Resveratrol is a bioactive phytochemical produced in the skins of grapes that is hypothesized to contribute to the health benefits of red wine. this molecule can exist in one of two isomeric forms, cis and trans. while both forms display bioactivity, the overwhelming constituent found in grape skins is the trans form, and this particular isomer is thought to have the greatest positive impact on health. in what follows, you will create a very simplified model of resveratrol isomerization equilibrium. you will need to make several assumptions for a solution of resveratrol molecules.

(1) each molecule can freely interconvert between the two states: the cis state or the trans state. neglect other possible conformations.
(2) kinetic energies and interactions with the solvent are neglected.
(3) the solution is dilute such that resveratrol molecules do not interact with one another.
(4) the conformational internal energy of a molecule in the trans state is 0 and that of a molecule in the cis state is -c. thus the total energy is given by e= - ne where n is the number of molecules in the cis state.

(a) develop an expression for the entropy of this simple model. begin by finding the number of microstates as a function of the number of cis molecules, n, and hence the total energy, e. simplify your answer using stirling's approxi- mation, and rewrite it in terms of the fraction of cis isomers x = n/n and the total number of particles n.
(b) find the fraction of cis isomers x as a function of temperature.