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It has been stated that the molar conductance($\Lambda_m$) of strong electrolytes is not affected to a greater extent on dilution and so to find the limiting value of molar concentration($\Lambda_m^0$) we can extrapolate the graph of molar conductance vs the concentration ($\sqrt c$). But as the concentration would near zero shouldn't the molar conductance decrease sharply to the range of say $10^{-8}$ which is the conductance of distilled water as the amount of solute that is the strong electrolyte would be negligible to consider?Also in this case extrapolating the graph should not be possible?

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The molar conductivity of a strong electrolyte $(\Lambda_m)$ is given as the ratio of measured conductivity $(\kappa)$ to the molar concentration $(c)$: $$\Lambda_m=\frac{\kappa}{c}$$

However, as you note, the value of $\Lambda_m$ is not independent of concentration. The reason for this is that $\kappa$ does not scale linearly with concentration.

$$\Lambda_m = \Lambda_m^\circ -K\sqrt{c}$$

In this equation, $\Lambda_m^\circ$ is the limiting or intrinsic conductivity of the electrolyte and $K$ is an empirical constant. The interpretation of $\Lambda_m^\circ$ is the molar conductivity at infinite dilution as $c\rightarrow0$.

$$\lim_{c\rightarrow 0}{\Lambda_m}=\Lambda_m^\circ$$

The infinite dilution scenario is different from the scenario when there were no ions in solution to begin with, just as mathematically $\lim_{x \rightarrow 0}{f(x)}$ does not always equal $f(0)$ (often because $f(0)$ is undefined).

In the limiting case of $c\rightarrow 0$, the ion concentration never reaches zero, but it does keep getting close. You can never dilute a solution containing a solute to zero concentration (i.e. pure solvent).

If you graph $\Lambda_m$ as a function of $\sqrt{c}$, you should get a straight line with a non-zero intercept $(\Lambda_m^\circ)$. This value is an empirical parameter used to model data. It can also tell you something about the intrinsic conductivity of ions (see this Wikipedia article). However, as you say, at exactly $c=0$, he conductivity should be negligible. At any other value of $c$, however the conductivity should follow from the equation given. Thus, the conductivity is not necessary linear in the infinitesimal range around $c=0$:

$$\Lambda_m=\begin{cases} 10^{-8}\approx0, &c=0\\ \Lambda_m^\circ - K\sqrt{c}, &c>0 \end{cases}$$

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On increasing the dilution the conductivity($\kappa$) does decrease as it is proportional to the number of ions in a unit volume. But the molar conductance is the product of $\Lambda_m=\kappa V$ where $V$ is the volume containing one mole of the dissolved electrolyte.It is proportional to the number of ions in $V$ volume of the electrolytic solution which includes all the ions in one mole of the electrolyte plus the additional contribution from water. Hence, on increasing the dilution $\kappa$ goes down but $V$ goes up in such a way that its product goes up. Therefore in increasing the dilution, the conductivity will fall to the level of distilled water but the molar conductance still includes the volume containing one mole electrolyte and hence will have a value higher than that of distilled water and the concentrated solution of the electrolyte.

[The general term Conductance (as defined by $C=\frac{1}{R}$) is different from the quantity molar conductance. The former, deals with a electrolytic setup with a definite cell constant and fixed condition. The other is equal to the Conductance of $V$ volume of electrolyte between two electrodes placed 1 UNIT APART.]

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Since molar conductivity is directly proportional to the number of ions ,reduction in concentration reduces the number of ions per unit volume hence reduction in conductivity.

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