Lecture 2. 1/07/15-1/09/15 Unversty of Washngton Department of Chemstry Chemstry 453 Wnter Quarter 2015 We are not talkng about truth. We are talkng about somethng that seems lke truth. The truth we want to exst Stephen Colbert explanng the meanng of truthness, 2005. A. Probablty and Entropy The Knetc Theory of Gases ala Boltzmann and Maxwell s a statstcal theory of the propertes of large scale gas phase systems. Can the statstcal approach be generalzed? The answer s yes. Everybody knows the meanngs of probablty and averages. How are these concepts so famlar to stock nvestors and gamblers applcable to large scale physcal systems? Large scale physcal systems have lots of parts. A lter of oxygen contans over 10 22 molecules. These molecules are constantly rearrangng themselves wthn ths volume as shown n fgure 2.. If the system s solated, t has a fxed energy E and fxed number of molecules N. Assume the volume V of the system does not change. Each snapshot of the system as t changes n tme s called a mcrostate. The number of mcrostates n an solated physcal system s a fundamental quantty that we wll show determnes the entropy. Fgure 2.1 An solated system of gas molecules wth fxed E, N, and V, changes of postons of the molecules n tme. Each snapshot s called a mcrostate. Macroscopc quanttes lke pressure and energy may be vewed as averages over these many mcrostates. But how do we take ths average for solated systems? An mportant prncple that governs performng such averages s called the Postulate of Equal A Pror Probabltes. Ths postulate states that n solated systems, all mcrostates are equally probable. If the number of mcrostates s W ( N, V, E ), then the probablty that each mcrostate occurs at any nstant s 1 P = W N, V, (2.1) E ( ) 7
In theory then, we would have to calculate the tme average usng (2.1) as the probablty of a mcrostate occurrng at any nstant. Such an average requres that the system be observed for a long enough tme that all mcrostates are observed. For large numbers of partcles the tme average can be an mpossble task. Josah Wllard Gbbs Jr. developed an alternatve averagng method based on ensembles. In the ensemble method, all possble mcrostates are enumerated and assembled for averagng purposes. Fgure 2.2: Left: J. Wllard Gbbs Jr. Yale mathematcal physcs professor who founded ndependent of Maxwell and Boltzmann the feld of statstcal mechancs. Gbbs verson s based on ensemble averagng. Rght: a mcrocanoncal ensemble as conceved by Gbbs. To keep N and V constant n ths mental construct, the wall between the mcrostates are rgd and mpermeable. To keep the energy constant the walls are adabatc and thus do not permt heat flow. An ensemble composed of solated systems each wth fxed E, V, and N and representng a unque mcrostate s called a mcrocanoncal ensemble. A second mportant postulate of statstcal mechancs s the Ergodc Postulate whch states that the tme average of a physcal property s equvalent to the ensemble average, f the members of the ensemble replcate the physcal state and surroundngs of the system. For an solated system that s ergodc the tme average and the mcrocanoncal ensemble average are equvalent. The Ergodc Postulate and the Postulate of Equal A Pror Probabltes are the two fundamental statements upon whch statstcal mechancs s bult. Entropy of Isolated System: It s well-known from Chemstry 452 that N A Vf Vf Sf S = Rln = kbln V V (2.2) N A or S = k lnv B For a sngle partcle, the number of mcrostates s proportonal to the volume V. N For N deal gas partcles the number of mcrostates W ( N, V, E) V. Substtutng ths nto equaton (2.2) we obtan Boltzmann s entropy equaton 8
( ) S = kb ln W N, V, E (2.3) The Boltzmann entropy equaton s the fundamental relatonshp between the thermodynamc propertes of large scale systems (.e. S) and the statstcal propertes of large scale systems (.e. W). The average of any physcal property assocated wth an solated ergodc system s the mcrocanoncal ensemble average: 1 X = PX = X W ( N, V, E) (2.4) Consequence of the Boltzmann Entropy Equaton: o Example 1: Suppose we have an deal gas confned to a sphercal contaner and separated from a second equal contaner by a stop cock. We open the stopcock. Consder two possble outcomes where the gas remans n one volume and n the other outcome the gas dstrbutes between the two volumes. Whch outcome s more physcally lkely? Fgure 2.3: The parttonng of deal gas molecules between two contaners of equal sze. Common experence shows the outcome wll be an equal parttonng o We know from common experence that the gas wll equally partton between the two contaner. How does equaton (2.3) predct ths? Let us smply the problem to sx molecules n two contaners where each contaner has sx stes that can be occuped. In Fgure 2.4, we enumerate all the ways of arrangng 6 molecules between 12 stes, wth sx stes n each contaner. There are 64 such mcrostates, as shown n Fgure 2.4. 9
