Which out of the following is a correct equation to show change in molar conductivity with respect to…

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 :
  1. \(\Lambda_{\mathrm{m}}-\Lambda_{\mathrm{m}}^0+\mathrm{AC}^{\frac{1}{2}}=0\)
  2. \(\Lambda_m^2 \mathrm{C}-\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\circ 2}+\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}} \Lambda_{\mathrm{m}}=0\)
  3. \(\Lambda^2{ }_m \mathrm{C}+\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}}^{\circ 2}-\mathrm{K}_{\mathrm{a}} \Lambda_{\mathrm{m}} \Lambda_{\mathrm{m}}^{\circ}=0\)
  4. \(\Lambda_{\mathrm{m}}-\Lambda_{\mathrm{m}}^{\circ}-\mathrm{AC}^{\frac{1}{2}}=0\)

Solution

$\begin{aligned} & \mathrm{HA}(\mathrm{aq}) \rightleftharpoons \mathrm{H}^{+}(\mathrm{aq})+\mathrm{A}^{-}(\mathrm{aq}) \\ & \mathrm{K}_{\mathrm{a}}=\frac{\alpha^2 \mathrm{C}}{1-\alpha} \\ & \alpha^2 \mathrm{C}+\mathrm{K}_{\mathrm{a}} \alpha-\mathrm{K}_{\mathrm{a}}=0 \\ & \left(\frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{m}}^{\infty}}\right)^2 \mathrm{C}+\mathrm{K}_{\mathrm{a}} \frac{\lambda_{\mathrm{m}}}{\lambda_{\mathrm{m}}^{\infty}}-\mathrm{K}_{\mathrm{a}}=0 \\ & \lambda_{\mathrm{m}}^2 \mathrm{C}+\mathrm{K}_{\mathrm{a}} \lambda_{\mathrm{m}} \lambda_{\mathrm{m}}^{\infty}-\mathrm{K}_{\mathrm{a}}\left(\lambda_{\mathrm{m}}^{\infty}\right)^2=0\end{aligned}$

Asked in: JEE Main 2024 (09 Apr Shift 2)

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