s01/3 Calculate C pm for a) nitrogen, b) water vapor? Experiment: N 2 (300 K): J K 1 mol 1 H 2 O (500 K): J K 1 mol 1

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1 [siolant] /38 Pressure of ideal gas fro the kinetic theory I Molecule point ass olecules of asses i, i,.., in a cube of edge L Velocity of olecule i is v i v i,x, v i,y, v i,z After elastic reflection: v i,x v i,x y A olecule hits the sae wall again after tie t L/v i,x Force change of oentu in a tie unit Moentu P v Change of oentu P x i v i,x Averaged force caused by ipacts of one olecule: x L Equipartition principle exaple Calculate C p for a nitrogen, b water vapor? Experient: 3 K: 9. J K ol H O 5 K: 35. J K ol 5/38 9. J K ol, 33.6 J K ol F i,x P x iv i,x iv i,x t L/v i,x L Pressure force of all olecules, divided by the area F i,x i v i,x p L L 3 Kinetic energy of one olecule i v i iv i iv i,x + v i,y + v i,z Pressure of ideal gas fro the kinetic theory II Kinetic energy of gas internal energy onoatoic gas In other words Assuptions: E kin i v i 3 i v i,x p i v i,x L 3 E kin 3 V pv 3 E kin! n eperature is a easure of kinetic energy Pressure is a result of averaged ipacts of olecules We used the classical echanics Once ore: n, k B R A A U E kin 3n 3 k B, C V, 3 R /38 Microstate, acrostate, enseble, trajectory 6/38 icrostate state, configuration instantaneous snapshot at given tie quantu description: state eigenstate wave function classical description: state positions and velocities of all particles at given tie, r,..., r, v..., v acrostate averaged action of all icrostates enseble set of all icrostates with known probabilities trajectory record of a tie developent of a icrostate icrostate acrostate enseble trajectory in fact, oenta ore later. here are states, hence we work with their probability density ρ ρ r,..., r, p,..., p. Boltzann constant pv n 3/38 [tche/siolant+.sh] 7/38 Microcanonical enseble and ergodic hypothesis Microcanonical enseble enseble of icrostates in an isolated syste which has developed in tie for a long tie Also denoted as VE const, V const, E const k B R A n A J K Ludwig Eduard Boltzann credit: scienceworld.wolfra.co/biography/boltzann.htl Ergodic hypothesis quantu: i const W W # of states Ergodic hypothesis classical: trajectory covers the space with unifor probability In other words: ie average over a trajectory t X t li Xt t t enseble average X W for any quantity X X, where t naely: the phase state of { r,..., r, p..., p } Equipartition principle 4/38 Expression E kin is coposed of f 3 ters of the for iv i,k, where k {x, y, z}. pv f 3 k B 3 E kin f nuber of echanical degrees of freedo. Average energy contribution per one degree of freedo: E kin f k B Generalization: any quadratic function in the Hailtonian Heat capacity in olar units A : U Ekin C V V fk B A 3 R deg. of freedo per olec. Mean value in the icrocanonical enseble X W 8/38 Exaple. You win $5 if you throw on a dice, you loose $ if you throw anything else. What is your ean ected win in this gae? Whole therodynaics can be built on the top of the icrocanonical enseble. But for const it is uch easier. Linear olecules: + rotations, C V 5 R but: hydrogen onlinear olecules: + 3 rotations, C V 3R Vibrations classically: + for each incl. E pot iprecise!

2 We want const: Canonical enseble Also V const, V const, const Ergodic hypothesis: E E + E E + do not interact E probability of any state with energy E E E E + E + E E const E α i βe th Law β is epirical teperature α i is syste-dependent a noralizing const. so that Deterining β: onoatoic perfect gas, per ato U 3 k B U EE v E v d v v d v Evaluation gives: U 3 β β 9/38 Boltzann probability 3/38... or the first half of statistical therodynaics. Probability of finding a state with energy E is proportional to [ E const E ] const E Exaples: a reacting syste can overcoe the activation energy E with probability E Arrhenius forula k A E the energy needed for transfering a olecule fro liquid to gas is vap H per ole, probability of finding a olecule in vapor is proportional to vaph Clausius Clapeyron equation integrated [ p p vaph R ] const vaph Deterining β U [start /hoe/jiri/vyuka/aple/beta.w] + /38 R 3 v v d v R 3 v d v v x + v y + v z e βv xdv x e βv ydv y e βv z dv z 3 e βv xdv x e βv ydv y e βv z dv z v x e βv xdv x e βv ydv y e βv z dv z e βv xdv x e βv ydv y e βv z dv z 3 v x e βv xdv x 3 e βv xdv x β β β We have used the Gauss integral: e ax dx and its derivate by paraeter a: 3 β a where a β Baroetric forula [siolant -I] + 4/38... Boltzann probability once again Potential energy of a olecule in a hoogeneous gravitational field U pot gh. Probability of finding a olecule in height h: U pot gh Mgh Probability density pressure: p p Mgh he sae forula can be derived fro the condition of echanical equilibriu + ideal gas equation of state dp dhρg, ρ Mp Which leads to the Boltzann probability x e ax dx d da e ax dx d da a a a Deterining β + /38 v x e βv xdv x e βv xdv x β Boltzann probability [cd tche; blend -g butane] 5/38 Exaple Energy of the gauche conforation of butane is by E.9 kcal/ol higher than anti. Calculate the population of olecules which are in the gauche state at teperature 7.6 K boiling point. cal 4.84 J. Solution: gauche : anti [ E/].9 Don t forget that there are two gauche states! gauche + anti [ E/] [ E/] ote: we assued that both inia are well separated and their shapes are identical. Better forula would be with G instead of E Mean value in the canonical enseble Generalization of the ean value ectation value: XE Xe α βe Xe βe e βe /38 herodynaics Internal energy Its sall change is U E 6/38 Boltzann factor: e E/k B du de + d E Exaple. You win $5 if you throw on a dice, you loose $ if you throw anything else. However, you have drilled a sall lead weight under opposite to so that the probabilities are. and.6. What is your ean ected win in this gae? $. de: energy level changed d: probability of state changed herodynaics: du p dv + ds p dv: A piston oved by dx. Change in energy de echanical work Fdx F/A dax p dv p pressure of state, pressure p p. ds: Change [V] change of the population of states with varying energies heat

