JEE Main 2024 — Electrochemistry Question with Solution

The dissociation of a weak electrolyte ($\mathrm{HA}$) in solution can be represented as:

$\mathrm{HA}(\mathrm{aq})\rightleftharpoons {\mathrm{H}}^{+}(\mathrm{aq})+{\mathrm{A}}^{-}(\mathrm{aq})$

The degree of dissociation ($\alpha$) for a weak electrolyte is given by:

$\alpha =\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}$

Where ${\Lambda }_{\mathrm{m}}$ is the molar conductivity at a given concentration $C$, and ${\Lambda }_{\mathrm{m}}^{\circ }$ is the molar conductivity at infinite dilution.

The dissociation constant ($K_{\mathrm{a}}$) is described by:

$K_{\mathrm{a}}=\frac{{\alpha }^{2}C}{1-\alpha }$

Rewriting this equation in terms of ${\Lambda }_{\mathrm{m}}$ and ${\Lambda }_{\mathrm{m}}^{\circ }$:

$K_{\mathrm{a}}=\frac{{\left(\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}\right)}^{2}C}{1-\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}}$

By multiplying out and simplifying the above expression:

${\left(\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}\right)}^{2}C+K_{\mathrm{a}}\left(\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}\right)=K_{\mathrm{a}}$

Substitute $\alpha =\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}$:

${\left(\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}\right)}^{2}C+K_{\mathrm{a}}\left(\frac{{\Lambda }_{\mathrm{m}}}{{\Lambda }_{\mathrm{m}}^{\circ }}\right)-K_{\mathrm{a}}=0$

Rewriting by multiplying through by ${\left({\Lambda }_{\mathrm{m}}^{\circ }\right)}^2$:

${\Lambda }_{\mathrm{m}}^{2}C+K_{\mathrm{a}}{\Lambda }_{\mathrm{m}}{\Lambda }_{\mathrm{m}}^{\circ }-K_{\mathrm{a}}{\left({\Lambda }_{\mathrm{m}}^{\circ }\right)}^2=0$

Thus, the correct equation that describes the change in molar conductivity with respect to concentration for a weak electrolyte is given by:

Option D:

${\Lambda }_{\mathrm{m}}^{2}C-K_{\mathrm{a}}{\Lambda }_{\mathrm{m}}^{\circ 2}+K_{\mathrm{a}}{\Lambda }_{\mathrm{m}}{\Lambda }_{\mathrm{m}}^{\circ }=0$