理想気体ideal gasの熱力学的基本関係式

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1 the equipartition law of energy ( )kt k Boltzmann constant 5 Longman Dictionary of Physics (/)kt q Bq (/)kt equipartition law of energy mol (3/)kT (3/)RT (3/)R (5/)R 3R kt - equipartition of energy The principle of equipartition of energy, based on classical statistical mechanics and enunciated by Boltzmann, states that the mean energy of the molecules of a gas is equally divided among the various degrees of freedom of the molecules. The average energy of each degree of freedom is equal to (/)kt, where k is the Boltzmann constant and T is the thermodynamic temperature. In the late nineteenth century the principle was extended to the vibrations of atoms in crystals and to electromagnetic radiation in a cavity (see black-body radiation). Some of the results were consistent with experiment within certain conditions; for ex-ample, the principle predicts *Dulong and Petit's law for the specific heat capacities of solids, which was verified for most substances at the temperatures that were then attainable.

2 In the case of radiation the principle led to difficulties and Planck proposed the quantum theory ( 9) to overcome these. This led to extensive research, for example, the case of the ernst and Lindemann vacuum calorimeter to measure specific heat capacities at low temperatures. At the time of the first Solvay Conference (9l) leading scientists agreed that the equipartition principle was untenable in general, although it is an admissible approximation in certain cases, especially at high temperatures. α ( x, y or z) E pα mvα (Eq.) m α m v α p α α m r m m r r m m m r, + + mmr ω E m ( rω ) + m ( rω ) ( m + m ) mm µ reduced mass m + m µ r ω moment of inertia I mm I mr m m r r i i µ + i l rm r + mm r m r m + m r r I ω ω ω µ ω ω l E Iω (Eq.) I 9 (X, Y, Z)

3 (Eq.) la Ea I aω a, a X, Y, Z (Eq.3) I a I a l a a Z p l, m I, v ω m p E + Kq (Eq.4) m p K q q Asin( ω t), p mq maω cos( ω t), f Kq p mω q y By T By ( ) P By A exp (Eq.5) kt y y + By A dy kt exp (Eq.5) By + By By exp dy + kt By By P( By ) dy + By exp dy kt 3

4 β + ln exp( β By ) + dy, β kt (Eq.6) exp( β By )dy Laplace I 5 Laplace 33p x x n n+ a, e ax e dx ax dx ( n ) n+ n! n a +!! π a n+ ( n )!! ( n )( n 3)( n 5), ( )!! (Eq.6) d π d By ln ln d B β β β dβ, β kt (Eq.7) By (/)kt (/)kt 4

5 BJ ( )( ) ( J + ) BJ J + J + exp J kt ε BJ ( ) ( J + ) J + exp J kt d ln dβ J ( J + ) exp[ β BJ( J + ) ] K J ( J + ) exp[ β BJ( J + ) ] J xe, dx β β Bx β B d ε ln β kt dβ β kt exp( β By )dy β + + β B y x exp( β By ) dy exp( x )dx + β B exp( x )dx β + 5

6 ideal gas (Helmholtz) (Gibbs) extensive variables intensive variables E de S E S E V E V, S, S, V E ds + V E dv + V, S, S, V temperature, pressure d electrochemical potential chemical potential chemical potential µ + µ + + µ r r µ i i i equations of state 6

7 3/ 5/ e s v E S V, e E, v V s S, e v s s + + cr ln R ln e v 7

8 / c e v ( s s ) / cr ln e v e / c v s s e e exp v cr e e s cr T e v e crt c v cv P T v s s + + cr ln R ln T v P v s s cr ln ( c ) R ln P v T P s s + ( c + ) R ln + R ln T P h P s s + + ( c )ln R ln h P /( c+ ) P s s h h exp P ( c + ) R h h s c + ( ) R T h h RT v P ( c+ ) P P f crt st crt ln( T / T) RT ln( v / v) g ( c + ) RT ( c + ) RT ln( T / T) + RT ln( P / P) st 8

9 97 (5) g g + RT ln P g T P g ( c + ) R ln + R s s T T ln P RT / P v P ( c+ ) / c / c V S S E E exp V cr c R E E SE e se µ + e + e + Pv st g c c R c cr T V S + s + c R ln R ln in Eq (I6) T v T + V S s + c R ln R ln T v E c RT V V v v 9

