THERMODYNAMICS. Temperature

Transcription

1 HERMODYNMICS hermodynamcs s the henomenologcal scence whch descrbes the behavor of macroscoc objects n terms of a small number of macroscoc arameters. s an examle, to descrbe a gas n terms of volume ressure temerature, number of artcles, and ther tye. he nterestng asect of ths s that s ossble at all. fter all, a macroscoc object contans ~10 5 artcles or more. Each would aear to need at least 6 numbers to descrbe t 3 oston and 3 momentum or velocty. So why are only a few (~ 10) needed n ractce? he subject s aroached n two very dfferent ways. he frst s the henomenologcal method stated above. he second s from the mcroscoc laws of Newton or quantum mechancs (f needed). hs second aroach s called statstcal mechancs. We wll use both, alternatng between them. emerature he concets of volume and ressure are already famlar, but temerature (n the scentfc sense) s not. We begn wth an objectve defnton of temerature we show how to measure t. Suose we have macroscoc objects, and B. Now suose we lace them n thermal contact so that t s ossble for energy to flow between them. In ractce, t s nearly mossble to revent such flow by radaton f no other way. If energy tends to flow from to B we say that the temerature of s hgher than that of B. If energy goes from B then B If there s no tendency for energy to flow we say that B Now add a thrd object C. hen the 0 th Law of hermodynamcs states that f B and

2 B C hen C hs seems obvous, but t contans a subtle assumton that only temerature affects the drecton of heat flow. hs turns out to be true exermentally but t dd not have to be that way. hus t s generally stated as an ndeendent law. emerature Scale Havng thus defned temerature t s a smle matter to obtan a scale and a method of assgnng a numercal temerature to an object. We observe that a very large number of hyscal roertes deend on a temerature (length, volume, color, resstance, etc.). We need merely ck a roerty and two fxed onts and we have a scale and a thermometer. For examle, a very convenent choce s an deal gas whch we wll dscuss n a moment. Frst we consder the famlar mercury n a tube thermometer. Here the roerty s the heght of the column of mercury. he two fnal onts are the meltng and bolng onts of water (under certan condtons). We lace the tube of mercury n the ce water mxture and mark ts heght. We then lace t n the bolng water and mark the new heght. We then dvde the sace between the marks nto equal ntervals. We now have a thermometer. If we take meltng ce to be 0 and bolng water to be 100, we have the famlar centgrade scale. If we take them to be 3 ad 1, we have the even more famlar Fahrenhet scale. If we take them to be 73 and 373, we have the Kelvn scale. Note that nether Centgrade nor Fahrenhet s ntrnscally better than the other. Kelvn on the other hand s fundamentally sueror as we wll see. Ideal Gas o get an dea of the fundamental mcroscoc meanng of temerature we consder an deal gas, defned to be a gas n whch there s no nteracton between the molecules excet at the nstant of collson. hs s never comletely accurate. It becomes arbtrarly close to realty as the densty of the gas 0. (Infntely far aart, they can t see each other.) Consder a cubcal box of sde L contanng N molecules. By the sotroy of sace we wll have equal numbers movng n each drecton. More accurately, the comonents of velocty n each drecton wll be the same. Furthermore, the molecules wll have some average magntude of velocty v. hen n one second each molecule movng n the x drecton wll make

3 v L collsons wth the wall at x = 0. When t does so ts momentum wll change by mv (mv n mv out). hus the total change n momentum n the x-drecton/sec s N v mv 3 L However, by Newton s nd law ths must be the force exerted on the gas by the wall. Hence F Nmv N KE 3 av 3L 3 V Where V = L 3 s the volume of the gas. By Newton s law however, ths s also the F/ exerted on the wall by the gas or the ressure. hus We now defne the deal gas temerature to be PV NKEav 3 KEav 3 k where k = Boltzman s constant = J/ K. Note that the lowest ossble temerature s the state of lowest ossble KE. In classcal hyscs ths would be KE = 0. Clearly ths works snce f KE = 0 so does dvm/dt and hence, P. he Kelvn scale smly starts wth 0 at the lowest ossble temerature whch haens to be -73 C. hs s why the Kelvn scale s ntrnscally sueror to the other two. hus we have an dea about the fundamental meanng of temerature at the mcroscoc level t s a measure of the random KE of the molecules makng u the materal. Next we wll turn to a statstcal mechancs descrton of an deal gas where we do not assume an average velocty at the outset. Statstcal Mechancs We need a way to characterze the mcrostate of a system. Begn by consderng a monatomc deal gas. hen each molecule can be located by gvng ts oston and momentum. hs wll requre 6 numbers 3 oston and 3 momentum. We thus magne a 6-dmenstonal sace. We slt ths sace nto nfntesmal boxes of volume

