The picture below shows a schematic structure of a micelle formed in water by th

Question

The picture below shows a schematic structure of a micelle formed in water by the detergent sodium dodecyl sulfate (SDS)*. The structure of a single molecule of SDS is also shown, where you can see that the headgroup is negatively charged and that the hydrocarbon chain consists of a 12-carbon chain. There is a positive Na+ counterion for each detergent molecule.

*Images are from http://www.chm.bris.ac.uk/motm/SLS/SLSh.htm, but this site is not relevant for the problem below

a) Studies of SDS micelles show that they contain about 64 SDS molecules, and have a radius of about 1.8 nm. Calculate the surface area of the micelle and estimate the area per hydrophilic headgroup located at the surface of the micelle for each SDS molecule.

b) From the results of part a, calculate the effective radius of each hydrophilic headgroup, and take twice the radius as an estimate of the separation distance between two negatively charged headgroups. Assuming that the headgroup region is fully hydrated, and hence has a dielectric constant near that of water (e=78), calculate the interaction energy for two negatively charged headgroups separated by the distance you just found. Express your answer as a multiple of kB T for T = 298 K.

The interaction calculated in part b is repulsive, and leads to a force that destabilizes the micelle structure by pushing the SDS molecules apart. A fully quantitative treatment is more complicated, but we will take the interaction energy calculated in part b as an estimate of the free energy cost of placing a single head group into the micelle.

In contrast, a free energy gain occurs from the hydrophobic effect. Assume that the surface area of each hydrocarbon chain is the surface area of a cone with the head group radius found in part b and height equal to the micelle radius of 1.8 nm. For a cone with base radius r and height h, the surface area (SA) is   

SA=pir(r+sqrtr^2+h^2)

In class we discussed an estimate of the free energy gained by removing a hydrophobic surface from water as =7kB T/nm2    for T 298 K. Use this value to find the energy gained as a multiple of kB T when the hydrocarbon tail of an SDS molecule is inserted into a micelle.

d) Based on the answers to part b and c, would this simple model predict that an SDS micelle will be stable, with the net free energy gain from the hydrophobic effect sufficient to offset repulsive electrostatic interactions of the head groups? Explain your answer. How important is the value of the dielectric constant of water in your analysis of the stability of the micelle? Explain

THANK YOU FOR ANY HELP

d) Based on the answers to part b and c, would this simple model predict that an SDS micelle will be stable, with the net free energy gain from the hydrophobic effect sufficient to offset repulsive electrostatic interactions of the head groups? Explain your answer. How important is the value of the dielectric constant of water in your analysis of the stability of the micelle? Explain.

Explanation / Answer

since the micelle is in a circular form surface area= 4*3.14*r2

  S.A= 4*3.14* (1.8)2

=40.69nm2

since there are 64 SDS molecules with one hydrophilic end group each, estimate the S.A per headgroup= 40.69/64= 0.635nm2

B) Effective radius of each hydrophillic headgroup

area= 0.635

4*3.14*r2=0.635

r2= 0.635/(4*3.14)

r= 0.22nm

thats all i can help you with!

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