! Summary Part 2 21.11.2018 Thermodynamic laws Thermodynamic processes Fys2160, 2018 1
1 U is fixed ) *,,, -(/,,), *,, -(/,,) N, 3 *,, - /,,, 2(3) Summary Part 1 Equilibrium statistical systems CONTINUE... Fys2160, 2018 2
A system in contact with a thermal and particle reservoir e g d, h(i, d) d The system can exchange energy and particles with a reservoir and it is in equilibrium at a fixed : and chemical potential ; Probability of the system being in a given microstate is proportional to the probability that the reservois is in any state that accomodate that particular microstate (hence the total number of microstates of the thermal bath corresponding to a given system s microstate) Probability ratio between two microstates (the system can exchange energy @A B = @E, and particles @G B = @G) H I J H I K = L M I J L M I K = N OM PJ QOM(PK) R = N STU MN VSWTX M = N VSTY N SWTX Probability of the system in a specific microstate a fixed T and ; \ Z [ = ^V _(`[V; a [ ) ](:, ;) Grand Partition function Ξ d, e = I N V S(Y PVW X P ) counts all the accessible microstates weighted by the Gibbs factor What is the microstate [? Fys2160, 2018 3
Non-interacting particle system in contact with a thermal and particle reservoir M _ L, `(a, L) L Each identical particle can occupy discrete energy states 3 4, 4 = 7, 8, is the state number For N identical particles, we can have @ 4 number of particles (occupation number) in the energy state 3 4 The energy of a specific microstate with @ E = 4 @ 4 particles is G E = 4 @ 4 3 4 H sum over all particles number @ E and over all the partitions of particles @ E in the quantum states with total energy G E Ξ L, M = N N U V W(X PVY O P ) = N U VW R O R(Z R VY) O P {O R } {O R } R O R TO P Ξ L, M = N O [ U VWO[ Z[VY N O ] U VWO] Z]VY N O^ U VWO^ Z^VY Fys2160, 2018 4
Occupation number of a state? k =, 7(l, =) = Probability of the system in a specific microstate a fixed T and 6 7 8 = 1 Ξ(=,?) AB C(D EBF G E ) = A BCG H I H BF A BCG K I K BF A BCG L I L BF GH A BCG H I H BF GK A BCG K I K BF GL A BCG L I L BF O P = Q BRS T U T B6 ST Q BRS T U T B6 Q BRS V U V B6 SV Q BRS V U V B6 Q BRS W U W B6 SW Q BRS W U W B6 O P = O(S T ) O(S V ) O(S W ) Probability for the occupation number S of the given state at fixed T and 6 O S = UB6 QBRS S Q BRS UB6, [ = \ Q BRS UB6 (]^_`a`abc decf`abc bd ^ PacghQ ibjq) S Fys2160, 2018 5
Non-interacting FERMIONS in contact with a thermal and particle reservoir b _ `, @(a, `) ` The occupation number for each quantum state is 3 = 5, 7 Probability for the occupation number 3 of the given energy state a fixed T and? @ A = BCDE FCG 1 + BCD FCG Average occupation number 3 of the given energy state M a fixed T and? FERMI-DIRAC distribution 7 3 (W) = Y 3Z5 3[ 3 = \C] WC? 7 + \ C] WC? 3 (W) = 7 \ ] WC? + 7 Fys2160, 2018 6
(! ", $(&, ") " Fermi Dirac distribution Fys2160, 2018 7
Non-interacting BOSONS in contact with a thermal and particle reservoir L p q, \(r, q) q The occupation number for each state is 2 = 4, 6, 7 = :;< >?@: A?B = C C?DEF GEH, IJK L < N (IJK >P>KQ N!) Probability for the occupation number 2 of the given energy state a fixed T and [ \ ] = 1 >?@ A?B >?@: A?B Average occupation number 2 of the given energy state N a fixed T and [ BOSE-EINSTEIN distribution = 2 (j) = k 2;4?n j?[ 2l 2 = 6 m k = 2;4 2 m?n2 j?[ 2 (j) = 6 m n j?[ 6 Fys2160, 2018 8
