Introduction to Statistical Methods

Transcription

1 Introducton to Statstcal Methods Physcs 4362, Lecture #3 hermodynamcs Classcal Statstcal Knetc heory Classcal hermodynamcs Macroscopc approach General propertes of the system Macroscopc varables 1

2 hermodynamc State emperature hermal Equlbrum 2

3 An Illustraton of the Zeroth Law of hermodynamcs emperature Scales hermodynamc Process 3

4 Heat ransfer Insulatng (adabatc) wall hermally conductng (dathermc) wall Work 4

5 Work W Fdx Work done by gas n a cylnder W pdv Work dfferental of work s nexact dfferental s not functon of state depends on the process 5

6 Work done by gas n the cylnder Work done by a gas n the contaner W Fdx pdv Work done by a gas n the contaner over a cycle 6

7 Work done n adabatc process does not depend on the choce of the path Internal Energy U 2 U1 Wad Fdx ad he Frst Law of hermodynamcs du Q W U U Q W 2 1 7

8 he Frst Law of hermodynamcs When a system changes from an ntal state 1 to a fnal state 2, the sum of work W and the heat Q, whch t receves from surroundngs s determned by the states 1 and 2, t does not depend n the ntermedate process. hermodynamc Process hermodynamc Process Reversble Irreversble 8

9 Reversble process If the system under consderaton changes from orgnal state 1 to fnal state 2 and ts envronment changes from state a to state b, then n some way t s possble to return the system from 2 to 1 and n the same tme return the envronment from b to a, the process (1,a) to (2,b) s sad to be reversble. Heat Engne produces useful work works through a cycle exchanges heat wth envronment Carnot Cycle for Ideal Gas 9

10 he Second Law of hermodynamcs Expermental Evdence of the Second Law of hermodynamcs Caratheodory s prncple: For a gven thermodynamc state of thermally unform system, there exsts another state whch s arbtrarly close to t but can not be reached from t by an adabatc change. 10

11 heorem of Caratheodory If dfferental form M x, y,... dx N x, y,... dy... has a property that n the space of ts varables every arbtrary neghborhood of pont P contans other ponts whch are naccessble from P along a path correspondng to the soluton of ts dfferental equaton, then an ntegratng denomnator for the expresson exsts. Clausus prncple A process whch nvolves no change other than the transfer of heat from a hotter to a cooler body s rreversble, or t s mpossble for heat to transfer spontaneously from a colder to hotter body wthout causng other changes. homson s (Kelvn s) prncple: A process n whch work s transformed nto heat wthout any other changes, s rreversble; or, t s mpossble to convert all the heat taken from a body of unform temperature nto work wthout causng other changes. 11

12 Prncple of the mpossblty of the a perpetuum moble of the second knd It s mpossble to devse an engne operatng n a cycle whch does work by takng heat from a sngle heat reservor wthout producng any other change. General Carnot Cycle wo sothermal and two adabatc processes Effcency W Q Q Q 1 Q Q Q Carnot s Prncple he effcency of a reversble Carnot cycle operatng between heat reservors R1 and R2 s unquely determned by the temperatures of the heart reservors and the effcency of any rreversble Carnot cycle operatng between the same heat reservors s less that the effcency of reversble Carnot engne

13 Reversble Carnot Cycle Q Q Q K Q, 3 Clausus s nequalty for arbtrary cycle When a system performs a cycle whle n contact wth envronment and absorbs heat from the envronment at temperature, then the followng holds Q 0 Entropy S 1 reversble Q ds Q, 13

14 Second Law of hermodynamcs 2 1 L Q S, Q ds hrd Law of hermodynamcs If one tres to reduce the temperature to absolute zero by repeatng the seres of operatons, each successve operaton yelds a smaller change of temperature and t appears that =0 wll never be reached. Nernst-Smon heorem S 0, 0 14

15 Infntesmal Reversble Process n the closed system du Q W, du ds pdv Infntesmal Reversble Process n the open mult-component system Chemcal potental U N S, V, N j du ds pdv dn Other hermodynamc Functons Enthalpy H U pv Helmholtz free energy Gbbs free energy Grand potental F U S G U pv S F N 15

16 Dfferentals of thermodynamc potentals dh ds Vdp dn, df Sd pdv dn, dg Sd Vdp dn d Sd pdv N d, Crtera for Equlbrum Q ds, du pdv ds, du pdv ds 0 Isolated System de=0, dv=0, dn=0 ds 0 S has ts maxmum at equlbrum 16

17 he closed Isothermal system d=0, dn=0, dv=0 df 0, F has ts mnmum at equlbrum he closed Isobarc system d=0, dn=0, dp=0 dg 0, G has ts mnmum at equlbrum he open Isothermal system d=0, d =0, dv=0 d 0, pv has ts mnmum at equlbrum 17

18 Frst Partal dfferental coeffcents of thermodynamc potentals Measurable Propertes of the system he coeffcent of volume thermal expanson 1 V Compressblty V X 1 V V p X Heat Capacty at constant volume C at constant pressure V C Q U S d p V V V Q H S d p p p 18

19 19 he Cross-Dfferentaton Identty 2 2 W W W W x y y x x y y x 1 V p U p V 1 p V H V p Maxwell s Relatons,,,. S V p S V p S V p S p V S p V S V p

20 hermodynamc system hermodynamc state Statstcal hermodynamcs 20

21 Ensemble Random Walk Problem Probablty of takng N steps wth n1 steps to the rght and n2=n-n1 to the left N! WN n1 p q n! n! n1 n

22 N! PN m p p N m N m!! 2 2 Nm Nm he mean value of dscrete varable u M 1 M 1 P u P u u M f u P u f u 1 22

23 Moments of the dstrbuton Random walk problem calculatons Probablty dstrbuton for large N 23

24 Gaussan Dstrbuton n W n 1 1 Npq W n n1 n1 2 exp, 2 2n1 12 n1 Np 2 exp 2Npq 2 Gaussan Dstrbuton 12 2 exp Npq P m m N p q 2 8Npq Gaussan Dstrbuton 2 1 x P xdx exp dx, p q Nl, 2 Npql 24