$K_{a_1}, K_{a_2}$ and $K_{a_3}$ are the respective ionisation constants for the following reactions

Question

$K_{a_1}, K_{a_2}$ and $K_{a_3}$ are the respective ionisation constants for the following reactions :
$
\begin{aligned}
& \mathrm{H}_2 \mathrm{~S} ightleftharpoons \mathrm{H}^{+}+\mathrm{HS}^{-} \
& \mathrm{HS}^{-} ightleftharpoons \mathrm{H}^{+}+\mathrm{S}^{2-} \
& \mathrm{H}_2 \mathrm{~S} ightleftharpoons 2 \mathrm{H}^{+}+\mathrm{S}^{2-}
\end{aligned}
$
The correct relationship between $K_{a_1}, K_{a_2}$ and $K_{a_3}$ is

Professor PC Thomas Classes · Answer

(a) : For the reaction : $\mathrm{H}_2 \mathrm{~S} ightleftharpoons \mathrm{H}^{+}+\mathrm{HS}^{-} ; K_{a_1}$ $ K_{a_1}=\frac{\left[\mathrm{H}^{+}ight]\left[\mathrm{HS}^{-}ight]}{\left[\mathrm{H}_2 \mathrm{~S}ight]} \ $ For the reaction : $\mathrm{HS}^{-} ightleftharpoons \mathrm{H}^{+}+\mathrm{S}^{2-} ; K_{a_2}$ $ \ K_{a_2}=\frac{\left[\mathrm{H}^{+}ight]\left[\mathrm{S}^{2-}ight]}{\left[\mathrm{HS}^{-}ight]} $ For the reaction : $\mathrm{H}_2 \mathrm{~S} ightleftharpoons 2 \mathrm{H}^{+}+\mathrm{S}^{2-} ; K_{a_3}$ $ \ K_{a_3}=\frac{\left[\mathrm{H}^{+}ight]^2\left[\mathrm{~S}^{2-}ight]}{\left[\mathrm{H}_2 \mathrm{~S}ight]} $ By multiplying eqn. (i) and eqn. (ii), we get $ K_{a_1} 	imes K_{a_2}=\frac{\left[\mathrm{H}^{+}ight]\left[\mathrm{HS}^{-}ight]}{\left[\mathrm{H}_2 \mathrm{~S}ight]} 	imes \frac{\left[\mathrm{H}^{+}ight]\left[\mathrm{S}^{2-}ight]}{\left[\mathrm{HS}^{-}ight]}=\frac{\left[\mathrm{H}^{+}ight]^2\left[\mathrm{~S}^{2-}ight]}{\left[\mathrm{H}_2 \mathrm{~S}ight]}=K_{a_3} $