Entropie statistique : exemple de la détente de Joule Gay

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Entropie)statistique):)
exemple)de)la)détente)
de)Joule)Gay7Lussac)
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1. Etude!par!la!thermodynamique!classique!
•"Présentation"de"la"détente"de"Joule"Gay3Lussac"
!*()+,#-,#()#( ./0'#(1( 2*3(!044*5(#4,(0-( )74&/47,7<(#$&+67%#-,*'( &#6%#,,*-,()#()74567%7-#6(47(0-( =*>(
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'"+-#6=7#(7-,#6-#()#(5#(=*>(-#()+&#-)(:0#()#('*(,#%&+6*,06#;(
D*-4(la"détente"de"Joule"3"Gay"Lussac8(0-(6+57&7#-,(𝐶!()#(C/'0%#(𝑉
!8(5/-,#-*-,(7-7,7*'#%#-,('#(=*>(@(
'*(,#%&+6*,06#(𝑇
!8(#4,(%74(#-(5/%%0-75*,7/-(@('"*7)#()"0-(6/?7-#,(*C#5(0-(*0,6#(6+57&7#-,(𝐶!()#(C/'0%#(
𝑉
!;(!#(6+57&7#-,(𝐶!(#4,(7-7,7*'#%#-,(C7)#;(
!#4(&*6/74()#4()#0$(6+57&7#-,4(4/-,(40&&/4+#4(67=7)#4(#,(&*6<*7,#%#-,(*,E#6%*-#4;(
(
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!('*(
,#%&+6*,06#()0(=*>()*-4(5#(-/0C#'(+,*,()"+:07'7?6#(,E#6%/)3-*%7:0#;(
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𝑇
!<𝑇 !8(%*74(5#,(#<<#,()#%#06#(,/0I/064(,694(<*7?'#;(
•"Calcul"de"la"variation"d’énergie"interne"
J/04(5E/74744/-4(5/%%#(434,9%#(,E#6%/)3-*%7:0#()"+,0)#('#(434,9%#(='/?*'(K=*>(L(C7)#M()+'7%7,+(
&*6('#4(&*6/74()#4(6+57&7#-,4(𝐶!(#,(𝐶!;(N(,6*C#64(5#4(&*6/748(7'(-"3(*(-7(,6*C*7'(A&*6/74(67=7)#4B(-7(,6*-4<#6,(
,E#6%7:0#(A&*6/74(*,E#6%*-#4B;(
!"*&&'75*,7/-()0(&6#%7#6(&67-57&#()#('*(,E#6%/)3-*%7:0#(5/-)07,()/-5(@(F(
∆𝑈=𝑊+𝑄!!⇔!!∆𝑈=0!!!(1)(
!*(détente"de"Joule"3"Gay"Lussac(4#(6+*'74#()/-5(@(énergie"interne"constante;(

