Lecture 6. Entropy of an Ideal Gas (Ch. 3)

Transcription

1 Lecture 6. Entropy o an Ideal Gas (Ch. oday we wll acheve an portant goal: we ll derve the equaton(s o state or an deal gas ro the prncples o statstcal echancs. We wll ollow the path outlned n the prevous lecture: Fnd Ω (... the ost challengng step (... Ω (... (... olve or (... o ar we have treated quantu systes whose states n the conguraton (phase space ay be enuerated. When dealng wth classcal systes wth translatonal degrees o reedo we need to learn how to calculate the ultplcty.

2 Multplcty or a ngle partcle - s ore coplcated than that or an Ensten sold because t depends on three rather than two acro paraeters (e.g.. Exaple: partcle n a onedensonal box -L L he total nuber o ways o llng up the cells n phase space s the product o the nuber o ways the space cells can be lled tes the nuber o ways the oentu cells can be lled Ω Ω space Ω p p x he total nuber o crostates: L px x p x L x -L L x p x -p x x Q.M. he nuber o crostates: L px x p x L p h x Quantu echancs (the uncertanty prncple helps us to nuerate all derent states n the conguraton (phase space: x p x h

3 Multplcty o a Monatoc Ideal Gas (spled For a olecule n a three-densonal box: the state o the olecule s a pont n the 6D space - ts poston (xyz and ts oentu (p x p y p z. he nuber o space crostates s: For olecules: x Ωspace here s soe oentu dstrbuton o olecules n an deal gas (Maxwell wth a long tal that goes all the way up to p ( / ( s the total energy o the gas. However the oentu vector o an average olecule s conned wthn a sphere o radus p ~ (/ / (/ s the average energy per olecule. hus or a sngle average olecule: he total nuber o crostates or olecules: However we have over-counted the ultplcty because we have assued that the atos are dstngushable. For ndstngushable quantu partcles the result should be dvded by! (the nuber o ways o arrangng dentcal atos n a gven set o boxes : Ω space Ω x y z p x y x p 4 π p p p p p Ω ΩspaceΩ p x p h x n p Ωndstngushable p! h z p

4 p z More ccurate Calculaton o Ω p p x p y Moentu constrants: partcle - partcles - px py pz p x p y pz px p y pz he accessble oentu volue or partcles the area o a -densonal hypersphere p / "area" π r! π Ω p! ( /! h / π / ( / r! [( /! π / ] 4π r he reason why atters: or a gven a olecule wth a larger ass has a larger oentu thus a larger volue accessble n the oentu space. Monatoc deal gas: ( degrees o reedo ( π / / e Ωndstngushable ( p h / ( Ω - the total # o quadratc degrees o reedo

5 Entropy o an Ideal Gas (onatoc (datoc 6 (polyatoc Monatoc deal gas: 4 ( / h π ( 4 / h ϕ π In general or a gas o polyatoc olecules: the acur- etrode equaton ( ( ϕ ( p Ω! /! / h π ( p h 4 ( / π an average volue per olecule an average energy per olecule

6 roble wo cylnders ( lter each are connected by a valve. In one o the cylnders Hydrogen (H at a C n another one Helu (He at a C. Fnd the entropy change ater xng and equlbratng. H : total For each gas: he teperature ater xng: H ( total H He : ( ( ( He ( He total.67 J/K

7 Entropy o Mxng Consder two derent deal gases ( ept n two separate volues ( at the sae teperature. o calculate the ncrease o entropy n the xng process we can treat each gas as a separate syste. In the xng process / reans the sae ( wll be the sae ater xng. he paraeter that changes s /: 4 ( / h π he total entropy o the syste s greater ater xng thus xng s rreversble. total / / / / total

8 Gbbs aradox I two xng gases are o the sae nd (ndstngushable olecules: ( / total ( h /!! Ω π h 4 ( / π Quantu-echancal ndstngushablty s portant! (even though ths equaton apples only n the low densty lt whch s classcal n the sense that the dstncton between erons and bosons dsappear. total because / and / avalable or each olecule rean the sae ater xng. total - apples only two gases are derent!

9 roble wo dentcal perect gases wth the sae pressure and the sae nuber o partcles but wth derent teperatures and are conned n two vessels o volue and whch are then connected. nd the change n entropy ater the syste has reached equlbru. h h 4 4 ( / / π π ( ( / α ( ( ( ( ( ( ( ( ( ( / / α α α at as t should be (Gbbs paradox ( ( / / α α - prove t!

10 n Ideal Gas: ro ( - to ( Ideal gas: ( degrees o reedo Ω / ( ( ( ϕ ( ( - the energy equaton o state - n agreeent wth the equpartton theore the total energy should be ½ tes the nuber o degrees o reedo. he heat capacty or a onatoc deal gas: C

11 artal Dervatves o the Entropy We have been consderng the entropy changes n the processes where two nteractng systes exchanged the theral energy but the volue and the nuber o partcles n these systes were xed. In general however we need ore than just one paraeter to specy a acrostate e.g or an deal gas oday we wll explore what happens we let the other two paraeters ( and vary and analyze the physcal eanng o the other two partal dervatves o the entropy: We are alar wth the physcal eanng only one partal dervatve o entropy: ( ( Ω When all acroscopc quanttes are allowed to vary: d d d d