(! ", $(&, ") " Bose Einstein distribution Fys2160, 2018 9
Classical limit W R S, U(V, S) S QUANTUM distributionfor the average occupationnumber of an energy state ; (=) = 1 A B CDE ± 1 High T limit ( E G HG 0) BOLZMANN distribution K L = M NO M NL ± M NO M NO Q K L = MDN LDO Fys2160, 2018 10
(! ", $(&, ") " Fys2160, 2018 11
Z * W, X(Y, W) W THERMODYNAMIC PROPERTIES AND DENSITY OF STATES Average energy * =, -., - /, - 0 1 (3) 3 6 7, 6 9, 6 7 = : ; < =6 7 : ; < =6 9 : ; < =6 > 3 1 = : ; < =3? 3 3 1 Density of states? 3 comes become we need to count all the quantum states at a given energy 3. Remember that the quantum state is given by the state of the wavefunction Number of states with energy between 3 and 3 + =3 Number of states with state number between 6 and 6 + =6 (positive quadrant) 3E? 3 d3 = 1 8 4J6K =6, 2E? 3 d3 = 1 2J6=6, 2E, (1E)? 3 d3 = =6 4 Energy 3 6 is determined by the quantum mechanics: Particle in a box 3 6 = MN OPQ N 6K Quantum harmonic oscillator 3 6 = 6ħS Relativistic particles 3 6 = hu = MV KQ 6 Fys2160, 2018 12
Densityof states 7 3 4, 5(6, 4) 4 Number of states with energy between! and! + #! Number of states with state number between % and % + #% 3' (! d! = + 2 %- #%, 2' (! d! = + %#%, 2', (1') (! d! = #% 2 FERMIONS: remember to multiply by factor 2 because there are two electrons per energy level (spin up and spin down) 3' (! d! = 2 + 2 %- #%, 2' (! d! = 2 + %#%, 2', 1' (! d! = 2 #% 2 PHOTONS : remember to multiply by factor 2 for the two transverse polarizations of the EM waves 3' (! d! = 2 + 2 %- #%, PHONONS: remember to multiply by factor 3 for the three polarizations of the sound waves 3' (! d! = 3 + 2 %- #% Fys2160, 2018 13
Thermodynamic properties and density of states / *,, K(.,,), Average energy 4 *(,,., /) = 2 56 7 6 8 6 = 2 3 3 4 56 7 6 6 9 : ;<= ± 1 Average number of particles 4 8(,,., /) = 2 56 7 6 8 = 2 3 3 4 56 7 6 1 9 : ;<= ± 1 Fys2160, 2018 14
DEGENERATE FERMIONS S? @, R(B, @) @!(#) = h' 8)* ' #', -.! /! = 0# ' /#, -.! = 0 2 8) h ' -!! 2 (3) = h' 8)* ' # ' 456 = h' 3 8)* ' 2 ' Average energy E F/!?(@, B,! 2 ) = C,!! D Average number of particles E F/! 3(@, B,! 2 ) = C,! D Fys2160, 2018 15
Photons _ 7 8, \(:, 8) 8! " = $% &' " )! *! = +"& *" )! =,+ - $%.!& Average energy 7 8, : = ; < = >? @?? A BC 1 = 8G : = hi J ;? J >? < A BC 1 = 8GK L8 M 15 hi J Planck distribution Average number of particles ` a = 8G h I J a J A Bbc 1 = Z 8, : = ; >? @? < 1 A BC 1 = 8G : = hi J ; >? <? [ A BC 1? = ha = hi/e Fys2160, 2018 16
! Summary Part 2 21.11.2018 Thermodynamic laws Thermodynamic processes Fys2160, 2018 17
First law of thermodynamics: CONSERVATION OF ENERGY./ = ($ + (& Reversible./ = )'*,'- Change in the internal energy of a system is due to heat or work exchanges with its surrounding!" = $ + & The change in the «stored» energy equal the sum of «energies in transit» The infinitesimal change in internal energy '" = ($ + (& Infinitesimal reversible process '" = )'*,'- Fys2160, 2018 18
First law of thermodynamics: CONSERVATION OF ENERGY 34 = $% + $5 Reversible 34 = )!" /!+ Clausius equality!" = $% &'( ) Heat capacities * + = $%!) + =,-,) +!- = * +!) * / = $%!) / = * + + 12 Fys2160, 2018 19
