Classical Thermodynamics I

8/7/2019 Classical Thermodynamics I

http://slidepdf.com/reader/full/classical-thermodynamics-i 1/21

. . • . ~

Qual Review, Meeting 1:

Classical Thermo'dynamics

1. n moles of an ideal monatomic gas, initially at volume v a and

temperature To, are enclosed in a cylinder of cross sectional area A

by a tight fitting, frictionless plunger of mass "M, as shown. The

system is in a gravitational field g. A piece of hot metal at tem-

perature T 1 > To with heat capacity C is gently pla<;ed in sweet

thermal contact with the cylinder. The system is thermally isolated

and quasis~tically reaches a new thermodynamic equilibrium.

la.What is the final temperature T f of the system?

lb. What is the total change in entropy of the system? (Youcan leave your "answer in terms of T f ),

Ie. What is ~h~;'work done on (or by - spe~ify which) the gas?"(Youcan 'still leave your answer in terms of T

f ), :"

An\ .

~1 ~1k s e - + " p f o / " P r o b / e . . . j.. ,

2. Suppose that in a certain range of temperature, when a rod

is stretched a distance x beyond its natural length, the rod exerts

a force given by F =.:-~(i.e. a temperature-modified Hookeanforce) where T is the temperature (in Kelvin) and A is a constant.

2a.What is the fundamental thermodynamic relationship be-tween the internal energy E, the entropy S, and the force F?

2b. Find an expression for (~ ; )T in terms of A , x, and 7 "

2c. Find an exp~ession for (~ ~ )T in terms of A, x~' and T.

2d. What is the change in the internal energy of the rod whenit is stretched at constant temperature an amount X o beyond itsnatural length ?

1



....•..

v

3. n moles of an ideal diatomic gas are cycled quasistatically

around the shown loop, beginning at point a. Step A is isothermal,

Step B is at constant volume (isochoric), and Step C is adiabatic.

Express all answers in terms of the initial pressure and volume, P o

and V O .

3a. Compute the work done W, the change in internal energy

tlE,and the heat Q added in step A.

3b. Do the same thing for step B.

3c. And again for step C.

3d. Compute the change in entropy in each of the three steps,

as well as the total change in entropy for the entire process.

4a. Starting from the first law of thermodynamics, prove that

4b. Use the expression from part a to calculate the change in

temperature of a Van der Waals gas undergoing free expansion from

V to 2V. The Van der Waals equation ofstate is ( P .+ v % " ) (v-b) =RT,

where v =V/n is the volume per mole of the gas.

2

\

" '



$,l...~-"AS . . J o ~iI'V '-Ptth~

J e - + - aft-' ~ 1k~J--Ffow (+ /~ 1 k o s /.~ ~t ~ &~.f..-Fltv '4 0 jo« -t -o r - 1k w e ) . " J .

fr

O ff£ . - t C Z " , , * 1 =-0 b , CSVt.S. PVivuy.

5if\!L1k- 'P i:s -t o l ' Ts t r e e - ic lMU~) y e - + - IS ~/MJ & !< A -+

b y ikfrt<55W"e- Ffk-r-sJ tM 7 p'7)l££$ 1k ft'5 ~.5

J L s -t - b e . f?,o b,.riC .

~'. Offs ;.46T'"L f C T f-T;;') ( & :o k - .{ ; - .. ,l T "I ' I ; b n w - -) . ~ Q ..J .; .I : : . C,6T Co - aT r - T,). ~

..",,> CrC~-lb') -I e c.l"lio-T; -=--0 . - - . . -

--, C'P' -- t c -T 2.V\ ~ \ D -\: c - l,7~ t = :.. 0 '= 2- F - - - -

~ -t C- 2 . . . V I - Q-\:-C-,~

6lE::TJ-S - p/)v.

