Which of the following statements is incorrect with respect to the plot of molar conductivity \((\Lambda _m)\) versus square root of concentration \((c^{1/2})\) for electrolytes in solution?

For strong electrolytes, a straight line with positive slope is obtained

The correct answer is option 3. For strong electrolytes, a straight line with positive slope is obtained.

Let us explore the relationship between molar conductivity \((\Lambda_m)\) and the square root of concentration \((c^{1/2})\) for both strong and weak electrolytes in detail, focusing on the graphical representation and the behavior of these electrolytes in solution.

Molar Conductivity and its Dependence on Concentration

Molar Conductivity (\(\Lambda_m\)) is defined as the conductivity of an electrolyte solution divided by the molar concentration of the electrolyte. It is given by:

\(\Lambda_m = \frac{\kappa}{c}\)

where \(\kappa\) is the conductivity of the solution and \(c\) is the concentration.

Strong Electrolytes:

Dissociation: Strong electrolytes (e.g., NaCl, HCl) completely dissociate into ions in solution.

Molar Conductivity Behavior: As the concentration decreases, the ions in the solution experience less inter-ionic attraction, which allows them to move more freely, increasing the molar conductivity. However, even at higher concentrations, the strong electrolytes are fully dissociated, leading to relatively high molar conductivities.

Kohlrausch's Law: The relationship between molar conductivity \(\Lambda_m\) and the square root of concentration \((c^{1/2})\) for strong electrolytes is described by Kohlrausch's Law:

\(\Lambda_m = \Lambda_m^o - A\sqrt{c}\)

where:

\(\Lambda_m^o\) is the limiting molar conductivity (molar conductivity at infinite dilution).

\(A\) is a constant that depends on the nature of the solvent and temperature.

\(c\) is the concentration of the electrolyte.

Graphical Representation: 

When plotting \(\Lambda_m\) versus \(\sqrt{c}\) for strong electrolytes, the graph is a straight line with a negative slope. The intercept on the \(\Lambda_m\) axis (when \(\sqrt{c} = 0\)) gives the limiting molar conductivity \(\Lambda_m^o\). The slope of the line is negative because as concentration increases (i.e., \(\sqrt{c}\) increases), the molar conductivity decreases due to increased ion interactions and decreased mobility.

Weak Electrolytes

Dissociation: Weak electrolytes (e.g., acetic acid, ammonium hydroxide) only partially dissociate in solution, meaning that not all of the electrolyte molecules break down into ions.

Molar Conductivity Behavior: For weak electrolytes, molar conductivity increases more significantly as the concentration decreases because the degree of dissociation increases with dilution. The relationship between \(\Lambda_m\) and \(c\) is more complex and does not follow a simple linear pattern.

Graphical Representation:

When plotting \(\Lambda_m\) versus \(\sqrt{c}\) for weak electrolytes, the graph is not a straight line. Instead, it shows a curved line that reflects the changing degree of dissociation with concentration. This non-linear behavior makes it impossible to directly extrapolate the limiting molar conductivity \(\Lambda_m^o\) from such a plot using a straight line.

Analysis of the Statements:

1. For strong electrolytes, a straight line with exploitable intercept of limiting molar conductivity \((\Lambda^o_m)\) is obtained.

The statement is correct. As explained above, for strong electrolytes, the plot of \(\Lambda_m\) versus \(\sqrt{c}\) gives a straight line, and the intercept at \(\sqrt{c} = 0\) gives the limiting molar conductivity \(\Lambda_m^o\).

2. For strong electrolytes, a straight line with negative slope is obtained.

The statement is correct. The slope of the line is negative due to the decrease in molar conductivity with an increase in concentration (as \(\sqrt{c}\) increases).

3. For strong electrolytes, a straight line with positive slope is obtained.

The statement is incorrect. The slope of the line for strong electrolytes is negative, not positive. Therefore, this statement is incorrect.

4. For weak electrolytes, a straight line is not obtained.

The statement is correct. The plot of \(\Lambda_m\) versus \(\sqrt{c}\) for weak electrolytes is not a straight line; it is curved due to the varying degree of dissociation.

Conclusion:

The incorrect statement is option 3: "For strong electrolytes, a straight line with a positive slope is obtained." For strong electrolytes, the line is indeed straight but has a negative slope, reflecting the decrease in molar conductivity with increasing concentration.