PracticeMidtermSolutions

PracticeMidtermSolutions - Questions on this exam concern...

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Unformatted text preview: Questions on this exam concern dilute solutes in a vesicle (which might be regarded as a crude model of a living cell), as sketched below. For purposes of counting, imagine dividing the vesicle into a lattice of M cells. A microstate is specified by detailing the presence or absence of solutes in every cell. The vesicle’s environment serves as a heat bath at temperature T. Quantum mechanical effects can be ignored throughout. Neglect as well contributions from solvent, particle velocities, and interactions among solute molecules unless explicitly stated otherwise. 1. Many molecules in biological cells are synthesized in very small numbers. In this question we consider an extreme case: chemical species that are either completely absent or present as single molecules. Specifically, our vesicle contains one particle of type A and one particle of type B. 8 pts Calculate the partition function Qunbound for the collection of microstates in which A and B move independently within the vesicle, as depicted below. Take the energy of such “unbound” configurations to be zero. Your answer may involve some (but not necessarily all) of the following quantities: T, M, and fundamental constants. (Dunbar/4d“: M (M“/) ‘3: 61 (ii) Calculate the partition function Qbound for the collection of microstates in which A and B are covalently bonded, as depicted below. Covalent bonding in this case lowers the energy by an amount EAB and constrains the two particles to reside in the same lattice cell. (The dimer AB can explore all cells within the vesicle.) Your answer may involve some (but not necessarily all) of the following quantities: T, M, EAB, and fundamental constants. 5A8 abound : M 5 A57. (iii) Calculate the total partition function Q for all microstates of this system (including both bound and unbound configurations). Your answer may involve some (but not necessarily all) of the following quantities: T, M , EAB, and fundamental constants. I pts (iv) Calculate the probability Pbound that A and B particles are present as the covalently bonded dimer. Sketch Pbound as a function of M (i.e.7 the vesicle’s volume). Q d M I! 7 boun e 5 Pboonfj - Q ~ Q ‘L a 62546 7,. pts (v) Sketch the heat capacity C associated with binding and unbinding of A and B particles as a function of M. You are not required to compute C explicitly but should explain the reasoning behind your plot. C 2. In this problem you will consider a single solute species that is present in large numbers (but is still dilute). This solute interacts favorably with the vesicle’s boundary. Specifically, a solute molecule residing in a lattice cell adjacent to the surface (shaded gray in the sketch below) has energy —€Surf; away from the boundary its energy is zero. Interactions between solute molecules are negligible. The number Msurf of lattice cells at the surface is large, but they constitute a small fraction of the vesicle’s volume (i.e., M >> Msurf >> 1, so you may approximate M — Msurf % M Let N be the total number of solute molecules. Nsurf of these molecules are located ‘ in lattice cells at the vesicle’s surface. The remaining Nin : N — Nsurf solutes reside in the vesicle’s interior (away from the surface). (i) Calculate the partition function Qin for the Nin indistinguishable'solute molecules in the vesicle’s interior. Your answer may involve some (but not necessarily all) of the following quantities: T, M, Msurf, N, Nin, Nsurf, €Surf, and fundamental constants. N, M II) QM M,” 'l‘ “ fl up. (ii) Calculate the partition function qurf for the Nsurf indistinguishable solute molecules at the vesicle’s surface. Your answer may involve some (but not necessarily all) of the following quantities: T, M, Msurf, N, Nin, Nsurf, Esurf, and fundamental constants. Mod” €50/%6 7" (iii) Combining results from parts and (ii), calculate the partition function Q for all N solute molecules in the vesicle. (Note that molecules at the surface can be distinguished from those in the interior, and that the two populations are , statistically independent.) Your answer may involve some (but not necessarily all) of the following quantities: T, M , Msurf, N , Nin, Nsurf, Esurf, and fundamental constants. (iv) From the dependence of Q on Nsurf, determine the most probable division of the N solute molecules between the vesicle’s surface and its interior. If you have worked the problem correctly, you should be able to express your answer compactly as a relationship between the interior density p E Nin/M and the surface density a E Nsurf/Msurf (Hints: (a) an is easier to work with here than Q itself. (b) Due tothe constraint N : Nin + Nin, Nin depends on Nsurf so that dNin/sturf 2: —1. (c) Useful properties of factorials are listed on the front page of the exam.) dA’éaf A8 T gsarf : ~/n{/\'1j+ /fl (Mun/F) + ‘87, .L /n NM _ /flMur/:O / or . :‘ n 4 —-f /n ______/U’/) + - dam“ ' O NSU’f M kgr A ‘5 g: wée/I [5 £27 Q. 6 l. -Mgf.'g& 5r sfou/c/ 56 >>fl / £0517 9-1464 5,13 :0 u 8 pts 3. This problem concerns a single solute species that can be exchanged between the vesicle and its surroundings. (i) For noninteracting, indistinguishable molecules (with no internal structure) that can each occupy M lattice sites, calculate the change in entropy, AS : 5(N — 1) — 8(N) when the number of molecules is reduced from N to N — 1. Express your answer as a function of the density p = N /M and fundamental constants. ' M’W /w” lfl ) "’ ’48 L4 (N! W. N! .7. ) (Lu... I )n A3 a MA] . ‘5 f Qts (ii) Consider a process in which one solute molecule is reversibly moved from inside the vesicle to its surroundings, as sketched below. Using your result from part (i), determine the net change in entropy resulting , from this exchange. Write your answer in terms of the solute density pin : N /M inside the vesicle, the corresponding density pout of solutes outside the vesicle, and fundamental constants. AS 3 45,}, + 4560,! 3* I23 //IJ0,',, - Ac cu} 8 pts (iii) How much heat (heV flows into the vesicle as a result of this reversible process? AS: ;V :5) qreu ': 8 pts (iv) Assuming that the vesicle’s internal energy is a function of temperature alone (E = E (T)), how much work wrev is done on the vesicle in the course of solute removal? Evaluate wrev in units of kcal/mol for the case pin : 2p0ut at room temperature. AE‘20 z) Wrev : "CZ/ed ...
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PracticeMidtermSolutions - Questions on this exam concern...

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