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Homework 7 Solutions Philip Powell October 16, sees 1. Prohlern 1 {KSEH 5.2} Here we consider the Earth‘s atmosphere, which we take to be eoropoeed of molecules of mass M. and
to be in thermal equilibrium at a temperature T = rfitg. First, we recall that the chemical potential of an ideal gas [without a gravitational potential} is [K351i 5.12a} ii = Tln [1} The deﬁnition of chemical potential is p = {ﬁFfﬂN}T__V, and the effect of the gravitational ﬁeld is
to decrease the energyr per molecule by an amount. Gmer {recall that the gravitational potential is
negative}, where m in the mass of the Earth. Thus, in the prmenee of a gravitational ﬁeld the chemical potential beoornoa nErj) _ GmM m 1i[r}=1'ln(nq T We can express the product Gm in terms of gravitational acceleration at the earth’s surface {g} by
noting: Gm. g 2 EE => Gm = QR: l3}
'I‘hua1 we have
2
to] = Tia — M33 (4}
Now, if the atmuapherc is in oqujl1'.l:n1'iu.rrlI the chemical. potential will he the same at all i", and will
equal a constant on:
111:?) JHQRE ___
Til] ( “Q j r — tan [5}
We can evaluate it; by setting 1" = R, the Earth’s radius1 so that we obtain
2
via a M33 = m (Tl—ml) — Mali {a}
“Q 1” no Solving this equation for nEr] gives _ m+ﬂafi
r r, [1r], 1“er = willie The total number of moleculm in the atmosphrne may he obtained by integrating the density over the
atmosphere’s vohnne: _ mT'nr W2
an}; a {a {a} l Sutetituting our expression for nﬁr} gives ‘2 W
N = Mamie—ﬁg j dr raeﬂLﬂ— {9}
R We can see that this!r integral divergm by melting the substitution u = M933!” so that we have MIR N=41m[R}Ie' “5“ (Mg‘qufa ' an (i) {10] T In the limit a —r [i the exponential approaehee l, ao the integral diverges thanlw to the a". . Problem 2 [KSaK 5.3} Here we eonaider a eolnmn of atoms of mam M at temperature T in a uniform
gravitational ﬁeld 9. As in the last problem: the ehenn'ral potential is guru by ,1: 2m +ng an Note that in this problem we nee expression U = ng for the gravitational potential, mauring the
potential enemas,r from ground level {and warming the gravitetirmal ﬁelrl is constant This stands in
eontraet to the last problem, where we chose the zero of potential energr to be inﬁnity. In equilibrium
the eheraieal potential must be constant throughout the eolumn. Thea, we may eat this expression
equal toita veloeaty=ﬂand write Tin + my = T In [12]
Solving for “(til we obtain
nit} = ﬂue—£1” {131' Now that we have the height diatrihution functionr we can empress the average height as _ it” dy wit!)
{it} — m [14} The numerator can be evaluated as follows: m: an
._ M
f demo} = 1mu day5 "i oE—g
o o T
a m
2 “WE dye—“i If I a
sale0
.r‘_"\.
QII—
Nu—a’ = 0—2 {15}
hileanwhihe1 the denominator is simply
do :Icl
f dentin = ﬁe] drew”
a e
_ ﬂ .
— I, [15)
Thus, we ﬁnd that the average height '15
1 'r
{a} — E = E {1?} Thus, the thermal average potential energy per atom is to = May} =— no The thermal average of the total energy [kinetic + potential} per atom is therefore 37' 57
" =_ =_u 1
{hi , +r 2 (a where the average kinetic energy per particle follows from the equipartition theorem. The heat capacity per atom is thus
_ 3E 05? => o=g gee] . Problem 3 {KS5K 5.8]
{3} Consider a model tor carbon monoxide poisoning in which we have N ﬁxed Heme sites, each of which can either be empty, oeenpied by an 0; molecule, or occupied by a CO molecule. The energy of
these states are taken to be l], 5,4 and s}; respectively. We ﬁrst ignore the CD, and consider a system only with 03. In this case the grand partition ﬂinetion is Z = Zea—err HEN
1 +eiitogE‘slr’f = 1 + nopeWT {21)
The probability of a site being eecupied by an ()3 molecule is therefore ., closet—"IT
Flog} _ 1+A[02]eEAE? If we tales MO?) = lﬂ’a, T = 3?“, and 13(02} = .9 we ﬁnd a, = —e.sr c‘y' [232! (b) Now we allow CC! to be present as well so that the grand partition function beeomes
z = l + noseW7 + nookE's” (24] [223 The probability of a site now being oocupied by so {)2 molecule is )LEDZ leg—=sz Ploil = 1 + Mﬂgle—“l’ + stoopsee (25] Using Alﬁ'ﬂ} = 10—7 and setting this probability equal to 10%, solving for 53 yields 83 = —e.ss or“ (as) . Problem 4 {KSsK 5.14) Consider a hemoglobin molecule, which is capable of binding up to four
0;; molecules. We denote the energy of a bound ()2, relative to free 02, as e and write the absolute
activity of the free oxygen molecule as A = el‘f’. The grand partition function of the hemoglobin is a = 1 + any” + eerie” + axle—3W + X‘s—4"” (27} where the coeﬂicionts {l,d,ﬁ,d,l} give the mnnber of ways which a given number of sites can be ocnlpierl.
The probability of only one D: molecule being adsorbed on a hemoglobin molecule is therefore rile—E” = its} Pl (1.4 Similarly, tho probability of mostly four polygon molecules are adsorbed is {2‘9} These probobilitics are plotted as a function of A on the next page1 with the “typical” value E m Ill EV.
AS we would expect, as A de oc {or ru —r on, which mmaponds to a large density of 02 molecules] the probability of four {)2 molecules being adsorbed approoohm 1. [LB _ as 11.4 112 ...I. 5. Probiem E (KﬂaK 6.1] Here wo studs.r the Fermi—Dirac distribution:
1 He} = m [an]
Diﬂ'ETEntiﬁting! we ﬁnd
{if 1 ate—xsirr
E = 7m {31}
Evaluating this expression at a" = ,u wo obtain
df 1
Te ‘32] 6. Problem 6 {K351i 5.2} Looking again at the Formi—Dimc distribution and writing E 2 p + 6 we can write 1
to + a: = W + 1 (as:
Letting :5 —r *6 we 3150 have 1
HF — 5] = m [34.]
Adding those two expressions gives
1 l
:5  = .—
fﬂu+ i+ftu L5} di_}.;l,,+l+ﬂ_“f+1
= {ta—WT +1]+{3“T + 1]
(Eif'r + 11(3—53'1" + _ 2"” + 3‘15” + 2
_ Ear“? + E—ﬁy’r + 2
= 1 {353
Thus, we can write
f{#+ﬂ'II=Iﬂ.ur5} {35] ...
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 Fall '08
 Flynn
 Physics, Atom, Energy, Kinetic Energy, Potential Energy, grand partition function, Gravitational potential

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