;rik7fs/ jE-,,-CvPT = = - f~[Z~T

:;) 6Sft's ~ C V \ A C # :)+ .N \{~ IA L ~ )+ v ; : N t < 'S I' ~) Y ! : : . . . ~ :? - > LV~C o r - J + ' )

_.. f Vo ~

6~=-Gr~(i) (prIJl$~1DlS"P&~CrDt).



r P - E ::-TJS,- F J 2 )( 7,.. . 11'----~

Wo,.\.(~ ~ ~

65~\ ; :C IA L ~ ')

o ~ S rm - = c . I A L ~ ) ~ Cr ~l~ ') ...

7)~~~:c c -~ L ~ ') ~ ~~RII\

:.----=

6E~ta -!SvJ.

!XJ<~" y,m-t;,").1IE~ & ,[TF-IO") -'/ (j,W~L C v - C v ) [Tr-Ti" ")

[~ w ':;V\Rffi-T;.).[

SfilCf-If- '7 To) te ,W " " 7 0 = - > 1 .J 6Ir / . < ~ ~ -tk fr"s .

; 6 ,'v J - = - ' P e N =-?Nr \If>') ~r' If - - " r , v0- '" 1 Z -T ,p - - V I . rz -~ = - " r< -tT 'F- - T b "y

1F= - - & - . 'lj1 6 + e - + c L '0 / CbWfl'eSS~ ik ~/ . ~ • &f'.f- -s{r,reS f!.AV"S1. , v . . if.$iVlu.- ~ IvOr\,( ~NL b r 1'k ~. /5 /#f).f.,w) lr<..- ~+ \

v-e-





1)~ ./o~ 'L-'ifS : G~iN \ < o =-~~ R·

L \E:00f :;.,. fa.::-b,vJ ,

6 w ::,\~\j ~~ " ! . ' 4 s T P{;: IJk~T I/~ )v l v . - : -~~/ !,jW ::: p . .V o 'v.1.~6C<J I

.~

i l b ,v J , ; .0 1

So 6E;;.~0(

!)Q=ev6T = - ~V'R(Tc-I:).

~i~,2-Vo- P b V b } .

I. , : '0 y ~¥ < p c . . L2V

pJ ;: p~V b =') 'P,,-;: 'P o 'L .

1

/ / - 6 G < "6E=-1?ov~LTY-t\-l)

/;/1..;.0 I.So 6 C . ~6W

tJ~=-Cv~- T c .. - ) ; : ~V\lllr.~~T;..') = t(yoyo - 'P < - ¥ ~'). /

\

S I - Y..•1 " \ S " I - y..• \'\

~/~S=- 2 . - < " ? o Vo

- f > o V o l .. . j = - 2.~ VoLI- 1. );~ L)\iJ, \

~ 6 E - " - bSr Jc-6~ -t /::.~ = -0 -



.6 5c

::- O' .

\ ~ \/ ~ ::-tv k C # = ; " ) ~ Nlt~ k L ~ ) ,

; : ' ) F S A ; :N'~tl\ 2.) DS~~-6SA I (~=-O).

.-

o S .6Sn ~ C -v H~.~ ) ~ " - f N k . ! 5 k[# :' ) .)-_ \ Iff -.. 'tr-' ( .< ' f" - ,

TV :.Cat'OT Af"~ L.) S<J klkV.j .=:T Vo)

__ -r

~ ' ) J c - : :. .2-)r" ': \

: ; 7 6$n = ~N\<~ 1 1 \ ( 2 . .~ - , , ) , : ~ N K(!, (1- "¥) l i t '2

' E 1 o < - + - ¥-C ,,-t N kr . ; : I1 '" ~ \< (J . =') 1-Y = - t:JU a Cv Cv ~ r - ll A l J ,

2-



\

,

W e - ~+ (~.

d- [ ~ ) v ~CvJ S o

So w-n~ frr:J~\ t)V~1"9') f ) 1 = "l"b\f )e ~~. '-.v

12r~l~\ !)v~t9~O k Cv

~7j E ::Cvbtr. Cv (?L\ f)V. L~v~

L < .~ J I E . : : - tiS-rO v -" .so

(tiS-~!)\I ~ C vJ2T~ G;l~:J E : J)V

o~~: f 1 S T S I \ '~ J I T = ~ wn~ J2S;l~)/)\~L~)~V

' ) T L ~1PTkl~)~PvJ _~J)V~Cv~T~4l~ :~

= - > ' T~), = - C v L~ ~) .