Fgure 2.4: There are 64 ways (.e. mcrostates) for arrangng 6 partcles between two equal-szed contaners. The mcrostates where partcles are equally parttoned between the bns are most numerous and wll domnate any averaged property. The number of ways of arrangng N partcles between two bns such that N 1 partcles are n bn 1 and N 2 partcles are n bn 2 s gven by the bnomal coeffcent expresson: N! W = (2.5) N 1! N 2! In fgure (2.4) equaton 2.5 s appled to the parttonng of 6 partcles between two bns. There are 64 mcrostates for ths system n total. There are exactly two mcrostates whch correspond to all sx partcles n one bn or the other. Applyng equaton 2.5 to 6 partcles n bn 1 and zero partcles n bn 2: 6! W6,0 = W0,6 = = 1 (2.6) 6!0! In contrast there are many more mcrostates for equal parttonng of partcles between two bns: 10
6! W 3,3 = = 20 (2.7) 3!3! Equatons 2.5-2.7show that molecules wll equally partton between the two bns because there are many more mcrostates that correspond to that physcal result than for other results. Therefore, as the system fluctuates through ts many mcrostates, most of the tme the system wll be n mcrostates correspondng to equal parttonng of partcles between the two bns. Any physcal property produced by averagng over mcrostates wll be domnated by the mcrostates that reflect equal parttonng. B. Alternatve Form for the Entropy: The Gbbs Entropy Equaton We can generalze equaton 2.5 to any number of mcrostates and n dong so obtan an alternatve form for the statstcal entropy called the Gbbs Entropy Equaton. Assume we have N deal gas molecules where N 1 are n mcrostate 1, N 2 are n mcrostate 2, etc. Then W s the number of ways of arrangng N partcles among the mcrostates such that N 1 are n mcrostate 1, N 2 are n mcrostate 2, etc. W s gven by: N! N! W = = (2.8) N 1! N 2! N M! N! = 1, M Take the log of both sde and then use Strlng s Approxmaton: lnw = Nln N N N ln N N = 1, M ( ) N = N ln N ( Nln N) = Nln = 1, M = 1, M = 1, M N (2.9) We now substtute the Boltzmann entropy equaton (2.3) nto (2.6) N N N S = kb Nln = kbn ln = 1, M N = 1, M N N (2.10) S N N = ln ( Pln P) kbn = = 1, M N N = 1, M The Gbbs entropy equaton gves the entropy as a functon of mcro-state probabltes. However, ths equaton can be appled more broadly than shown here, as we wll see. Example 2: Back to the Two Bn System.. Assume an solated system composed of N dentcal partcles that can be dstrbuted between two bns such that: 11
. W ( N!, M N ) = (2.11) M!( N M )! The entropy s S = ln N! ln M! ln( N M)! kb S M M M M = ln 1 ln 1 Nk + B N N N N = fln f + 1 f ln 1 f ( ( ) ( )) (2.12) where we desgnate f as beng the fracton of molecules n the left bn. Note the entropy s maxmum at equlbrum when ds = 0 (2.13) df * Solvng 2.13 we fnd that f = 0.5. At equlbrum the entropy s maxmum, there s equal probabllty of fndng the bns populated, and W s also maxmum. The macrostate wth the largest number of mcrostates occurs at equlbrum. The varaton of the equaton 2.12 expresson for entropy wth f s shown n fgure 2.5. The maxmum entropy s S ( * * ( * ) ( * ln 1 ln 1 )) ln 1 = f f + f f = ln 2 0.693 Nk = = (2.14) B 2 Fgure 2.5: Plot of equaton 2.12 showng the entropy of a two bn l system as a functon of the fracton of N partcles that occupy the left bn. 12
C. Summary remarks: There are two postulates n statstcal mechancs:. Tme averages are hard to calculate. The Ergodc Postulate says tme averages are equvalent to ensemble averages.. The Postulate of Equal A Pror Probabltes says each mcrostate wthn the mcrocanoncal ensemble occurs wth equal probablty An solated system changes constantly n tme between ts many mcroscopc confguratons called mcrostates. Macroscopc physcal propertes lke thermodynamc state functons, pressure, etc. are tmes averages over these mcrostates A mcrocanoncal ensemble s a mental construct used to descrbe statstcally the thermodynamc propertes of an solated system. A mcrocanoncal ensemble s composed the very large number of mcrostates, each wth the same N, V, and E. Entropy has the statstcal forms: S = kb lnw...boltzmann s Entropy Equaton S Pln P kn =...Gbbs Entropy Equaton B 13