(Solved): Consider the photoisomerization of Retinal from all-trans-retinal to 11 -cis ...

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Consider the photoisomerization of Retinal from all-trans-retinal to 11 -cis retinal, comtained in \( \mathrm{N} \) rhodopsin protein complexes (one retinal molecule per complex) in retinal cells in the back of the eye. Considering a two state system with the cis form having an internal energy of \( -\varepsilon \) in the trans state and 0 in the cis state such that the total energy is given by \( \mathrm{E} \) \( =-\mathrm{n \varepsilon} \), where \( \mathrm{n} \) is the number of molecules in the cis state. a) Develop an expression for the entropy of this simple two state model and first find the number of microstates as a function of cis molecules, \( \mathrm{n} \) and thus the total energy, E. Simplify this using Stirling's law and write it in terms of the fraction of cis isomers \( \mathrm{x}=\mathrm{n} / \mathrm{N} \) and the total number rhodopsin complexes, \( \mathrm{N} \). b) Find an expression for the fraction of cis isomers, \( \mathrm{x} \) as a function of temperature