.~ Tl~\-'?::-Cv~ \

~1(3I-\ ~-(TL*\ ~'P')Lev)E- - .

L - v "

~ ~* .JCE-TS)"- -sPT ---<'\:)9\f~) G ~'\ : :( ~ \I 4V) ,. l(>j \ ),;

~ ' ) \ C ~ : ~ E : :- [T(l7)~-1>"



C-j ( l E . ) ,_ ~ 2 ) \ - - . . ~

." 7:>\1 E - - Cv WI.



Qual Review, Meeting 2:

Statistical Mechanics

1. "The Dreamer" For a photon gas, show that:

la. ~=aT 4, where a is some constant, T is the temperature, U is

the internal energy, and V is the volume of the system.

lb. p =k ~ , where p isthe pressure.

lc. Find the adiabatic equation of state for this photon gas.

2. "The Law" A zipper has N > > 1 links. EaCh link has twostates: state 1, in which it is closed and .has energy 0, and state 2,in which it is open with energy €. The zipper can only unzip from

the left end and the 8th link can'not open unless all the links to itsleft (1,2, ..., 8 - 1) are already open.

2a. Find the partition function for the zipper ..

2b. Find the mean number of open links. Evaluate your result inboth the high and low temperature limits.

3. "The Heartbreaker" Consider a system of N non-interacting

quantum mechanical osdllators in equilibrium at temperature T.

The energy levels of a single oscillator are given by Em =( m +~)~,where A is a constant and the volume V is one-dimensional. You

may assume the oscillators are distinguishable.

ga. Find U and C v as functions of T.

3b. Determine the equation of state for the system.

3c. What is the average number of p~rticles in the mth level?

1



4. "The Boss" Two dipoles, with dipole moments M1 and

M2, are held apart at a separation R, but are allowed to rotatefreely. They are in thermal equilibrium with the environment at

temperature T. Compute the meanforce F between the dipoles for.

the high temperature limit ~t~j « 1. (Hint: the potential energy

between two dipoles is

i f > = 3(M1 • R)(M2 • R) - (MI' M2)R2

)

R5

5. "The Non-Interacting Isin~ Model" Consider a systemof N identical non-interacting spin 1/2 magnetic ions with magneticmoment J . L o in a crystal attemperature T in a magnetic fieldH. Forthis system calculate:5a. The partition function Z.

5b. The entropy S.5c. The average energy U. .5d. The average magnetic moment M and the fluctuations in the

magnetic moment, b..M =J «(M - (M ))2 ).

5e. The crystal is initially in thermal equilibrium with a reservoirat T =1 K in a magnetic field, Hi =10, 000 G. The crystal is

then thermally isolated from the reservoir (and everything else inthe known universe) and the field is then reduced to HI =100 G.What happens to the temperature of the crystal?

2

\

" '



~ l

.\. ~. E. S4h ~ ( 1 ' 1 t . .'> = - 2 - - ~--::6f.-_-\-

f. £

~) E - = : Ii-jilt.~Ll-~'a- \ ~ L e ; - ( J £ . - - - \ ' ~fk~ VzO}

E = : ) ~t.\~I!; _\ JtJ ~ ( ) I {c - ) i5 1k I k s ; + t ol . s . . { . ; , / e s .



.

~l<k:.+s: t 9 : . Q = - » s - ; - { )~/ !)E= - ~'Pf).\}, 1).•.+ f- = - : : } r v :-)c 9 E = ' ! Y / J v . / , S V 'J ) p

- ; ." ) 7 : > ' 1J r = ' - L fp f)V = - '> rk - : .- 'i - pv _ ' f J / - /3 - r, p 3V~/ 1 'J -~I'

,', :...-;

~'.. '.

