Monatomic ideal gas: partition functions and equation of state.

Transcription

1 Monatomic ideal gas: partition functions and equation of state. Peter Košovan Dept. of Physical and Macromolecular Chemistry Statistical Thermodynamics, MC260P105, Lecture 3, If you find a mistake, kindly report it to the author :-)

2 Systems of non-interacting particles For non-interacting particles a, b, c, d,... we can decompose the system Hamiltonian to individual contributions: Ĥ Ĥa + Ĥb + Ĥc + Ĥd +... Consequently, the energy is a sum of individual particle energies E ε a + ε b + ε c + ε d +... Complex many-body problem approximated as a superposition of simpler one-body and two-body problems. Often encountered in Physics Allows for systematic corrections e. g. Separation of a single molecule Hamiltonian Ĥ Ĥtranslation + Ĥrotation + Ĥvibration + Ĥelectronic + Ĥnuclear +... ε ε translation + ε rotation + ε vibration + ε electronic + ε nuclear +... P. Košovan Lecture 3: Monatomic ideal gas 1/24

3 Systems of non-interacting particles Suppose we can decompose Hamiltonian to individual contributions. Non-interacting distinguishable particles a, b, c,... : Q(N, V, T ) = e βe j = e β(εa k +εb l +εc m +... ) j k,l,m,... ( )( )( ) = e βεa k e βεb l e βεc m = q a q b q c... k k, l, m,... are indices of (quantum) states a, b, c,... are inidces of particles Single-particle partition function: q i (V, T ) = j l e βεi j m P. Košovan Lecture 3: Monatomic ideal gas 2/24

4 Distinguishable vs. indistinguishable particles If all particles are the same but distinguishable, q a = q b = q c = q : Q(N, V, T ) = e β(εa k +εb l +εc m +... ) = q a q b q c = q N (V, T ) k,l,m,... The N-body problem reduces to a one-body problem! P. Košovan Lecture 3: Monatomic ideal gas 3/24

5 Distinguishable vs. indistinguishable particles If all particles are the same but distinguishable, q a = q b = q c = q : Q(N, V, T ) = e β(εa k +εb l +εc m +... ) = q a q b q c = q N (V, T ) k,l,m,... The N-body problem reduces to a one-body problem! But identical particles are indistinguishable: Q(N, V, T ) = k,l,m,... e β(ε k+ε l +ε m+... ) Permutation of k and l yields the same quantum state of the system. Correct for double-counting: N! permutations. P. Košovan Lecture 3: Monatomic ideal gas 3/24

6 Indistinguishable particles For indistinguishable particles: Q(N, V, T ) = qn (V, T ) N! where q(v, T ) = j e βε j P. Košovan Lecture 3: Monatomic ideal gas 4/24

7 Indistinguishable particles For indistinguishable particles: Q(N, V, T ) = qn (V, T ) N! where q(v, T ) = j e βε j We silently ignored multiply occupied states: ε i = ε j : P. Košovan Lecture 3: Monatomic ideal gas 4/24

8 Indistinguishable particles For indistinguishable particles: Q(N, V, T ) = qn (V, T ) N! where q(v, T ) = j e βε j We silently ignored multiply occupied states: ε i = ε j : Bosons: ψ symmetric with respect to permutation any number of particles can occupy a given quantum state. P. Košovan Lecture 3: Monatomic ideal gas 4/24

9 Indistinguishable particles For indistinguishable particles: Q(N, V, T ) = qn (V, T ) N! where q(v, T ) = j e βε j We silently ignored multiply occupied states: ε i = ε j : Bosons: ψ symmetric with respect to permutation any number of particles can occupy a given quantum state. Fermions: ψ antisymmetric with respect to permutation two particles cannot occupy the same quantum state. P. Košovan Lecture 3: Monatomic ideal gas 4/24

10 Indistinguishable particles For indistinguishable particles: Q(N, V, T ) = qn (V, T ) N! where q(v, T ) = j e βε j We silently ignored multiply occupied states: ε i = ε j : Bosons: ψ symmetric with respect to permutation any number of particles can occupy a given quantum state. Fermions: ψ antisymmetric with respect to permutation two particles cannot occupy the same quantum state. We will show later that except for low temperatures or high densities the number of available states N N. In such cases multiply occupied states are rare and the equation above is an excellent approximation P. Košovan Lecture 3: Monatomic ideal gas 4/24