10 T V S + s + c R ln R ln R ln T v / V / V V V s + crln ( T / T) + Rln ( v/ v) [ ] entropy of mixing S m Sm / sm R ln R xlnx G( TP,,, ) µ ( TP, ) + µ ( TP, ) + RT ln + RT ln + + µ ( T, P), µ ( T, P) G( T, P,, ) µ ( T, P) + µ ( T, P) RT + RT ln ln ln { / ( + / ) } ln, + + x, x, x + x + +

11 G µ ( T, P, x) µ ( T, P) + RTln x, x x G µ ( T, P, x) µ ( T, P) + RTlnx µ µ ( T, P, x) ( T, P) x RT µ ( T, P, x ) µ ( T, P) + RTlnx colligative Properties m M / M ( ) M m { m /[ / M ] } x [ / M ] /[ / M ] m / m m m m /[ / M ] k + m [ / M ] m m m mol Kg µ µ + RT ln k + RT ln m m ( ) / P.W. Atkins, Physical Chemistry, fourth edition, p8(99, Richard Clay Ltd) Herbert B. Callen, Thermodynamics and an Introduction to Thermostatics, Second edition (985, John Wiley Sons, Inc.)

12 ideal van der Waals fluid P R T BT CT DT v + ( ) v + ( ) v + ( ) v + 3 second virial coefficient third virial coefficient ideally f v T P f( T v) f T v RT BT ( ) CT ( ) DT ( ), ideal (, ) v v 3v f d( BT) d( CT ) d( DT) s sideal R T v dt v dt 3v dt db dc dd e eideal RT v dt v dt 3v dt s ideal, e ideal molar heat capacity at constant volume de Tds c v dt at v constant c v s f cv T T T T v v d ( BT) d ( CT) d ( DT) cv cv, ideal RT v 3 dt v dt 3v dt

13 3 ε 3 BT ( ) b ( R ) exp, b A kt π σ 3 Molecular Theory of Gases and Liquids (Hirschfelder, Curtiss & Bird, John Wiley & Sons, Inc., p.58) 3 BT ( ) b b R ε / kt ( ) squarewell potential Lennard-Jones (6-) potential:.8.56 (Hirschfelder et al. p59) BT ( )/ b CT ( )/ b exp( ε / kt ) σ, b, R, ε/ k (Hirschfelder et al. p6) σ( 6 3 m) b ( m ) ε / k( K) real gas van der Waals 873 3

14 P R T a or P a v v b RT v b v T +, ( / )( ) P R a R b b a T v / v v T v v v v T R v v b a b T v BT ( ) b a/ T, CT ( ) b ( ) 3 4π σ 3 ε b b 4 A, a b( R ) 3 k excluded volume effect 4 π σ 3 3 van der Waals ds T de P + T dv 4

15 s s v e e v P v T e e T v R a v T e e v b v T v a v e T v ( / v) T e ( e/ a) T v cr cr / a T e e/ a cr / a cr ( y) T e/ a+ / v e + a / v ideal van der Waals fluid P R acr T v b ev + av ds cr de a dv + R dv e+ a/ v v e+ a/ v v b crd ln( e + a / v) + Rd ln( v b) van der Waals e+ a/ v v b s s + crln + R ln e + a/ v v b 5

16 / c a v b s s a esv (, ) e + exp v v b cr v c a V b S s a E / / / e + exp v v b cr V van der Waals T v b s s + crln + R ln T v b P+ a/ v v b s s + crln + ( c+ ) Rln P + a/ v v b T P + a / v s s + + ( c ) R ln R ln T P + a / v van der Waals 6

17 general ideal gas simple ideal gas general ideal gas ernst theorem T e e c ( T ' ) dt ' + T v cv ( T) dv ds dt + R T v T c v ( T ) v s s + + dt R ln T T v T T cv ( T ) crln dt T T T T c v ( T ) v S + + s dt R ln R x ln x T T v 7

18 T E e + cv ( T ) dt T G µ µ RT[ φ T + P + x ] φ T ( ) ln ln e s T ( ) + ln P ln ( ) RT R + T RT R T T ' ' cv T dt T T c v ' ( T ) dt ' T ' partial molar Gibbs potential ( ) e c RT s φ ( T ) + ( + c )[ ln( T / T )] ln P RT R g g + RT ln P P Px ( ) g RT ( + c ) ln( T / T ) s [ ] ln P R 8