\ ,



00--- --



.

J)J T~~ 1 ~ t - : (3f..>'/ I .

\i)k-i: 1c3[~I.

{i3~1 e -o f I ~ "

<tJ):=- (jL,/ ~ - (1J'f t)e-- 1 {)t{~-t"-~ - 1

'= -'> \ > _ ((\f/f?f { S r .D ~ (.}f. ) ---~ f3~LI-+O~~))

~ a . . l -IJ£ $0 vJ _ I t2 -'f = pL.>"? • z-"/ e . . ; : : :q , "olt I a o f . J c - [ y - u , r e - r + .

- fJ f.. ..J!>d ! o J -tI') I ..{.}E..../3tl tJ' 4')

0/ ,eN")~ e _(N-H) e . h,r N ~ : ; > \ ) e ~~ e . . . · .

\

"



1

-------- '! - -1-~-~

,6\;: T c.sckC~), £iVllL-ilo~l~k 0$- J L ~ ~ '-s ~ J L h

l = - : J . - , f ' J = - 2~ c s J t L - r . - ).."'toT

£: _~ ~ I t - - t f '' t -<;J:{~~::~ N :.~ 1 i.t, ,)tscL.{~) .ZV 0)'- . z .V c.'Jc -kL~")

E=- ~ c b ! U 'E - \ 2 ,\1 .J,'





(Z

~ ~

~/R =: I t I\ 2 - ~ s 0 -, J H 1-' R . - zH z - ( L e o s < 9 - 1 . .

-------



rr

J ~ $1\<-0, c - r - - r A if), P - f ) J ) r f , ! # 1 . - = It, , ' l - \ 'i,,~() ~ :- l6Tr'l.

. 0

: 1 f ~

\ SM f),usf)\ Q G - 1~ ) s~ Q C •••(f');.0.

o (7

' 2 ' \ l

J ) U 7 S ( J I - f L J P f I = = ( ; ) . Wl,.•~ N e e Q M:)c1- mAr.

t:)

\

"



r ~ ~ '

~ 5 . " , 3 & / ) & : ; ~ f ~ ) aNj,-tfrJ P f . fN z ~ 2-rr~ S D e = i f J J ' l./~I¥ v 0

"1f' ,

~s ,,,$~"Q !J~= ' ~ f So (J):: 2'f-lf~'Z.~ 3~~Z-

, "

7, ~ / 6 'L ~ 2 ..(3 2 rr " - fG T I 't .)7/ C~~ 1 \ -+ /\l ~ .-t ~

= - Ibll'L-+ 1:. L f - ~ T r '- - " /6 ,, ' l . ( [ -+ ~ 't )

- _ _ 0 / 1 \ ~ : - ~ d ~ _ - \ ~jft{ I rt.t{l..t~1 = - .CJIoJ) E - "df!:> ~"dt:) - JG t1 t'-< f) '"3 " lrz.') \.(~T

. 1 -" , ,: I I ~ 1 - n ( b l . - - k A & 1 J So ~f'



.

£ _- d { 1 \ 1: ""1 H I It - ~t3:=' -/ j" < "W A li t X

:

f~ ; : - < fo ) . H - " \ <~') = I f , ,~ k > < 1

«~ _L J-\»)1 .);<l-'\~> - (1'\;7..- o'L-tlfo~~~ '\

J \fk-[/7 iJJJL -7.~ i& b ;" -~ J So~5~+ 5"

SMtL S~S(i-)=-/ u ; - - J - + x ='/ U>I ' -~+e~ .

" ~7!:t"'~ : ; - '1 1 ; :c l~O' " I \ . ( = : 1 0 ' 2 . - I t <

T----...L-

\ ,

Classical Thermodynamics I

Documents