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1. Consider a dilute solution of molecules at fixed temperature T. These molecules have access to a surface that has a total of B binding sites where molecules can bind. To count states in this system, we will divide space into small cells that each can hold a single molecule. There are a total of B cells that have a binding site, and a total of M cells that do not have binding sites. The overall number of cells is thus B+M This situation is schematically illustrated in the drawing below. molecule e surface binding site molecule bound to surface When a molecule is bound to a binding site (ie., when it occupies a cell with a binding site), it has an attractive energy-e (i e., e is a positive number). When a molecule is not bound to the surface (when it occupies a cell that does not have a binding site), its energy is zero. Molecules do not interact with each other. For this problem, we will ignore fluctuations of the velocities of the molecules. To further simplify things, we will only consider a single molecule. Later in the course, we will revisit this problem and deal with the case of many molecules (a) *D1* Imagine that the system contains only a single solute molecule. How many microstates does the system have? (1 pt) (b) *D1* How many microstates exist that have the molecule bound to a surface binding site? (1 (c) *D1* How many microstates exist that have the molecule free in solution? (1 pt) (d) *DI* What is the probability P(Vbound) of one particular microstate Vbound that has the molecul<e bound to a surface binding site? Your answer should include the temperature T (or inverse temperature 3), the interaction energy e, and a yet-to-be-determined normalization constant c. (2 pts)(e) *D1* What is the probability P(Vfree) of one particular microstate Vfree that has the molecule free (f) *D1* By summing up the probabilities of all microstates, determine the normalization constant (g) *D1* Calculate the probability Pbound to find the molecule bound on the surface, i.e., in any of (h) *D1* Calculate the probability Pree to find the molecule free in solution, г.е., in any of the M (i) *D1* Using a computer and setting B 10 and M-100, make a plot of Pbound as a function in solution? (2 pts) c. (2 pts) the B surface cells. (2 pts) cells away from the surface. (2 pts) of the binding energy є. It is convenient to measure є in units of kBT: practically, this amounts to setting kBT-1 in your equations. Explain the behavior of the molecule for small and large binding energy in terms of energy and entropy (or multiplicity). (2 pts) (j) In the same diagram, plot Pound as a function of the binding energy є for M = 1000, Explain the differences between the two curves (for M - 100 and M 1000) in terms of energy and entropy (or multiplicity). (2 pts) (k) Using a computer and setting B = 10 and M = 100, make a plot of Pound as a function of the temperature T. It is convenient to measure T in units of e/kB; practically, this amounts to setting e/kB -1 in your equations. Explain the behavior of the molecule for small and large temperature in terms of energy and entropy (or multiplicity). (2 pts)

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-Sorution ao hese is gingle mole cuie in he cen, and there ze total 0+ m cens 90 one molecule has otcuPy onry one Ceil at a time and has oniy miczostales coxzes ponding to one cell hevefore there are total numbei of micto Stotes the system have in the same way, these are 0 binding ceis exist here fore, he numbes of micsa states that have the molecule bora) bound to the suface binding Site are 6 5imi lariy, m numbes of micoostdas exists Fo the mole cole in fee state The Pobabilty of one particulaz bound microstai to the total num bed of bound micaostates is calculated by the max wens Boltzmann Sta tistics xelation of Pão bability gieBE Hexe q degenesacy 5o, Paobability is given b P: 8e

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