12 herodynac Identty or d( ( onotonc as a uncton o ( quadratc degrees o reedo! ay be nverted to gve ( d d d d copare wth μ pressure checal potental hs holds or quas-statc processes ( μ are well-dene throughout the syste. d d d d μ μ shows how uch the syste s energy changes when one partcle s added to the syste at xed and. he checal potental unts J. - the so-called therodynac dentty or

13 he Exact Derental o ( d d d d μ he coecents ay be dented as: ( d d d d μ gan ths holds or quas-statc processes ( and are well dened. d d d d μ ype o nteracton Exchanged quantty Governng varable Forula theral energy teperature echancal volue pressure dusve partcles checal potental μ connecton between therodynacs and statstcal echancs

14 Mechancal Equlbru and ressure Let s x and but allow to vary (the ebrane s nsulatng pereable or gas olecules but ts poston s not xed. Followng the sae logc spontaneous exchange o volue between sub-systes wll drve the syste towards echancal equlbru (the ebrane at rest. he equlbru acropartton should have the largest (by ar ultplcty Ω ( and entropy (. eq he stat. phys. denton o pressure: - the sub-syste wth a saller volue-per-olecule (larger at the sae wll have a larger / t wll expand at the expense o the other sub-syste. In echancal equlbru: ( - the volue-per-olecule should be the sae or both sub-systes or s the sae ust be the sae on both sdes o the ebrane. /

15 he ressure Equaton o tate or an Ideal Gas Ideal gas: ( degrees o reedo he energy equaton o state ( : he pressure equaton o state ( : - we have nally derved the equaton o state o an deal gas ro rst prncples! ( ( ϕ

16 Quas-tatc rocesses d d d d δ Q δw (quas-statc processes wth xed hus or quas-statc processes : δq d (all processes δ Q d d Coent on tate Functons : δ Q - s an exact derental ( s a state uncton. hus the actor / converts δq nto an exact derental or quas-statc processes. Q Quasstatc adabatc (δq processes: d sentropc processes he quas-statc adabatc process wth an deal gas : const const - we ve derved these equatons ro the st Law and R On the other hand ro the acur-etrode equaton or an sentropc process : / const const

17 roble: (all the processes are quas-statc (a Calculate the entropy ncrease o an deal gas n an sotheral process. (b Calculate the entropy ncrease o an deal gas n an sochorc process. You should be able to do ths usng (a acur-etrode eq. and (b δq d d d d Ω / ( ( δq d δ Q d d d / [ g( ] const const Let s very that we get the sae result wth approaches a and b (e.g. or const: nce δq δw d δq (r..4

18 roble: body o ass M wth heat capacty (per unt ass C ntally at teperature s brought nto theral contact wth a heat bath at teperature.. (a how that << the ncrease n the entropy o the entre syste (bodyheat bath when equlbru s reached s proportonal to (. (b Fnd the body s a bactera o ass - g wth C4 J/(g K K.K. (c What s the probablty o ndng the bactera at ts ntal or t - s over the lete o the nverse (~ 8 s. (a body δq Cd C < heat bath δq Cd > C (b total body total heat bath C C 4 C J / K. α α C ( α α... > J / K

19 roble (cont. (b Ω or the (non-equlbru state wth bactera.k s greater than Ω n the equlbru state wth bactera K by a actor o Ω Ω J / K total 4 exp exp.8 / e J K 6 he nuber o ps trals over the lete o the nverse: 8 hus the probablty o the event happenng n trals: 6 (# events( probablty o occurrenceo an event

20 r... non-quasstatc copresson. cylnder wth ar ( - K a s copressed (very ast non-quasstatc by a pston (. F x -. Calculate δw δq and. W Q δ δ holds or all processes energy conservaton quasstatc and are welldened or any nteredate state quasstatc adabatc sentropc non-quasstatc adabatc [ ] J a ( x x x x d const d W δ [ ] d W δ J a he non-quasstatc process results n a hgher and a greater entropy o the nal state. const along the sentropc lne * δq or both Cauton: or non-quasstatc adabatc processes ght be non-zero!!! n exaple o a non-quasstatc adabatc process

21 Drect approach: adabatc quasstatc sentropc δ Q ( ( / const δ W / adabatc non-quasstatc - a K - J/K W J a - J

22 δ Q J o calculate we can consder any quasstatc process that would brng the gas nto the nal state ( s a state uncton. For exaple along the red lne that concdes wth the adabat and then shoots straght up. Let s neglect sall varatons o along ths path ( << so t won t be a bg stae to assue const: δ Q J J (adabata J/ K K Q J otal gan o entropy: he entropy s created because t s an rreversble non-quasstatc copresson. For any quas-statc path ro to we ust have the sae. Let s tae another path along the sother and then straght up: sother: ( d - - x a x K straght up : Q J J / K K J / K J / K J / K d J / K

23 he nverse process sudden expanson o an deal gas ( also generates entropy (adabatc but not quasstatc. ether heat nor wor s transerred: δw δq (we assue the whole process occurs rapdly enough so that no heat lows n through the walls. ecause s unchanged o the deal gas s unchanged. he nal state s dentcal wth the state that results ro a reversble sotheral expanson wth the gas n theral equlbru wth a reservor. he wor done on the gas n the reversble expanson ro volue to : δw rev he wor done on the gas s negatve the gas does postve wor on the pston n an aount equal to the heat transer nto the syste Q rev W rev Qrev Wrev x > x J/K hus by gong we wll ncrease the gas entropy by J/K