This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Questions on this exam concern dilute amphiphilic molecules in (and at the
surface of) aqueous solutions. One such molecule (denoted “A” for “amphiphile”),
consisting of a polar head and a nonpolar tail, is sketched below. <———polar head A: <—rnpﬂartai_l . These amphiphilic solutes can shield their hydrophobic tails from solvent by forming micelles, roughly spherical clusters of solutes whose head groups point outward %
stab micelle (cross sectim) Each micelle (denoted An) is made up of n amphiphile molecules. We will not
specify a value for n, except to say that it is much greater than one. Let in be the chemical potential of an individual amphiphile A that is well
separated from other solutes. Let an be the chemical potential of an entire micelle
An. Similarly, we will denote the number of unassociated amphiphiles per unit
volume as p1, and the number of micelles per unit volume as pn. Finally, let N be
the total number of amphiphiles (including those in micelles), so that c : N / V :
p1 + npn is their total concentration. Temperature T and pressure p will be ﬁxed throughout this problem. (i) What is the change AG” in Gibbs free energy when n amphiphiles are brought
together to form a micelle, i.e., resulting from a single instance of the reaction nA —> An. Your answer should involve some (but not necessarily all) of the
following quantities: the chemical potentials ,u1 and pm, the size n of a micelle, T,
p, N, c, and fundamental constants. AG;I;M4—HM, (ii) What is the value of AGn when the densities p1 and ,0n take their equilibrium
values? Explain your answer in terms of the second law of thermodynamics. AJ msﬂrr'vm mu, 5“,! so 44,, :'0 I #e :9” /aw 7’6”: «28
{4‘03} 7/52 37V,“[é/lam sip/Mtg I}: a MMQWM 0"” 14:96 (”“37 (iii) Assuming the solution to be ideally dilute in both A and An, write an equation
relating the equilibrium densities p1 and pn. In addition to T, p, n, N , c, and
fundamental constants, your answer may involve the following quantities: a reference density p0, the chemical potential MED) of individual amphiphiles when p1 : p0, and the chemical potential #5?) of micelles when pn 2 pg. (0)  J0, w f4
f1 (:69 +As7lrn 45:): ”did fAéf/n :02" (iv) Consider a choice of standard state density p0 for which #720) 2 WE”. You should then be able to express the law of mass action from part (iii) in a very
compact form. Write an equation relating ,51 : p1 / p0 and ﬁn E pn/po, i.e., the
densities of A and An relative to the reference value. n/nﬁ?/xzﬂ / ﬂ:/5;" Extra credit (v) From the result of part (iv), describe the position of the equilibrium nA “é An
when 0 < ,00. Similarly, describe the relative densities of A and An When 0 > p0.
Note that for a large exponent m7 the quantity cam is nearly zero when :6 < 1 and is
very, very large When :16 > 1. [Hint: You might begin by calculating the derivative dE/dﬁl : 1 + dp‘n/dﬁl and use
it to make a plot of p1 vs. 0.] A: WAM swat” 711:": ‘00 WA“ Ia’jr #1:? 9 very very hr 6 ,hcreasc'ds («’9 idly d’L/Q 74) Md/@ #1ch ”95/ 7495 Q?(/ [It‘d Iii/M ﬂea/5' ﬁlw‘ﬁf M [CE 1%? J Ac/e E 3:}: juvs 71L» ﬁliuynlxléh'u» 1411de that; icjuql amfhilphayﬂs
J 10 pts (Vi) Plot the osmotic pressure of the amphiphile—micelle solution as a function of
total concentration c. Your score will be based on the behavior in the limits of
small and large c, where your intuition for chemical equilibria should tell you What
densities of A and An to expect. Indicate the value of the slope d7? / dc in these
limits. 2. In the remaining problems we will consider amphiphiles that reside exclusively at
the interface between solution and air (perhaps because their hydrophobic tails are
very long), as sketched below. A lateral force of strength Fext is exerted on this collection of N surface molecules,
using a block that limits the interfacial area A = X Y they can explore. Moving the
block laterally by an amount dX does work on the system dw : _ exth = _fextd~’47
where f = F/ Y is the force per unit width of the liquid—air interface. 7 pts (i) Write an equation for dA, the differential change in Helmholtz free energy
A z E — TS, in terms of changes dT, dA, dN, and quantities like T, ,u, S, N, E, andf' 0/4: {army/v— 14M 7 pts (ii) Write the analog of the Gibbs—Duhem equation (relating differential changes dT,
df, and d/r) for this interfacial system. SJT+No/M ~A0/1f50 7 pts 7 pts (iii) Write a Maxwell relation for the derivative (8 f / 8T) A, N. Is the sign of this
derivative known? Explain. , L(ﬁ/} ;(~J JA d5
CIT J/l TWAIN W{JI:)AIN)TIN  ELM (iv) Is the Sign of the derivative (8 f / 8/071, N known? Explain. ‘4': é/c 74¢}, are Conjufak. Varth/éuﬁ 3. This problem concerns phase transitions in the system of surfacebound amphiphiles
described in problem 2. At a given temperature To, increasing the force per unit width to a value f0
produces a discontinuous change in the area per molecule a = A/N, as shown
below. adaise asparse a We will denote the change in a accompanying this transition as Aa : asparse—adense.
