Molar Conductivity Variation and Micelle Formation of Sodium Stearate

Sodium stearate ($\text{CH}_3(\text{CH}_2)_{16}\text{COONa}$) is a surface-active agent (surfactant) that behaves as a strong $1:1$ electrolyte at low concentrations and forms aggregated colloids (micelles) above a threshold concentration known as the Critical Micelle Concentration (CMC).

1. Behavior Below CMC ($c < \text{CMC}$):

At very low concentrations ($c < \text{CMC}$), sodium stearate completely dissociates into individual sodium ions ($\text{Na}^+$) and stearate ions ($\text{C}_{17}\text{H}_{35}\text{COO}^-$): $\text{C}_{17}\text{H}_{35}\text{COONa (aq)} \rightarrow \text{C}_{17}\text{H}_{35}\text{COO}^- \text{(aq)} + \text{Na}^+ \text{(aq)}$

As a strong electrolyte, its molar conductivity ($\Lambda_m$) follows the Debye-Hückel-Onsager equation: $\Lambda_m = \Lambda_m^\circ - A\sqrt{c}$ where $\Lambda_m^\circ$ is the limiting molar conductivity at infinite dilution and $A$ is a constant. Thus, as the concentration $c$ (or $\sqrt{c}$) increases, interionic attraction increases moderately, causing $\Lambda_m$ to decrease linearly with a characteristic mild negative slope.

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2. Behavior At and Above CMC ($c \ge \text{CMC}$):

When the concentration reaches the CMC, the hydrophobic hydrocarbon chains ($\text{C}_{17}\text{H}_{35}-$) aggregate together to minimize contact with water, forming spherical clusters called micelles.

Upon micelle formation:

1. Reduction in Free Ions: A large number of individual stearate ions (typically $50-100$) aggregate into a single micellar species, drastically reducing the total number of independent charge-carrying particles per mole of surfactant.
2. Counter-Ion Binding: A significant fraction of $\text{Na}^+$ ions bind to the charged hydrophilic heads ($\text{-COO}^-$) on the surface of the micelle due to strong electrostatic attraction. This reduces the effective concentration of free $\text{Na}^+$ ions available for electrical conduction.
3. Decreased Ionic Mobility: The large size of the micelle reduces its overall mobility compared to individual small monomer ions.

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Conclusion:

Because of the sudden decrease in the effective number of free charge carriers and counter-ion binding at concentrations exceeding the CMC, the molar conductivity ($\Lambda_m$) drops much more sharply as $\sqrt{c}$ increases beyond the CMC.

Therefore, the graph of $\Lambda_m$ versus $\sqrt{c}$ shows a linear decrease with a moderate negative slope below CMC, followed by a steeper negative slope above CMC.

This behavior is correctly depicted in Option (C).