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An Introduction to Thermal Physics

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Unformatted text preview: § 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 firm/2 3M2 N anU 3N/2 _ = _—___ .2 _ l_. l k ln(N! hsN (3N/2)!(2mU) an +ln hz ) lnN. ln[(3N/2)] 27rmU 3’2 . 3N 3N 3N =Nan+Nln( h, ) —N1nN+N-_2_1I_1(..2_)+_2_ V' 21rmU 3/” 3N 3/2 5 V 47rmU 3#2 5 _N[lnfi+ln( hz ) 447) J7] ’N[ln(7v“(3Nh2) )+§]' In the second line I’ve used Stirling’s approximation twice, in the form of equation 2.16 which omits the merely “large” factor of v21rN. The final expression is the Sackur—Tetrode result, equation 2.49. 27rmUA N 27rmUA ( h, ))=Nln( h, )—2(NlnN—N) 27rmUA A 21rmU ' H mm, W M. E Q‘xgclem 2.3 . For argon at room temperature and atmospheric pressure, the volume per , mo Bl ' i l i ' a l l . —23 1%— : kg. 2 :4_14 X 10‘26 m3, while the energy per molecule is U 3 3 7v— : EkT = §(1.38 x 10—23 J/K)(300 K) = 6.21 x 10'21 J. The mass of an argon atom is 40 u or 6.64 x 10‘26 kg, so the argument of the logarithm. the Sackur—Tetrode equation is In}; V 47mm} 3/2 . _ 4n 6.64 x 10—26 k 6.21 x 10-21 J 3/2 :I-V—( :3th) = (4.14 x10 26 m3)( ( EX =1‘02X10fl 3(6.63 x 10-34 J s)2 The entropy of a mole of argon under these conditions is théefez S = R[1n(1.02 x 107) + g] = R[18.64] = 155 J/K. more mass has more momentum, resulting in a larger “hypersphere” 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 = Nl‘c-ln TE, where V,- and Vf are the initial and final volumes. But the heat input during this proc was computed in equation 1.31 as V! NkT V, Q: _w=+/ TdV—Nlen—‘Z. 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 -“ therefore AS is most definitely 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 fluids) 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 fiXed 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 Stirling’s approximation: ASmixing N k[NlnN—N——NAlnNA +NA —N31nNB = k[NlnN~—NAlnNA —NBlnNB]. Now substitute N A = (1—-:1:)N and NB 2 :er: ASmjxjng = k[NlnN — (1—x)Nln[(1—:I:)N] — :len[:vN]] = Nk[lnN ~— (1—x)ln(1-.—:c) — (1—1c)lnN -— :vlna: — :vlnN]. The lnN terms now cancel, leaving us with the same expression as in Problem 2.37, ASmixing = -Nk[:vln:v + (1—1v)ln(1—:v)] . Problem 2.41 . (Irreversible processes.) When you stir salt into a pot of soup, the sodium and chlorine ions can roam through— out the entire volume of the liquid. They can then have many more possible arrange— ments than when they are locked into crystals. More (arrangements means higher multiplicity and hence higher entropy. And as we all know, it’s 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 “yolk molecules” and “white molecules” can mix among each other. In addition, cook- ing the egg “denatures” the protein molecules, undoing their special folded patterns and stretching them out into long chains that can flop around randomly. (c) Humpty Dumpty’s 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 king’s horses and the king’s men just knew the second law of thermodynamics, they wouldn’t 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 definitely 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 it’s 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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08p340hw4 - 3 roblem 2.3 Starting from equation 2.40 for...

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