As the area changes from asparse to adense, a total amount of heat qshrink > 0 is
observed to ﬂow out of the system. 5 pts (i) On the graph above, sketch f as a function of a for a lower temperature, To + (5T
(where 6T < 0). 10 pts (ii) Assuming 6T to be small, calculate the new transition force f0 + 6 f . Write
your answer for 6 f in terms of (ST and some (but not necessarily all) of the following
quantities: To, f0, Aa, qshrink, N , and fundamental constants. [Hints: Begin by considering relationships among the chemical potentials
asparse( f0 +6 f, T0+6T) and Mdense(fo+5f, T0+6T). Then perform Taylor expansions
for small 6 f and (ST. Some of the results from problem 2 should also be useful] ,9 N. 7. g“ m. (7E3 C m 3., «:v «I: 3
Mam W + 3;: , w on M a. i + A ; k V
“ )s  ﬂ :2” , x A e _V a, A
( ”mama ”AS: sazwvss j J T “i i P “W A i ”it
p, ___________________ r_ WNW—r .......... ’
—<i$‘l’\.w/ W: ”020‘“
/’ ”‘“""‘:::;. ___ _ g“
e: l ﬁghjeh V El [55‘ "if
Wigwam. Sis—"\fyz VA “1;?“ ,M
‘3 {A i 8 53‘“ «NS“W“ Mmmwmmw “MK '{E ‘ ‘3
bl C VW Mas6‘“ f 4:31” ff“ 5 f : 13M: 4 :2“ E
i, “ 3 \A“ A ' j 7 t i
Mar 5% 3.4an g :1 MW in r: W”. 0* J W J 4‘: v *
a , i2 lw ‘ w ‘ ‘“ O J ., g m
_ i . i 1 C 3:13? err—WW" LHWMK,“ v  , .. ‘
WWW W: Maw. y 9549“ i ‘ a 10 pts (iii) Draw a phase diagram for the surface—bound amphiphiles in the plane of f
and T. Label all phases, and indicate where they coexist. Your diagram should be
consistent with previous parts of this problem and with the following information: (a) Through a series of reversible changes in f and T, it is possible to switch
between the sparse state at (f0, To) to the dense state at (f0, To) without a
discontinuous change in a. The highest values of f and T accessed in this procedure are f1 and T1, respectively. (b) When the force is increased beyond f0 (still at ﬁxed T = To) to a value f2,
the area per molecule again decreases abruptly. The change in a for this transition
is signiﬁcant, but smaller than Aa for the sparse—to—dense transition. In the new,
ordered phase, surfacebound amphiphiles all slant in the same direction, as
sketched below. This transition is also accompanied by a release of heat into the
environment. 8 pts (iv) Small amounts of a second, distinct amphiphile are added to the surface. These
molecules mix with the original amphiphiles in the disordered dense phase, but are
excluded from the ordered phase. How should addition of this second component ‘2
change the force at which the ordered phase and dense disordered phase coexist at ck”: ;\
’ 4
temperature To? In other words, Will the ordering force at To be greater than, less sf” F “
than, or equal to f2? Explain your reasoning in terms of chemical potentials. 3‘ ”l” » 9}} avail,
Mamwml We ; "Pym 3 x 34: s“? 32 33'
i u; p f i , inl'L / W E
ii i
w E M¢ v.7. w» ,., .Hn “ iiﬁ Wain/w”; {was 3unlcl l0 u” WWW. 44%. $1323 WEE/xi ifvmilﬁ. l ‘i ’ 9 "‘ l ,V i
’ a . ~ ’ xi $3. Oi: ,
4’ 4163:!in r C‘\ Wow Qumgg/ngy {a 
’ \ _,~. \ ﬁr
fi/Qié/V‘ EC Li (3 “Lem L M» is» (Ci/g 'i/i/‘St dJJJKr i» gal/”w“ “J” 10 ...
View
Full Document
 Spring '09
 JAMESAMES
 Physical chemistry, pH

Click to edit the document details