4 We now defne a dstrbuton functon, f, by dxdydzdxdyd z # of molecules n box x, y,z,,, dx d x y z z macroscoc state s then gven by secfyng the occuaton of each box. For the case of a monatomc deal gas or any other for that matter we must recognze three dstnct ossbltes. Case 1 Classcal In classcal mechancs each artcle s dstngushable from all the others we can kee track of whch s whch. Case QM Ferm Drac In quantum mechancs dentcal artcles are not dstngushable we can t kee track of whch s whch. here turn out to be two dfferent classes of artcles and hence two dfferent statstcs. One s the Ferm-Drac artcles and the other Bose-Ensten. Case 3 QM Bose-Ensten We wll, for now, only consder the classcal case. We now make the fundamental assumton of statstcal mechancs that all ossble mcrostates are equally lkely and therefore that the most robable macrostate wll be that one whch arses from the largest number of mcrostates. Hence we need to know how many mcrostates corresond to the same macrostate. macrostate s secfed by gvng the number of artcles n each nfntesmal box. Let n be the number of artcles n the th box, and N be the total number of artcles. How many ways can we have n artcles n the 1 st box? We care whch artcles they are, but not he order n whch they were chosen. he 1 st artcle can be chosen n N way, the second n N-1 ways, etc. Hence there are N N 1... N n 1 ways to do ths. However, there are n 1! orders n whch they can have been chosen. Hence the number of dstnct mcrostates s N N 1... N n1 n! 1

5 Now take the next box. he number of ways s Hence the total number of ways s N n... Nn n n! N N1 1 N! n!n 1! n! n! hus the robablty of the macrostate secfed by the n s roortonal to or N! n! P q N! n! We now assume that all the n and N >> 1 (smly choose large enough boxes). hs wll not work for quantum statstcs, but s fne here. hen t s easest to use lnp nstead of P. hus We now use Strng s roxmsaton hus np nq nn! nn! nn! N nn N np nq N n N N n n n n nq N n N n n n 1 Next we must mose constrants on the system the total number of artcles s fxed and the total energy fxed. In other words, we are dealng wth an solated system. here are other ossbltes, but ths wll do for now. Here N n E n

6 where s the energy of artcles n the th box. We then fnd the macrostate as the one for whch lnp s greatest consstent wth the constrants. n np0nnn n nn 1n nnn n snce n 0and We use the method of Lagrange Multlers dd these equatons together to get n, n 0 n 0, n 0 nn n 0 nn n 0, Snce ths must be true for any choce of n we must have nn, n e e, subject to the condtons, e e N, e, e E In classcal mechancs we relace the sum by an ntegral snce the boxes can be made nfntesmally small (ths s not true n QM systems). hus N e dxdz e, x1z

7 or e N, xz dxdz e hus far we have not used the fact that we have a monatomc deal gas. For that case the only energy s knetc hus e 3, x y z m N x y z m m m x y z dxdydz d e d e d e N V e md he Gaussan ntegral n [ ] can be easly evaluated as follows: Hence x y e dx e dx e dy e dxdy 1/ x x y / r 1/ 1/ 1/ r r rdrde re dr e 0 1/

8 N e 3/ m V o determne β we calculate the ressure ths system would exert on a wall of the contaner. We do t as before, excet now the artcles n box have the velocty /m rather than the average velocty. hus the ressure exerted by the artcles n box s n Now we must add ths u for all the boxes. y and z can be anythng, but x must be > 0 (else the artcles won t ht the wall. Further, the box must be wthn x /m of the wall or the artcles can t reach the wall n a second. Hence 0 y z x x0 y z x m x0 x x m x x x m x y z P e e dxd N m 3/ m V x m d e dx x N 1 m N x d 1/ x e d x e m d m 1/ m m V d 0 mv m 0 1/ 1/ N d N 1 N m N 1/ 1/ 3/ m m V V mv d m m mv m But we know that the deal gas temerature s gven by Hence PV Nk Nk N 1 V V k z

9 hus we arrve at the famous Boltzmann Dstrbuton. f e k wth k e dx d N z Degrees of Freedom Next we consder what haens when we do not have a monatomc deal gas.