19 ν A stoichiometric coefficient d d ~ d ν ν d ~ d ~ ν d dg SdT + VdP + µ + µ + ~ ~ SdT + VdP + νµ d + νµ d + SdT + VdP + d ~ ν µ dg d νµ νµ heat of reaction chemical potential G H G TS G T T + P,,, 9

20 d dh ~ dg ~ dg ~ dh ~ d ~ d T ~ d d d T d P,,, dh ~ T ν µ d T dh d endothermic reaction exothermic reaction ν ln x ν ln P ν φ ( T) ln KT ( ) νφ( T) mass action law ν x P K( T) ν ν ν K( T ) ( Px ) P ( dh ~ T RT ν φ + RT ν ln P + RT ln ν ln x d T d ν µ RT ν φ ( T ) dt νµ dh RT d ln KT ( ) d dt van t Hoff relation

21 ln K ( T ) dh ~ + const RT d ν φ ( T ) G / RT dh / d H H ~ ~ ~,, a, b, + c ( ) ( ) A B C A B H c HC a HA b HB H / chc aha bhb H H ln K ( T ) + const RT G RT ln K, G H T S H S ln K + RT R C

22 Einstein Model Dulong-Petit law H m p m ω + q harmonic oscillator fixed points collective vibrational motion Debye

23 ε ( ν+ / ) ω h h ε ω ε n ω ω ω ω ω ω ω ω ω ( 3 + E / ω)! ( 3 + E / ω)! Ω ( 3 )!( E / ω)! ( 3)!( E / ω)! S kln Ω Stirling approximation ln( M!) Mln M M if ω ln Ω ( 3 + X) ln( 3 + X) 3 ln 3 X ln X 3 ( + Y) ln 3( + Y) ln3 Yln3Y 3 ( + Y) ln( + Y) YlnY 3 ln( + Y) + Yln( + / Y) e e e s 3R ln 3R ln e e e 3

24 e 3 A ω / T s/ e 3R e + ln T e e 3 A ω e exp( ω / kt ) 3Rexp( ω / kt ) ω c v [ exp( ω / kt ) ] kt ω / k ΘE Einstein cv 3Rf E ( Θ E / T ) x exp( x) f E () x [ exp( x) ] x x e x ( + x + x / + ) x e ( + x + x / + ) ( ) microcanonical ensemble ω ω ω 4

25 , S kln Ω microcanonical ensemble ω ω ω ε av n n ω exp n exp ( n ω / kt ) ( n ω / kt ) ex x x e + e + e + + x n e e x nx d nx e ne e x n dx n e ( ) x 5

26 ω ε av ω exp kt e 3 A ε av Debye 9 Θ D e Θ D + 3RTD 8 T 3 3 x ξ dξ D( x) 3 x ξ e Θ D 3Θ D / T cv 3R 4D T exp( Θ / T ) D Θ D Debye Debye 6

27 p h n n x y nz ε + + n 8 x, ny, nz 3,,, m m lx l y lz p ( ) hn hn x y hnz ε, p px, p y, pz,, m lx l y lz nx, ny, nz,,,,,, h / l, h / l, h / l ( ) x y h h h l x l y l z z 3 h, V V l x l y l z 3 V / h V Vdpxdp ydpz p dp 3 xdp ydpz 3 h h dxdydzdp xdpydpz 3 dxdydzdp dp dp / h x y z Landau Lifshitz Quantum Mechanics (third Edition,Pergamon Press) (48.7) 7

28 q qs p p s / π ( ) s E ε + ε + + ε ( P + p + + p ) m p i ( p xi, p yi, pzi ) pxi me ξi, pyi me ξ, p me i zi ξ + + i ξ + ξ + + ξ 3 me Γ ( E) ~ V Γ( E) h V 3 h ( me) 3 pi me i 3 / dp 3 ξi i x dp y dp z dp dξ dξ dξ x 3 dp y dp z 3 / (me) V C 3 3 h C 3 Wm. G. Hoover, Computational Statistical Mechanics (Studies in Modern Thermodynamics, Elsevier, 99) p.7 8

29 + + + / dx dx dx exp( r ) + dx exp( x ) π d( C R )/ dr C R d( C R ) C R dr π / C Γ ( r ) dr C r exp( r ) C r exp ( / ) / C Γ + / C π / Γ + Γ( n) n! n n n Γ π 3 / C π / Γ( 3 / + ) 3 3 3! / e ! Γ +! Γ( 3 / + ) 3 ±! 3 ± / e 3 ± 3 3 e 3 / C 3 dr 9