11 Some useful results from quantum mechanics (QM) Time-independent Schrödinger equation: Ĥψ = Eψ Particle in a well of length a: 2 Ĥ = h2 2m x 2 ε n = h2 n 2 8ma 2 n = 1, 2,... Particle in a 3d box of length a: ε nx,n y,n z = h2 8ma 2 (n2 x + n 2 y + n 2 z) n y n x Number of states with ε nx,n y,n z < ε: ( ) 4πR 3 Φ(ε) = = π ( 8ma 2 ε 6 h 2 ) 3/2 P. Košovan Lecture 3: Monatomic ideal gas 5/24

12 Degeneracy of translational energy levels Density of energy levels from previous slide: Φ(ε) = π 6 ( 8ma 2 ε h 2 ) 3/2 Number of levels between ε and (ε + ε): ω(ε, ε) = Φ(ε + ε) Φ(ε) = π 4 ( ) 8ma 2 3/2 ε 1/2 ε + O(( ε) 2) Let s plug in some numbers: ε = 3k B T/2, T = 300 K, m = g, a = 10 cm, ε/ε = 0.01: ω(ε, ε) O(10 28 ) Degeneracy of translational energy levels far exceeds the number of particles at sufficiently low density and high temperature. P. Košovan Lecture 3: Monatomic ideal gas 6/24 h 2

13 Ideal monatomic gas At high temperature and low pressure (Boltzmann limit): Q(N, V, T ) = qn (V, T ) N! Further decomposition of q: q(v, T ) = q tr q el q nucl Nuclear partition function: q nucl = ω n Except for rare cases just a constant. P. Košovan Lecture 3: Monatomic ideal gas 7/24

14 Ideal monatomic gas At high temperature and low pressure (Boltzmann limit): Q(N, V, T ) = qn (V, T ) N! Further decomposition of q: q(v, T ) = q tr q el q nucl Nuclear partition function: q nucl = ω n Except for rare cases just a constant. Electronic partition function: q elec = ω e1 + ω e2 e β ε 1, Zero energy at electronic ground state Can be degenerate Higher excited states typically several ev At 300 K : 1eV 40 k B T First two or three terms usually suffice P. Košovan Lecture 3: Monatomic ideal gas 7/24

15 Quantitative view of electronic levels Table from McQuarrie, Statistical Mechanics, University Science Books (2000) P. Košovan Lecture 3: Monatomic ideal gas 8/24

16 Quantitative view of electronic levels Fraction of electrons in second excited state: f 2 = ω e2 e β ε 1,2 ω e1 + ω e2 e β ε 1, Table from McQuarrie, Statistical Mechanics, University Science Books (2000) Higher electronic states need to be considered at higher temperatures. P. Košovan Lecture 3: Monatomic ideal gas 9/24

17 Translational partition function Translational energy levels: ε nx,n y,n z = q tr = n x,n y,n z=1 h2 8ma 2 (n2 x + n 2 y + n 2 z), n x, n y, n z = 1, 2,... e βεnx,ny,nz ( βh 2 n 2 ) ( x βh 2 n 2 ) ( x βh 2 n 2 ) x = exp 8ma 2 exp 8ma 2 exp 8ma 2 n x=1 n x=1 n x=1 ( ( βh 2 n 2 ) ) 3 ( ( βh 2 n 2 ) ) 3 = exp 8ma 2 8ma 2 dn n=1 Alternative approach q tr = ω(ε)e βε exp n=0 ( ) 8ma 2 3/2 ε 1/2 e βε dε n=0 ω(ε)e βε dε = π ε=0 n=0 4 h 2 P. Košovan Lecture 3: Monatomic ideal gas 10/24

18 Translational partition function We evaluate ( q tr = exp n=0 ( βh 2 n 2 8ma 2 ) 3 ( ) 2πmkB T 3/2 dn) = h 2 V and introduce thermal de Broglie wavelength Λ ( h 2 ) 1/2 Λ = such that q tr = V 2πmk B T Λ 3 Translational energy and momentum per particle: ( ) ln ε tr = k B T 2 qtr = 3 T 2 k BT, p = (3mk B T ) 1/2, Λ h p P. Košovan Lecture 3: Monatomic ideal gas 11/24

