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3 / roblem 2.3 . Starting from equation 2.40 for the multiplicity, we have for the entropy
of a ' gas 38).
40 HN (231,525,345 S 1 VN ļ¬rm/2 3M2 N anU 3N/2
_ = _ā___ .2 _ l_. l
k ln(N! hsN (3N/2)!(2mU) an +ln hz ) lnN. ln[(3N/2)]
27rmU 3ā2 . 3N 3N 3N
=Nan+Nln( h, ) āN1nN+N-_2_1I_1(..2_)+_2_
V' 21rmU 3/ā 3N 3/2 5 V 47rmU 3#2 5
_N[lnļ¬+ln( hz ) 447) J7] āN[ln(7vā(3Nh2) )+Ā§]' In the second line Iāve used Stirlingās approximation twice, in the form of equation 2.16
which omits the merely ālargeā factor of v21rN. The ļ¬nal expression is the SackurāTetrode
result, equation 2.49. 27rmUA N 27rmUA
( h, ))=Nln( h, )ā2(NlnNāN) 27rmUA A 21rmU '
H mm, W M. E Qāxgclem 2.3 . For argon at room temperature and atmospheric pressure, the volume per , mo Bl ' i
l
i
' a
l
l . ā23
1%ā : kg. 2 :4_14 X 10ā26 m3, while the energy per molecule is U 3 3
7vā : EkT = Ā§(1.38 x 10ā23 J/K)(300 K) = 6.21 x 10'21 J. The mass of an argon atom is 40 u or 6.64 x 10ā26 kg, so the argument of the logarithm. the SackurāTetrode equation is In}; V 47mm} 3/2 . _ 4n 6.64 x 10ā26 k 6.21 x 10-21 J 3/2 :I-Vā( :3th) = (4.14 x10 26 m3)( ( EX =1ā02X10ļ¬ 3(6.63 x 10-34 J s)2
The entropy of a mole of argon under these conditions is thĆ©efez S = R[1n(1.02 x 107) + g] = R[18.64] = 155 J/K. more mass has more momentum, resulting in a larger āhypersphereā of allowed momentum states for the gas and hence a larger multiplicity. /Pfoblem 2.3/21. The increase in entropy during quasistatic isothermal expansion of
' meuted in equation 2.51 as
V! AS = Nlāc-ln TE, where V,- and Vf are the initial and ļ¬nal volumes. But the heat input during this proc
was computed in equation 1.31 as V! NkT V,
Q: _w=+/ TdVāNlenāāZ. Dividing this expression by T gives the preceding expression for AS, so indeed, AS = Q/
For the free expansion process, however, AS is still given by the same expressron but Q -ā
therefore AS is most deļ¬nitely not equal to .Q / T. ' * W the unmixed state, this system could have quite a bit of entropy due
.to m ecular energies and (for ļ¬uids) configurations. When we allow the system to mix,
assuming that the mixture is ideal, the only change is that molecules of different types
can now switch places with each other at random (with no inherent tendency to prefer
like or unlike neighbors). Therefore, to compute the mixing entropy, we can ignore the
initial entropy and pretend that the molecules are initially frozen in place. Upon mixing, molecules randomly switch places with each other but still occupy the same collection of
N ļ¬Xed sites. The increase in multiplicity due to mixing, therefore, is the number of ways of assigning the two species of molecules to the, N sites, that is,'."th_e number of ways; of
choosing N A of the sites to be occupied by molecules of type A: - N
9mixing = - The entropy of mixing is then In times the natural log of this expression: N N!
ASmixing = kln = k . Assuming that both N A and N B are large, we can approximate the factorials using Stirlingās
approximation: ASmixing N k[NlnNāNāāNAlnNA +NA āN31nNB = k[NlnN~āNAlnNA āNBlnNB].
Now substitute N A = (1ā-:1:)N and NB 2 :er: ASmjxjng = k[NlnN ā (1āx)Nln[(1ā:I:)N] ā :len[:vN]]
= Nk[lnN ~ā (1āx)ln(1-.ā:c) ā (1ā1c)lnN -ā :vlna: ā :vlnN]. The lnN terms now cancel, leaving us with the same expression as in Problem 2.37, ASmixing = -Nk[:vln:v + (1ā1v)ln(1ā:v)] . Problem 2.41 . (Irreversible processes.) When you stir salt into a pot of soup, the sodium and chlorine ions can roam throughā
out the entire volume of the liquid. They can then have many more possible arrangeā
ments than when they are locked into crystals. More (arrangements means higher
multiplicity and hence higher entropy. And as we all know, itās not at all easy to
reverse the process and get the salt out of the soup. Scrambling an egg mixes the yolk with the white, so that creates mixing entropy as the
āyolk moleculesā and āwhite moleculesā can mix among each other. In addition, cook-
ing the egg ādenaturesā the protein molecules, undoing their special folded patterns
and stretching them out into long chains that can flop around randomly. (c) Humpty Dumptyās fall itself is reversible (to a good approximation, neglecting air
resistance), but when he lands and breaks into many pieces, his entropy suddenly
increases because there are many more ways for him to be broken than whole. If the
kingās horses and the kingās men just knew the second law of thermodynamics, they
wouldnāt have wasted their time trying to put him back together again! ((1) There are many more ways for the sand to be scattered about than for it to be sculpted
into a sand castle, so the action of the wave most deļ¬nitely increases the multiplicity and entropy of the sand. (e) You can cut the tree in many places, at many angles, and it can fall in many directions,
so there are many more ways for it to be cut down than for it to remain standing.
Hence its entropy has increased. And of course, we all know that itās pretty much
impossible to undo the cutting. (f) When you burn gasoline, not only do you convert a smaller number of relatively large
hydrocarbon molecules into a larger number of relatively small exhaust gas mole-
cules, but you also release a great deal of thermal energy (converted from chemical
energy) into' the environment. This energy can arrange itself in many ways among the
surrounding atoms, so the entropy of the environment increases a great deal as this
thermal energy spreads farther and farther. ...

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