Biomembranes fulfill many important functions in the living cell. Membranes in plant and animal cell — Analytical Chemistry Chemistry Question
Micelles
Biomembranes fulfill many important functions in the living cell. Membranes in plant and animal cells contain 40 – 50 % proteins. Phospholipids, which are key constituents in biomembranes, have hydrophobic fatty acid tails and polar hydrophilic head groups. Such structures are commonly called amphiphiles. Knowledge of membranes is obtained from studies of the aggregation behavior of amphiphiles with a simple(r) molecular structure. Typical aggregates are micelles, mono-, and bilayer structures and vesicles (liposomes). Single-tailed surfactant molecules like sodium n-dodecylsulfate (SDS) and n-dodecyl-trimethylammonium bromide (DTAB) cooperatively form micelles upon dissolution in water above the critical micelle concentration (CMC). [VISUAL] The structure of micelles is pictured in the figure. In these micelles, a central hydrophobic core can be recognized and a layer containing head groups and some counter ions (Guoy-Chapman layer). For micelles of SDS the central core has a radius of 16.6 Å and the Stern layer has a thickness of 4.6 Å.
| Amphiphile | CMC (mmol dm⁻³) | Relative micelle mass (g mol⁻¹ ×10³) |
|---|---|---|
| SDS | 8.1 | 18.0 |
| DTAB | 14.4 | 15.0 |
Calculate the volume of the Stern-layer in this micelle of SDS.
Model Answer
Volume SDS micelle = 4/3 * pi * (16.6 + 4.6)³ = 39911.33 ų
Volume of the core = 4/3 * pi * 16.6³ = 19160.77 ų
Volume of the Stern layer = volume of the SDS micelle – volume core = 20750.56 ų
In a simplified model, Micelle formation can be expressed by the equilibrium:
n S + n B <==> M
wherein S is the amphiphile, B is counter ion and n is the number of molecules involved. The standard Gibbs energy of micelle formation per S is expressed by:
ΔG_M = (RT/n) * ln(K_M)
KM is the equilibrium constant. At the critical micelle concentration [M] = 0. Furthermore, assume that [S] is approximately equal to [B]. R is the gas constant (8.314 J mol⁻¹ K⁻¹).
Calculate GM for the micelle formation of SDS and of DTAB.
Model Answer
The equilibrium constant K_M = [M] / ([S]^n * [B]^n)
Substitution in ΔG_M:
ΔG_M = (RT/n) * ln([M] / ([S]^n * [B]^n)) = (RT/n) * (ln[M] - n ln[S] - n ln[B])
At the CMC there are no micelles: [M] = 0 and [S] = [B], thus: ΔG_M = 2 RT ln[S]
For SDS (using CMC = 8.1 mmol dm⁻³ = 8.1 * 10⁻³ mol dm⁻³):
ΔG_M = 2 * 8.314 * 298 * ln(8.1 * 10⁻³) = – 23.86 kJ mol⁻¹
For DTAB (using CMC = 14.4 mmol dm⁻³ = 14.4 * 10⁻³ mol dm⁻³):
ΔG_M = 2 * 8.314 * 298 * ln(14.4 * 10⁻³) = – 21.01 kJ mol⁻¹
Calculate the number of amphiphile molecules in the micelles of SDS and that of DTAB.
Model Answer
Average number of amphiphiles per micelle = relative micelle mass / relative amphiphile mass
For SDS (Mr = 288): n = 18×10³ / 288 = 62.5
For TDAB (DTAB) (Mr = 308): n = 15×10³ / 308 = 48.7