Surfactants, amphiphilic molecules with a hydrophilic head group and a hydrophobic tail, have been u — Physical Chemistry — Kinetics Chemistry Question
Surfactant Micelles
Surfactants, amphiphilic molecules with a hydrophilic head group and a hydrophobic tail, have been used for washing since 2500 B.C. In aqueous solutions, they self-assemble, i.e. organize spontaneously into aggregated structures, so-called micelles. This concept of structuring is not only widely found in nature and in many every-day applications but it has recently become of interest for the controlled design of more complex structures in the nanometer size range as well.
Self-assembly takes place above a certain concentration, the so-called critical micelle concentration (cmc).
Micellar aggregates are separated from solutions of varying initial surfactant concentrations c0, and the surfactant concentration in the remaining solution c1 is determined.
| c0 (g dm^-3) | 0.5 | 0.75 | 1 | 1.5 |
| c1 (g dm^-3) | 0.5 | 0.75 | 0.75 | 0.75 |
What is the cmc of the surfactant?
Model Answer
The cmc is c = 0.75 g dm^–3.
Why do amphiphilic molecules aggregate in aqueous solution?
Model Answer
Amphiphilic molecules contain a hydrophilic part which is “water-soluble” and a hydrophobic part which is “water insoluble”, i.e. the free energy for the dissolution of the hydrophilic part in water is negative, while it is positive for the hydrophobic part.
When micellar aggregates are formed, exposure of hydrophobic parts of the molecule to the aqueous phase is avoided (“hydrophobic interaction”). In addition, hydrophilic head groups can interact with water (negative hydration energy).
Sketch the osmotic pressure as a function of surfactant mass concentration and indicate the cmc.
Model Answer
[VISUAL] The osmotic pressure increases linearly with concentration up to the critical micelle concentration (cmc). Above the cmc, the osmotic pressure remains almost constant, increasing only with a very small positive slope, because any additional surfactant added forms micelles rather than free monomers.
Determine a relationship between K, c0, N and c(A), where there is a general aggregation equilibrium of N molecules of A in an aggregate B with equilibrium constant K, c(A) and c(B) are the molar concentrations of monomers and aggregates, and c0 is the total concentration of monomers in the solution.
Model Answer
K = c(B) / c(A)^N and c(A) + N * c(B) = c0
Relationship: K = (c0 - c(A)) / (N * c(A)^N)
N = 50 and K = 10^90 L^49 mol^-49 (L = dm^3) are values of self-assembly of a typical surfactant. Calculate c0, c(A) and c(B) if the fractions f = c(A)/c0 of surfactant molecules present as monomers are 0.9999, 0.5, 0.01, 10^-3 and 10^-4 respectively.
Model Answer
When c(A) = f * c0:
c(A) = [ (1 - f) / (f^N * N * K) ]^(1 / (N - 1))
Calculated values:
- For f = 0.9999:
c(A) = 0.011 mol dm^-3
c0 = 0.011 mol dm^-3
N * c(B) = 1.11 * 10^-6 mol dm^-3
c(B) = 2.23 * 10^-8 mol dm^-3
- For f = 0.5:
c(A) = 0.013 mol dm^-3
c0 = 0.027 mol dm^-3
N * c(B) = 0.013 mol dm^-3
c(B) = 2.69 * 10^-4 mol dm^-3
- For f = 0.01:
c(A) = 0.015 mol dm^-3
c0 = 1.477 mol dm^-3
N * c(B) = 1.462 mol dm^-3
c(B) = 0.029 mol dm^-3
- For f = 10^-3:
c(A) = 0.015 mol dm^-3
c0 = 15.481 mol dm^-3
N * c(B) = 15.466 mol dm^-3
c(B) = 0.309 mol dm^-3
- For f = 10^-4:
c(A) = 0.016 mol dm^-3
c0 = 162.265 mol dm^-3
N * c(B) = 162.249 mol dm^-3
c(B) = 3.245 mol dm^-3
Depending on the surfactant architecture, micelles can have different shapes. In this context, surfactant molecules are characterized by the area a of their head group, the length l of the molecule and the volume v of the molecule, being combined in the so-called packing parameter v∙(a∙l)^-1. [VISUAL]
Based on geometrical considerations, determine conditions for the packing parameter so that the amphiphile can form:
a) spherical aggregates
b) cylindrical aggregates (disregard end caps)
c) flat aggregates (bilayers)
Model Answer
a) For spherical aggregates with radius l and aggregation number N, micelle volume V and micelle surface A are:
(I) V = N * v = (4/3) * pi * l^3 (v = volume of surfactant molecule)
(II) A = N * a = 4 * pi * l^2 (a = head group area of surfactant molecule)
Division of (I) by (II): v/a = l/3, or v / (a * l) = 1/3 (specifically, for a sphere of radius R <= l, v / (a * l) <= 1/3)
b) For cylindrical aggregates with radius l we consider a part of the cylinder with length b and the aggregation number N. Micelle volume V and micelle surface A are:
(I) V = N * v = pi * l^2 * b
(II) A = N * a = 2 * pi * l * b
Division of (I) by (II): v/a = l/2, or v / (a * l) = 1/2 (specifically, 1/3 < v / (a * l) <= 1/2)
c) For a flat bilayer with the thickness 2 * l we consider a part of the size (area) x and aggregation number N. Micelle volume V and micelle surface A are:
(I) V = N * v = x * 2 * l
(II) A = N * a = 2 * x
Division of (I) by (II): v/a = l, or v / (a * l) = 1 (specifically, 1/2 < v / (a * l) <= 1)
For sodium dodecyl sulfate (SDS):
V = 0.35 nm^3, a = 0.57 nm^2 and the (maximum 'liquid') length l = 1.67 nm.
Which shape do SDS micelles in aqueous solution have? Calculate. (Hint: Are the ideal values calculated in 31.5 lower or upper values?)
Model Answer
Packing parameter calculation:
v / (a * l) = 0.35 nm^3 / (0.57 nm^2 * 1.67 nm) = 0.37
Since 1/3 < 0.37 <= 1/2, cylindrical micelles form (specifically slightly elongated micelles or 'short cylinders').
The value calculated for the spherical geometry is an upper value. Concerning larger values of the packing parameter, the volume of the hydrophobic part of the molecule is too large to fit into a sphere. Cylinders (or slightly elongated micelles) can form, although the geometric conditions are not ideal. (Note also that the surfactant length given refers to the maximum extension of the hydrocarbon chain: conformations with shorter extensions may form, but there is no conformation with longer lengths.)
What do you think will form after the addition of a base to the SDS micellar solution?
Model Answer
Spherical micelles. Note: After the addition of a base, protolysis increases, the charges on the head groups (on average) increase and thus the effective head group area increases due to electrostatic repulsion. Hence, the value of the packing parameter v∙(a∙l)^-1 = 0.37 calculated in a) decreases. Since 0.37 is not much higher than the limit for spherical micelles, the regime of spherical aggregates (<= 1/3) can be reached.