19 Translational partition function We evaluate ( q tr = exp n=0 ( βh 2 n 2 8ma 2 ) 3 ( ) 2πmkB T 3/2 dn) = h 2 V and introduce thermal de Broglie wavelength Λ ( h 2 ) 1/2 Λ = such that q tr = V 2πmk B T Λ 3 Translational energy and momentum per particle: ( ) ln ε tr = k B T 2 qtr = 3 T 2 k BT, p = (3mk B T ) 1/2, Λ h p Boltzmann applies at Λ 3 V, i. e. when quantum effects diminish. P. Košovan Lecture 3: Monatomic ideal gas 11/24

20 Summary single-particle partition function q(v, T ) = q tr q el q nu ( 2πmkB T q tr = h 2 ) 3/2 V = V λ 3 q el = ω e1 + ω e2 e β ε q nu = ω n The above holds for particles without internal structure (monatomic ideal gas) Vibrational and rotational terms arise for composite particles (next lecture) P. Košovan Lecture 3: Monatomic ideal gas 12/24

21 Thermodynamic functions of an ideal monatomic gas A(N, V, T ) = k B T ln Q = k B T ln N! Nk B T ln q(v, T ) [ (2πmkB ) ] T 3/2 V e = Nk B T ln h 2 Nk B T ln ( ω e,1 + ω e,2 e β ε ) 1,2 N ( ) ln Q E = k B T 2 T N,V In both above cases, the first term is dominant. ( ) ln Q p = k B T = Nk BT V V N,T = 3 2 Nk BT + Nω e,2 ε 1,2 e β ε 1,2 q el + Equation of state (EOS) of an ideal gas. P. Košovan Lecture 3: Monatomic ideal gas 13/24

22 Thermodynamic functions of an ideal monatomic gas Entropy: ( ) ln Q S = k B T T N,V = 3 2 Nk B + Nk B ln + k B ln Q [ (2πmkB T h 2 ) ] 3/2 V e N + Nk B ln ( ω e,1 + ω e,2 e β ε ) ( ωe,1 + ω 1,2 e,2 e β ε ) 1,2 + Nk B Sackur-Tetrode equation: [ (2πmkB ) ] T 3/2 V e 5/2 S = Nk B ln h 2 + S el N q el P. Košovan Lecture 3: Monatomic ideal gas 14/24

23 Quantitative view of entropy Sackur-Tetrode equation: [ (2πmkB ) ] T 3/2 V e 5/2 S = Nk B ln h 2 + S el N Table from McQuarrie, Statistical Mechanics, University Science Books (2000) Higher electronic states have to be considered at higher temperatures. P. Košovan Lecture 3: Monatomic ideal gas 15/24

24 Thermodynamic functions of an ideal monatomic gas Chemical potential: ( ) ln Q µ = k B T = k B T ln q N T,V N [ (2πmkB ) ] T 3/2 V = k B T ln h 2 k B T ln q el q nu N [ (2πmkB ) ] T 3/2 k B T = k B T ln h 2 p k B T ln q el q nu + k B T ln p p where = µ 0 (T ) + k B T ln p p µ 0 (T ) = k B T ln [ (2πmkB T h 2 ) 3/2 k B T p ] k B T ln q el q nu P. Košovan Lecture 3: Monatomic ideal gas 16/24

25 Summary Separation of different contributions to Ĥ, E, Q and q Fermi-Dirac and Bose-Einstein statistics Boltzmann as high-temperature and low-density limit Thermal wavelength EOS of an ideal gas from first principles (Boltzmann limit) P. Košovan Lecture 3: Monatomic ideal gas 17/24

26 Summary Separation of different contributions to Ĥ, E, Q and q Fermi-Dirac and Bose-Einstein statistics Boltzmann as high-temperature and low-density limit Thermal wavelength EOS of an ideal gas from first principles (Boltzmann limit) What comes next Multiply occupied states Fermi-Dirac, Bose-Einstein vs. Boltzmann statistics Polyatomic gas molecules vibration and rotation Condensed matter dense gases and liquids (classical limit) Distinguishable particles crystals, magnets P. Košovan Lecture 3: Monatomic ideal gas 17/24