Second law of thermodynamics Irreversible heat flow Entropy increase Irreversible heat transfer is smaller than the reversible heat exchange at a given T Clausius inequality:!" $% & Entropy of the Universe:!" I Clausius inequality $% :;;<= & < % ;<= & CD $% & Isolated system Entropy tends to increase as the system sponteneously finds its equilibrium state CD I Fys2160, 2018 20
Example of (reversible) transformations of an ideal gas: G 1. Isothermal process:! " =! $ & " = & $ ((& = ) *+, +. *+, = /) 6 $7 6 $ ) 123 = 5 86 = 9:! ;< 6 " 6 " (= =. *+,! = 9: ;< 6 $ 6 " I J I K H 2. Isobaric process: 7 " = 7 $ = 7 (> 6 8! =!8? 786) 6 $7 ) 123 = 5 86 = 7 6$ 6 " 6 " F. *+, (? = 5 @ABCD*@E!! $ 8! = > 7 5! "! = > 7;<! $! " Fys2160, 2018 21
Example of reversible transformations of an ideal gas: 3. Adiabatic process:!" #$% = ' )* =,)- G. - )/ =,)-. - )/ / = 01 )- - / 2 / 3 = - 3-2. 4. - 53-2, 6 789 = : )- =.- (/ 2 / 3 ) - 3 I J I K H => = ' 4. Isochoric process: - 3 = - 2 (. - )/ = /)?) - 2, 6 789 = : )- = ' - 3!" #$% =? = : @ABCDB#@C / / 2 )/ =. - : / 3 / =. - EF / 2 / 3 Fys2160, 2018 22
Second law of thermodynamics: In search for the most probable state Boltzmann s formula Δ0 = 2 ln Ω 45678 Ω 9:9;9<= Things that are more probability, tend to occur more often Ω P9:<= Ω 9:9;9<= Entropy cannot decrease Δ0 0 An isolated system will spontaneously evolve towards the most likely equilibrium state, i.e. The macrostate with maximum multiplicity O S Fys2160, 2018 23
Two weakly interacting ideal gases Total multiplicity: Ω "#"$% = ' ( ) * + *, - (/ + /, ) 12 3 (, * +, / + (, *,, /, Ω 4$5 "#"$% = ' ( ) 6 ) )- 7 ) 8- States near the most likely state by varying U / = / + + /, * = * + + *, / + = 7 ) + :, /, = 7 ) : =>?h : //2, while * + = *, = 6 ) le Ω "#"$% 3( 2 ln / 2 ) : ) 3( ln / 2 3( 2 2: / ) The width scales as W X = Y X Y Ω "#"$% / + = Ω 4$5 "#"$% exp 3( 2 Y Z[ = X Z[ Y ] 2 / ) / + / 2 ^_ [ ) //2 Fys2160, 2018 24
Two weakly interacting ideal gases 2, & ', ) ' 2, & 6, ) 6 Macrostates near the-most-likely macrostate (aka THE EQUILIBRIUM STATE) Ω "#"$% & ', ) ' = Ω +$, "#"$% exp 2 2 & 4 & ' & 2 4 exp 32 2 2 ) 4 ) ' ) 2 4 ) = ) ' + ) 6 & = & ' + & 6 The number of microstates away from the equilibrium macrostate decay very rapidly, hence they are extremely unlikely 2 ) ' & ' ) ' ) ' & ' & ' Fys2160, 2018 25
Liquid-gas phase transition Van der Waals equation of state for fluids! + #$% & % & $( = $*+! = $*+ & $( #$% & % Maxwell s equal-area construction: The phase transition happens at constant PRESSURE! We find the pressure on the P-V diagram by the equal areas construction of a given isotherm Phase transition from liquid to vapor at a constant Gibbs free energy -(!, +, $) 1-2 = 1-3 Fys2160, 2018 26
Liquid-gas phase transition Phase transition from liquid to vapor at a constant Gibbs free energy!(#, %, &) (! ) = (! + = Clausius Clapeyron relation - ) (% + / ) (# = -_+(% + / + (# (# = - +1- ) = 23 = 5, 5 67 89: );8:<8 9:;8 of the phase transition (% / + 1/ ) 24 % 24 > Entropy jumps going from a liquid to a gas Volume expansion going from a liquid to a gas Clausius Clapeyron relation tells us how much the phase transition pressure changes with changing temperature How does the boling temperature of water depends on the altitude (pressure)? Fys2160, 2018 27