30 3 / πe C 3 3 Γ ~ ( E) 3 / ~ 4πemE Γ( E) V 3h ~ 3 / ~ 3 4πem 3h 3 / E ( E de) E V de Γ( ) Γ + 3 / 3 4πem 3 / E V 3h! ( / e) 3 / 3 / 3 4πm 5 / E V Ω( E, V, ) e 3h 5 3 3h 3 E V S( E, V, ) k ln + k ln + k ln 4πm 5 3 3h S( E, V, ) R ( ln ) A ln 4πm 3 E V + R ln + R ln A 6. 3 h (Js) 5 R ln 3 3h 3 ln 4.458R + R ln M 4 m π ( A ) w 4.458R 37.66(JK - ) 3

31 canonical ensemble S( E, V, ) kln Ω( E, V, ) thermal reservoir subsystem tot tot tot res tot res tot res tot Ω E E Ω E res ( tot ) tot ( tot ) total system 3

32 Ω res ( Etot E ) f Ω E tot ( ) tot f { S ( E E )/ k} exp res tot exp { S ( E )/ k} tot tot S E S E + S E E ( ) ( ) ( ) tot tot res ( Etot E ) S ( E E ) S ( E E + E E ) res tot res tot S res tot S ( ) ( ) res E E E E S res ( Etot E) + tot E E S res ( Etot E) + T f exp{ β[ E TS( E) ] βe }, β kt Helmholtz βf βe f e e exp βf βe f e e canonical partition sum Z e E β β e F Z, F kt ln Z ( ) 3

33 canonical ensemble f βe e / βe e - Maxwell-Boltzmann distribution ν ν Eν Etot Eν ν! Ω( ν, ν, ) ν! ν! () () () () ( n) ( n) Ω Ω( ν, ν, ) + Ω( ν, ν, ) + + Ω( ν, ν, ) + S k ln Ω( ν, ν, ) 33

34 W! W!! n +! W () n W n! + n! n ( )( ) ( ( n ) ) ( + )( + ) ( + n ) W ln Wn ( ) lnw + ln( ) + ln( ) + + ln( ( n ) ) ln( + ) ln( + ) ln( + n ) n / lnw n ln W e ( ) W( n) W exp( n / ) n / x Stirling 34

35 ( )! / e W W ( / e)!! Stirling! e π W π /π lnw ln + ln ln π ln ln ln ln / ln ln ln ln ln Stirling Stirling S / k νln ν/ e νln ν / e νln ν / e δ ( S / k) ν + δν ln ν + δν ν lnν { ( ) ( ) } ( lnν + ) δν δν E δν ( ln ν ) δν L 35

36 ( lnν + α + βe ), ( lnν + α + βe ) ( ln ν + α + βe ) δν L lnν + α + βe, ν exp α βe ( ) f exp βe / exp βe ( ) ( ) E f E ln exp( βe ) β F F F E F TS F T T T V T T kt kt β, F kt ln ( ) exp βe kt F( T, V, ) kt ln Z( T, V, ) ln Z β 36

37 S( E, V, ) kln Ω( E, V, ), β Ω βe Z( T V, ) Z( ) ( E, V, ) e de βe Z T, V, Ω E, V, e ( ) ( ) Laplace transform Inverse Laplace transform c+ i c i pe Ω ( E) Ω( E V, ) Z( p) e dp, d E E E f ln Z( β ) dβ 37

38 ε ( p + p + p ) x y z m dxdydzdp xdpydpz dxdydzdp dp dp 3 / h x y z transl dxdydz dpxdp ydpz exp β px + p y + p z / 3 z [ ( ) m] h V 3 h πmkt [ ] 3 / π α / e αx dx ( α ) ( z ) transl Z z V 3 / transl πmkt 3!! h 3 / V πmkt F kt ln Z kt ln kt h E Z kt β ln 3 E F 5 V 3 π S k + k ln + ln mkt T h 38

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untitled 1 Physical Chemistry I (Basic Chemical Thermodynamics) [I] [II] [III] [IV] Introduction Energy(The First Law of Thermodynamics) Work Heat Capacity C p and C v Adiabatic Change Exact(=Perfect) Differential