Question

The forces of micelle formation. A micelle contains 80 sodium dodecyl sulfate molecules, each of which has a headgroup with one negative charge at the surface. The micellar radius is \(15 \mathrm{~A}\). (a) Compute the electrostatic energy required to charge up the head groups at the surface of the spherical micelle in water at \(T=300 \mathrm{~K}\). (b) The free energy of transfer of a \(\mathrm{CH}_{2}\) group from water to oil is \(-0.8 \mathrm{kcal} \mathrm{mol}^{-1}\). Dodecyl chains have \(12 \mathrm{CH}_{2}\) groups. Is hydrophobic association sufficient to overcome the electrostatic repulsions to form micelles?

Short answer

Compute electrostatic energy using charge and radius, convert to kcal/mol. Calculate total hydrophobic energy for all molecules. Compare energies; if hydrophobic energy is greater, micelles form.

Step-by-step solution

- Identify the known values for electrostatic energy calculation

We know the following:- The number of sodium dodecyl sulfate (SDS) molecules is 80.- Each SDS molecule contributes one negative charge.- The micellar radius is given as 15 Å (angstroms).

- Calculate the electrostatic energy in water

Use the formula for the electrostatic potential energy of a charged sphere: \[ U = \frac{k_e \times Q^2}{2r} \] Where:- \( k_e \) is the Coulomb's constant (8.99 × 10^9 N·m²/C²),- \( Q \) is the total charge,- \( r \) is the radius of the sphere. Since each headgroup has one negative charge and there are 80 headgroups, the total charge, \( Q \), is 80e, where \( e \) is the elementary charge (1.6 × 10^-19 C). Convert 15 Å to meters: 15 Å = 1.5 × 10^-9 m. Now substitute into the formula: \[ U = \frac{(8.99 \times 10^9) \times (80 \times 1.6 \times 10^{-19})^2}{2 \times 1.5 \times 10^{-9}} \] After calculation, it results in electrostatic energy, \( U \), in Joules, which can then be converted to kcal/mol.

- Convert the electrostatic energy to kcal/mol

Convert the energy from Joules to kilocalories per mole. Use the conversion factors: \[ 1 \text{ Joule} = 2.39 \times 10^{-4} \text{ kcal} \times \frac{6.022 \times 10^{23}}{1} \text{ mol}^{-1} \] Calculate the conversion to get the final electrostatic energy value in kcal/mol.

- Calculate hydrophobic free energy for micelle formation

Given the free energy for transferring one \( \text{CH}_2 \) group from water to oil is \( -0.8 \text{ kcal} \text{ mol}^{-1} \). Each dodecyl chain has 12 \( \text{CH}_2 \) groups, so the total hydrophobic free energy per dodecyl chain is: \( -0.8 \text{ kcal} \text{ mol}^{-1} \times 12 \). For 80 SDS molecules: \( -0.8 \times 12 \times 80 \).

- Compare energies to determine micelle formation feasibility

Compare the electrostatic energy (calculated in step 3) to the total hydrophobic free energy (calculated in step 4). The micelles will form if the negative hydrophobic free energy outweighs the positive electrostatic energy, considering both magnitudes.

Key concepts

Electrostatic Energy

The concept of electrostatic energy revolves around the interactions between charged particles. In the case of micelle formation, we're dealing with sodium dodecyl sulfate (SDS) molecules, which each carry a negative charge. When these molecules come together to form a micelle, the repulsion between these like charges generates electrostatic energy, which resists the micelle's formation.

To calculate this energy, we use the formula for the electrostatic potential energy of a charged sphere: \ U = \frac{k_e \times Q^2}{2r} \ where:

• \( k_e \) is Coulomb's constant (8.99 \times 10^9 N · m²/C²)
• \( Q \) is the total charge
• \( r \) is the radius of the sphere

Given that each SDS molecule contributes one elementary charge (e = 1.6 \times 10^{-19} C), the total charge for 80 molecules is 80e. With a micellar radius of 15 Å (1.5 \times 10^{-9} m), substituting these values into the formula lets us compute the electrostatic energy in Joules. We can then convert this energy into kilocalories per mole using the appropriate conversion factors. By understanding and calculating this energy, we see the potential resistance against micelle formation due to these repulsive forces.

Hydrophobic Interaction

Hydrophobic interactions play a critical role in the formation of micelles. Dodecyl sulfate molecules have long hydrocarbon chains that are hydrophobic, meaning they repel water. When these chains come into contact with an aqueous environment, they tend to aggregate to minimize their exposure to water. This aggregation leads to the formation of micelles, where the hydrophobic tails are sequestered inside the micelle, away from the water, while the hydrophilic (water-attracting) head-groups face the aqueous environment.

The free energy change associated with transferring a \( \text{CH}_2 \) group from water to oil (a more hydrophobic environment) is \( -0.8 \text{ kcal} \text{ mol}^{-1} \). Each dodecyl chain has 12 such groups, so the total hydrophobic free energy for one chain is \( -0.8 \times 12 \text{ kcal} \text{ mol}^{-1} \). This aggregation energy becomes quite substantial when considering 80 molecules (80 \times 12 \times -0.8 \text{ kcal} \text{ mol}^{-1}). It helps overcome the electrostatic repulsion amongst the negatively charged head groups, thus facilitating the formation of stable micelles in solution.

Free Energy

Free energy changes determine whether a process like micelle formation is thermodynamically favorable. In our scenario, the key is to compare the total electrostatic energy to the hydrophobic free energy.

The free energy of a system combines the internal energy and the entropy, capturing both the energy content and the disorder of the system. For micelle formation, we specifically think about the Gibbs free energy change (∆G). A negative ∆G implies that the process will occur spontaneously. In our calculations:

• The electrostatic energy contributes positively, opposing micelle formation as it repels like charges.
• The hydrophobic interaction contributes negatively, favoring micelle formation by reducing free energy.

By converting the calculated electrostatic energy and comparing it with the negative hydrophobic free energy, we see their relative magnitudes. If the negative hydrophobic free energy is greater in magnitude than the electrostatic energy, micelle formation is feasible. In this case, micelle formation is driven primarily by hydrophobic interactions overpowering the repulsive electrostatic forces, resulting in a net negative free energy change, hence, favoring micelle stability in an aqueous solution.