Which out of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning:

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Solution

Q.
Which out of the following is a correct equation to show change in molar conductivity with respect to concentration for a weak electrolyte, if the symbols carry their usual meaning: [2024]

1 $Λ_{m} - Λ_{m}^{0} + {AC}^{1 / 2} = 0$

2 $Λ_{m} - Λ_{m}^{°} - {AC}^{1 / 2} = 0$

3 $Λ_{m}^{2} C + K_{a} Λ_{m}^{° 2} - K_{a} Λ_{m} Λ_{m}^{°} = 0$

4 $Λ_{m}^{2} C - K_{a} Λ_{m}^{° 2} + K_{a} Λ_{m} Λ_{m}^{°} = 0$

Ans.
(4)

Consider solution of a weak electrolyte AB with concentration C mol/L, undergoing dissociation in water with degree of dissociation $α$ .

$AB(aq) ⇌ A^{+} (aq) + B^{-} (aq)$

$t = 0$    $C$

   $t = \infty$    $C (1 - α)$    $C α$    $C α$

$K_{a} = \frac{[A^{+}] [B^{-}]}{[AB]} = \frac{C α \times C α}{C (1 - α)} = \frac{C α^{2}}{1 - α} - I$

$α$ is related to $Λ_{m}$ and $Λ_{m}^{°}$ as $α = \frac{Λ_{m}}{Λ_{m}^{°}} - II$

Put $α$ from II in I

$K_{a} = \frac{C {(\frac{Λ_{m}}{Λ_{m}^{°}})}^{2}}{1 - (\frac{Λ_{m}}{Λ_{m}^{°}})}$

$C {(\frac{Λ_{m}}{Λ_{m}^{°}})}^{2} = K_{a} - K_{a} (\frac{Λ_{m}}{Λ_{m}^{°}})$

   $C {(\frac{Λ_{m}}{Λ_{m}^{°}})}^{2} + K_{a} (\frac{Λ_{m}}{Λ_{m}^{°}}) - K_{a} = 0$

$Λ_{m}^{2} C + K_{a} Λ_{m} Λ_{m}^{°} - K_{a} {(Λ_{m}^{°})}^{2} = 0$

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