PHYSICS · Multi-part · Fermi University Physics · Thermal, Statistical & Complex Systems · Condensed Matter, Materials & Soft Matter · Thermodynamic potentials and the formal structure of thermodynamics
Soft Matter at the Thermal Scale, Ch Amphiphiles and self-assembly
Problem
In the closed-association model, amphiphile molecules in solution are either free monomers, present at mole fraction X1, or joined into aggregates that all contain exactly N molecules (the simplest version of the model), the amphiphile in such aggregates having total mole fraction XN. Treat each aggregate size as an ideal dilute species (the law of mass action for ideal mixtures you have met), so that a molecule belonging to an N-aggregate has chemical potential μN=μN∘+NkBTln(NXN), where μN∘ is the standard chemical potential per molecule in an N-aggregate, and XN/N (not XN) is the mole fraction of the aggregates themselves, since each one carries N molecules. A free monomer has μ1=μ1∘+kBTlnX1. Define α≡(μ1∘−μN∘)/kBT, the free energy in units of kBT gained per molecule on joining an N-aggregate rather than staying free.
((a)) Using the condition that a molecule at equilibrium gains nothing by moving between the monomer state and the aggregate state, μN=μ1, derive XN=N(X1eα)N.
((b)) Check the formula by setting N=1 (a monomer is itself an "aggregate" of size one) and show that consistency requires α=0 in that case; explain in one sentence why this must be so.