Under ambient conditions, which among the following surfactants will form micelles in aqueous solution at lowest molar concentration?

For ionic surfactants dissolved in water, micelles start forming only after the concentration of monomeric surfactant reaches a particular threshold called the critical micelle concentration (CMC). Empirical studies give the relation

$$\log(\text{CMC}) = A - B\,n$$

where $$A$$ and $$B$$ are positive constants for a given head-group type and $$n$$ is the number of carbon atoms in the hydrophobic chain. We immediately notice two facts:

1. The term $$-B\,n$$ is negative because of the minus sign, and it becomes more negative as $$n$$ increases. 2. Therefore, as $$n$$ becomes larger, $$\log(\text{CMC})$$ decreases, and so the CMC itself decreases exponentially.

In simpler words, a longer hydrocarbon (non-polar) chain gives a lower CMC because stronger hydrophobic interactions drive the molecules to aggregate (form micelles) even at very small concentrations.

Next, the ionic nature of the head group can also influence CMC, but for surfactants belonging to the same broad class (ionic head, single hydrocarbon tail) the chain-length effect dominates. Hence we first rank the four options by the length of their hydrocarbon tails:

• Option A: $$CH_3(CH_2)_{11}\overset{\oplus}{N}(CH_3)_3 Br^-$$ has $$12$$ carbon atoms in the tail (one $$CH_3$$ plus eleven $$CH_2$$).
• Option B: $$CH_3(CH_2)_{13}OSO_3^-Na^+$$ has $$14$$ carbon atoms in the tail.
• Option C: $$CH_3(CH_2)_{15}\overset{\oplus}{N}(CH_3)_3 Br^-$$ has $$16$$ carbon atoms in the tail.
• Option D: $$CH_3(CH_2)_8COO^-Na^+$$ has $$9$$ carbon atoms in the tail.

Now we compare these counts:

$$n_C(\text{Option C}) = 16 \gt n_C(\text{Option B}) = 14 \gt n_C(\text{Option A}) = 12 \gt n_C(\text{Option D}) = 9$$

Because the CMC decreases as $$n$$ increases, the order of CMC (from lowest to highest) will be the reverse:

$$\text{CMC}(\text{C}) \lt \text{CMC}(\text{B}) \lt \text{CMC}(\text{A}) \lt \text{CMC}(\text{D})$$

Therefore, among the given surfactants, Option C with the longest C16 tail will form micelles at the lowest molar concentration under ambient conditions.

Hence, the correct answer is Option 3.