6 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1},\) \(\lambda_{\left(\mathrm{Br}^{-}\right)}^{\circ}=8 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1}, \)
\( \lambda_{\left(\mathrm{NO}_3^{-}\right)} =7 \times 10^{-3} \mathrm{Sm}^2 \mathrm{~mol}^{-1},\)

Suppose the solubility of \(\mathrm{AgBr}\) in \(10^{-7} \mathrm{MAgNO}_3\) is \(\mathrm{mol} / \mathrm{L}\)
\(
\begin{aligned}
& \underset{s \mathrm{~mol} / \mathrm{L}}{\mathrm{AgBr}} \rightleftharpoons \mathrm{Ag}^{+}+\mathrm{Br}^{-} \\
& \mathrm{AgNO}_3 \rightleftharpoons \mathrm{Ag}^{+}+\mathrm{NO}_3^{-} \\
& 10^{-7} \mathrm{M} \quad 10^{-7} \mathrm{M} \times 10^{-7} \mathrm{M} \\
& \text { Total }\left[\mathrm{Ag}^{+}\right]=\left(s+10^{-7}\right) \mathrm{M} \\
& K_{\mathrm{sp}} \text { of } \mathrm{AgBr}=\left[\mathrm{Ag}^{+}\right]\left[\mathrm{Br}^{-}\right] \\
& 12 \times 10^{-14}=\left(s+10^{-7}\right)(s)=s^2+10^{-7} s \\
& s^2+10^{-7} s-12 \times 10^{-14}=0\end{aligned}
\)
On solving \(s=3 \times 10^{-7} \mathrm{M}\)
Hence, \(\left[\mathrm{Br}^{-}\right]=3 \times 10^{-7} \mathrm{M}=3 \times 10^{-7} \times 10^3\)\(\mathrm{~m}^3=3 \times 10^{-4} \mathrm{~m}^3\)
\(
\left[\mathrm{Ag}^{+}\right]=\left(3 \times 10^{-7}+10^{-7}\right) \mathrm{M}=4 \times 10^{-7} \times\) \(10^3 \mathrm{~m}^3=4 \times 10^{-4} \mathrm{~m}^3
\)
\(
\left[\mathrm{NO}_3^{-}\right]=10^{-7} \mathrm{M}=10^{-7} \times 10^{+3}=\) \(1 \times 10^{-4} \mathrm{~m}^3
\)
\(
\because \lambda=\frac{k}{C} \text { or } k=\lambda \times C
\)
\(\therefore k_{\mathrm{Br}^{-}} =3 \times 10^{-4} \times 8 \times 10^{-3} \mathrm{Sm}^{-1}\) \(=24 \times 10^{-7} \mathrm{Sm}^{-1} \)
\( k_{\mathrm{Ag}^{+}} =4 \times 10^{-4} \times 6 \times 10^{-3} \mathrm{Sm}^{-1}\) \(=24 \times 10^{-7} \mathrm{Sm}^{-1} \)
\( k_{\mathrm{NO}_3^{-}} =1 \times 10^{-4} \times 7 \times 10^{-3}=7 \times 10^{-7} \mathrm{Sm}^{-1} \)
\( k_{\text {Total }} =k_{\mathrm{Br}^{-}}+k_{\mathrm{Ag}^{+}}+k_{\mathrm{NO}_3^{-}} \)
\( =\left(24 \times 10^{-7}+24 \times 10^{-7}+7 \times 10^{-7} \mathrm{Sm}^{-1}\right. \text { ) } \)
\( =55 \times 10^{-7} \mathrm{Sm}^{-1} \)
\( \left.=\mathbf{5 5} \mathbf{S m}^{-1} \text { (in term of } 10^{-7} \mathrm{Sm}^{-1}\right)\)