Write the given proportionality

According to the statement, $\text{Conductivity }(S) \propto \frac{A\,c}{l}$ where:

• $A$ = area of vessel, unit $m^2$
• $c$ = concentration, unit $\text{mol } m^{-3}$
• $l$ = length, unit $m$

So, $S = k\frac{Ac}{l}$ where $k$ is the proportionality constant.

Find the unit of the proportionality constant

Rearranging, $k = S\cdot \frac{l}{Ac}$

Substitute units: $[k] = S\cdot \frac{m}{(m^2)(\text{mol }m^{-3})}$

Simplify the units

First simplify the denominator: $(m^2)(\text{mol }m^{-3}) = \text{mol } m^{-1}$

Hence, $[k] = S\cdot \frac{m}{\text{mol } m^{-1}}$

$[k] = S\cdot \frac{m\cdot m}{\text{mol}} = S\,m^2\,\text{mol}^{-1}$

Match with the options

The required unit is $S m^2\, mol^{-1}$

This corresponds to Option B.

Interpretation

This proportionality constant is actually the molar conductivity $\Lambda_m$, whose unit is indeed: $S m^2 mol^{-1}$