Understanding Molar Conductivity Above CMC

The question asks about how the molar conductivity of sodium dodecylsulphate changes when its concentration in an aqueous solution goes above the Critical Micelle Concentration (CMC). It also asks about the state of the solution in this concentration range.

Let's break down what happens as the concentration of a surfactant like sodium dodecylsulphate increases in water:

• Below CMC: Sodium dodecylsulphate exists primarily as individual ions (monomers): $\text{Na}^+$ and $\text{dodecylsulphate}^-$ ions. These ions are mobile charge carriers, and the solution behaves like a typical electrolyte. As concentration increases below CMC, the molar conductivity slightly decreases due to increased interionic interactions, which is a common behavior for strong electrolytes.
• At CMC: This is the specific concentration where surfactant molecules start aggregating to form micelles. Micelles are clusters of surfactant molecules, with the hydrophobic tails facing inward and the hydrophilic heads facing outward towards the water.
• Above CMC: As the concentration is increased further above the CMC, almost all additional surfactant molecules added to the solution aggregate to form more micelles. The concentration of free monomers remains relatively constant at the CMC value.

Conductivity and Micelle Formation

Conductivity in a solution depends on the number of charge carriers and their mobility. Molar conductivity ($\Lambda_m$) is defined as:

$\Lambda_m = \frac{\kappa}{C}$

where $\kappa$ is the conductivity and $C$ is the molar concentration of the surfactant.

Above the CMC, the charge carriers in the solution are the free counterions ($\text{Na}^+$) and the charged micelles (formed by the aggregation of dodecylsulphate ions, which also bind some counterions to their surface). Compared to the individual dodecylsulphate monomers, micelles are large, heavy aggregates.

• The mobility of these large micelles is much lower than the mobility of individual monomeric ions.
• Although micelles carry charge, their contribution to the total conductivity per formula unit of surfactant is significantly less than that of the highly mobile monomers.
• As the total concentration of surfactant increases above CMC, the *additional* conductivity comes mainly from the formation of more low-mobility micelles and some counterions, while the total concentration used in the $\Lambda_m$ calculation keeps increasing.

Because the highly mobile species (monomers) are no longer increasing significantly with concentration, and the new species (micelles) have low mobility, the conductivity ($\kappa$) does not increase proportionally with the total concentration ($C$). This leads to a sharp decrease in the molar conductivity ($\Lambda_m$) above the CMC.

Solution State Above CMC

Micelles typically have a size in the range of 1 nm to 100 nm (or slightly larger, up to 1000 nm depending on the system). Particles in this size range dispersed in a medium form a colloidal solution. Therefore, the presence of micelles above the CMC means that the solution becomes a colloidal solution.

Analyzing the Options

Let's evaluate the given options based on our understanding:

• Option 1: molar conductivity increases sharply, but the solution does not remain colloidal - Incorrect. Molar conductivity decreases sharply, and the solution becomes colloidal.
• Option 2: molar conductivity decreases sharply, but the solution remains colloidal - This aligns with our understanding. Molar conductivity drops sharply above CMC, and the solution is colloidal due to micelles.
• Option 3:

  molar conductivity decreases sharply and dissociation into monomers also occurs sharply

  - Incorrect. Molar conductivity decreases sharply, but the concentration of monomers stabilizes or slightly decreases above CMC as they form micelles, rather than increasing sharply.
• Option 4:

  molar conductivity increases sharply with large loss of entropy

  - Incorrect. Molar conductivity decreases sharply.

Based on the behavior of surfactants above the Critical Micelle Concentration, the molar conductivity decreases sharply due to the formation of low-mobility micelles, and the solution transitions into a colloidal state because of the size of the micelles.