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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 91464 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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 Spring '09
 JAMESAMES
 Physical chemistry, pH

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