O-(=*>(&*6<*7,(/?+7,(@('*(&6#%796#('/7()#(./0'#8(5"#4,1@1)76#(:0#(4/-(+-#6=7#(7-,#6-#(-#()+&#-)(:0#()#('*(
,#%&+6*,06#(4#'/-('*(6#'*,7/-(F(
𝑈𝑇=𝐶!𝑇!!⇔!!∆𝑈=𝐶!∆𝑇(
∆𝑈=0!⇔!!∆𝑇=0(
P/06( 0-( =*>( &*6<*7,8( '*( )+,#-,#( )#( ./0'#( 1( 2*3( !044*5( 4#( <*7,( )/-5( @( ,#%&+6*,06#( )0( =*>( 5/-4,*-,#(
𝑇
!=𝑇 !;(Q/0,#(%/)7<75*,7/-()#('*(,#%&+6*,06#(*0(5/064()#('*()+,#-,#()0(=*>(#4,()/-5(0-#(&6#0C#()0(
5*6*5,96#(-/-(&*6<*7,()0(=*>(+,0)7+;(
•"Calcul"de"la"variation"d’entropie"
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)+,#-,#()#(./0'#(2*31!044*5(4"+567,(F(
∆𝑆=𝑛𝑅 ln
𝑉
!+𝑉 !
𝑉
!
!!(2)(
!#(434,9%#(-"+5E*-=#*-,(*050-(,6*-4<#6,(,E#6%7:0#(*C#5('"#$,+67#068('"#-,6/&7#(+5E*-=+#(*0(5/064()#(
'*( )+,#-,#( )#( ./0'#( 2*31!044*5( #4,( -0''#( #,( '"*&&'75*,7/-( )0( 4#5/-)(&67-57&#( )#( '*( ,E#6%/)3-*%7:0#(
5/-)07,(@(F(
∆𝑆=𝑆!+𝑆!!!⇔!!𝑆!=∆𝑆!(
⇔!!𝑆!=𝑛𝑅 ln
𝑉
!+𝑉 !
𝑉
!
!(
⇔!!𝑆!>0!!!3!(
!*()+,#-,#()#(./0'#(2*31!044*5(#4,()/-5(0-#(transformation"irréversible;(
2. Etude!par!la!thermodynamique!statistique!
P/06( +,0)7#6( '*( )+,#-,#( )#( ./0'#( 2*31!044*5( )0( &/7-,( )#( C0#( )#( '*( ,E#6%/)3-*%7:0#( 4,*,74,7:0#8( #,(
*7-47(7''04,6#6('"7-,#6&6+,*,7/-(4,*,74,7:0#()#('"#-,6/&7#8(7'(<*0,(<*76#(*&&#'(*0$(-/,7/-4(4,*,74,7:0#4()"+,*,(
%*56/45/&7:0#(#,()"+,*,(%756/45/&7:0#;(
H-(,E#6%/)3-*%7:0#(4,*,74,7:0#8(/-(*&&#''#(F(
1 état"macroscopique(A/0(macroétatB()0(434,9%#8(0-(+,*,()0(434,9%#(4&+57<7+(@(0-(7-4,*-,()/--+(
&*6('*(5/--*744*-5#()#4()7<<+6#-,#4(C*67*?'#4()"+,*,4(%*56/45/&7:0#4(%#406*?'#4(A&*6(#$#%&'#( '*(
&6#447/-(#,('*(,#%&+6*,06#()0(434,9%#B(
1 état"microscopique( A/0(microétatB( )0(434,9%#8( 0-(+,*,()0( 434,9%#(4&+57<7+(@( 0-(7-4,*-,()/--+(
&*6('*(5/--*744*-5#()#4()7<<+6#-,#4(C*67*?'#4(%756/45/&7:0#4()0(434,9%#(A&*6(#$#%&'#8('*(&/47,7/-8(
C7,#44#8(+-#6=7#()#(5E*50-#()#4(&*6,750'#4(5/-4,7,0*-,('#(434,9%#B(
1 états" accessibles8( '#4( +,*,4( %756/45/&7:0#4( )0( 434,9%#( 5/%&*,7?'#4(*C#5( '#4( 5/-,6*7-,#4(
%*56/45/&7:0#4(A&*6(#$#%&'#(#-(,#6%#4()#(C/'0%#(/0()"+-#6=7#B(7%&/4+#4(*0(434,9%#;(
•"Postulat"fondamental"de"la"thermodynamique"statistique"
!#(&67-57&#(406('#:0#'(4#(</-)#('*(&E347:0#(4,*,74,7:0#(A#,(,/0,#('*(,E#6%/)3-*%7:0#(:07(#-()+5/0'#B(
&#0,( 4"#$&67%#6( ,694( 47%&'#%#-,(F( tous" les" états" microscopiques"accessibles"d’un" système"
thermodynamique"isolé(et"à"l’équilibre"sont"équiprobables.(
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6+*'74*-,(5#(%*56/+,*,()/--+(#,('#(-/%?6#(,/,*'()"+,*,4(*55#447?'#4()0(434,9%#(
Conséquence":(!#(%*56/+,*,('#(&'04(&6/?*?'#()0(434,9%#(#4,(5#'07(&/44+)*-,('#(&'04()#(%756/+,*,4;(

•"Irréversibilité"de"la"détente"de"Joule"Gay3Lussac"
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1 Ω!(𝑉
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𝑃=
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!!(4)(
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•"Entropie"statistique"
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'"entropie"statistique(𝑆(*(+,+(#<<#5,0+(&*6(!0)T7=(U/',>%*--(#-(VWXX(F(
𝑆=𝑘!ln Ω!!(5)(
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𝑘!=1,38×10!!"J.K!!(
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𝑃=Ω!𝑉
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⇔!𝑃=
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∆𝑆=𝑛𝑅 ln
𝑉
!+𝑉 !
𝑉
!
⇔!∆𝑆=
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⇔!∆𝑆=𝑘!ln
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⇔!∆𝑆=𝑘!ln
Ω!
Ω!
•"Interprétation"statistique"de"l’entropie""
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de"désordre"particulaire"du"système"F('#(434,9%#(#4,()"*0,*-,(&'04()+4/6)/--+(A#,()/-5(4/-(#-,6/&7#(
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•"Questions"
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