27 Fermi-Dirac and Bose-Einstein statistics Sometimes we have to explicitly account for multiple occupancy of quantum states At low T and high density Partition function in terms of molecular energy levels (occupation numbers n k ): Q(N, V, T ) = j e βe j = {n k } e β i ε in i Evaluation of the restricted sum is awkward. Convenient solution: grandcanonical ensemble (GCE). *Constrained sum: N = k E j = k n k ε k n k P. Košovan Lecture 3: Monatomic ideal gas 18/24

28 Fermi-Dirac and Bose-Einstein statistics Formulation in GCE: Ξ(µ, V, T ) = e βµn Q(N, V, T ) N=0 Absolute activity: λ = e βµ µ = k B T ln λ Rearrangements (* is the restricted sum from the previous slide): Ξ(µ, V, T ) = e β i ε in i = λ i n i e β i ε in i λ N N=0 {n k } ( = N=0 {n k } k ) λe βε nk k N=0 {n k } Crucial step (now each n k ranges over all possible values): Ξ(µ, V, T ) = n max 1 n 1 =0 n max 2 n 2 =0 ( k ) λe βε nk k P. Košovan Lecture 3: Monatomic ideal gas 19/24

29 Fermi-Dirac and Bose-Einstein statistics We continue rearranging Ξ(µ, V, T ) = = n max 1 n max 2 n 1 =0 n 2 =0 = k 1 n max n 1 =0 ( λe βε 1 k n max n k =0 ( k ) λe βε nk k 2 ) n1 n max n 2 =0 ( ) λe βε nk k ( ) λe βε n2 2 Last equation is a general result. Special cases: Fermions, n max k = 1: Ξ FD = ( ) 1 + λe βε k k Bosons, n max k = : Ξ BE = ( λe βε k k n k =0 P. Košovan Lecture 3: Monatomic ideal gas 20/24 ) nk

30 Fermi-Dirac and Bose-Einstein statistics Use j=0 xj = (1 x) 1 for x < 1 to rewrite BE formula Ξ BE = k ( ) λe βε nk ( 1 k = 1 λe k) βε for λe βε k < 1 n k =0 and cast both FD and BE formula in the same form ( ±1 1 ± λe k) βε for λe βε k < 1 Ξ F D BE = k These are the only exact distributions in statmech. k Low T and high density: FD or BE statistics apply, q(n, V, T ) has no meaning (symmetry requirements of the wave function) High T and low density: both degenerate to Boltzmann statistics Nobel prize in Physics 2001: experiment Bose-Einstein condensate. P. Košovan Lecture 3: Monatomic ideal gas 21/24

31 Thermodynamic quantities from FD and BE statistics Number of particles: N = N = ( ) ln Ξ n k = k B T µ k V,T ( ) ln Ξ = λ = λe βε k λ V,T 1 ± λe βε k k Energy E and average energy per particle, ε: E = Nε = k n k ε k = k pv = k B T ln Ξ = ±k B T k λε k e βε k 1 ± λe βε k ( ) ln 1 ± λe βε k Number of particles in state k: λe βε k n k = 1 ± λe βε k P. Košovan Lecture 3: Monatomic ideal gas 22/24

32 The limiting case of Boltzmann For λ 1 (equivalent βµ 0): λe βε k n k = 1 ± λe βε k we can write approximate relations n k = λe βε k, N = k and using q(n, V, T ) = k e βε k n k N = n k = λ k e βεk k e βε k = e βεk q(n, V, T ) e βε k i. e. both FD and BE reduce to Boltzmann statistics. P. Košovan Lecture 3: Monatomic ideal gas 23/24

33 The limiting case of Boltzmann Similarly, for small λ we obtain: E = k λε k e βε k 1 ± λe βε k j Using ln(1 + x) x for small x: so that pv = k B T ln Ξ ( ±k B T )( ±λ j βpv = ln Ξ = λq Ξ = e λq = (λq) N = N! N=0 λε j e βε j, ε = E k n k e βε j j ε je βε j j e βε j ) = λk B T e βε j = λk B T q j λ N Q(N, V, T ), where Q(N, V, T ) = qn N! N=0 P. Košovan Lecture 3: Monatomic ideal gas 24/24