3 Boltzann equation for entropy [jkv pic/boltzannob.jpg] 7/38... or the nd half of the statistical therodynaics E α i βe E β [α i ln ] de d β [α i ln ] d ln β On coparing with ds: d ln S k B ln { /W for E E Microcanonical enseble: for E E Boltzann equation: S k B ln W Property: S + S + S k B lnw W k B lnw + Boltzann H-theore Second Law + /38 Feri golden rule for the transition probability caused by a perturbing Hailtonian H pert in an isolated syste: d W h H pert ρ final W W Change of the population of state aster equation: d W W W [ ] Rate of entropy change: ds k d ln k ln W [ ] rick: swap and su: ds k W [ln ln ][ ], he entropy of an isolated syste never decreases Loschi paradox: Irreversibility fro reversible icroscopic laws Exaple: Ideal solution Energies of neighbors: All configurations have the sae energy Mix olecules of + olecules of : W!! S k B ln W k B ln + ln S R x ln x + x ln x cf. S k B ln We used the Stirling forula, ln ln : per partes [ ] ln ln i ln x dx x ln x x ln More accurately: ln asypt. ln + ln /38 [traj/ice.sh] 9/38 Exaple: Residual entropy of crystals at Crystal: icrostate S k B ln 3rd Law 3rd Law violation: CO, O, H O. ot in the true equilibriu, but frozen because of high barriers Exaple : Entropy of a crystal of CO at K S k B ln A R ln Exaple : Entropy of ice at K S k B ln.57 A 3.4 J K ol Pauling derivation: 6 4 orientations of a water olecule then an H-bond is wrong with prob. A bonds in a ole S k B ln 6 A 3.37 J K ol A Maxwell Boltzann distribution of velocities he probability that a olecule is found in: /38 a tiny box dxdydz with coordinates in intervals [x, x + dx, [y, y + dy a [z, z + dz AD with velocities in intervals [v x, v x + dv x, [v y, v y + dv y, [v z, v z + dv z, is proportional to the Boltzann factor E pot + E kin Epot v x v y v z he probability that a olecule is found with velocities in intervals [v x, v x + dv x, [v y, v y + dv y, [v z, v z + dv z irrespective of E pot is proportional to v x v y v z [tche/mbfunkce.sh] Maxwell distribution historical approach + 3/38 Assuptions: is isotropic is coposed of independent contributions of coordinates, li v v x, v y, v z v x, v y, v z v x v y v z he only function satisfying these conditions is Exaples of functions: v x const const v x. x + y is isotropic, but is not a product, bad liit. x y is a product, is not isotropic, bad liit x + y is a product, is not isotropic, good liit 4. 3 x / y / good! Assuption: velocity is a su of any sall rando hits Central liit theore Gauss distribution Exaple: Inforation entropy of DA /38 Experiental verification 4/38 Assuing rando and equal distribution of base pairs. Doppler broadening of spectral lines Per one base pair: S k B ln 4, per ole: S R ln 4. Corresponding Gibbs energy at 37 C: G ln kj ol o be copared to: AP ADP standard: r G 3 kj ol in usual conditions in a cell: r G 57 kj ol olecular bea: Stern, Zartan 9: Pt wire covered by Ag a slit 3 screen λ λ v x λ c Laert 99 vapor of Bi or Hg? credit: a other literature: Sn oven credit:

4 [tche/mbe.sh] 5/38 Pseudoeriental verification and consequences oralized distribution in one coordinate: v v x x σ v σ, σ v v x k B v M Distribution of velocities, i.e., probability density that a particle is found with v v in interval [v, v + dv: v 4v v v v x v y v z σ 3 v σ v v x /s. -. tche/mbe.sh: - switch to VE: e - record: F - stop recording: F - end: ESC ESC 3 K 5 K K v x /.s - v/s v/.s - 3 K 5 K K Seiclassical partition function Integrals over positions and oenta are separated Integrals over oenta can be evaluated: p,x /k B After 3 integrations we get: Q Z Λ 3, de Broglie theral wavelength: Λ h kb Λ de Broglie wavelength at typical particle velocity at given requireent: Λ typical ato ato separation V/ /3 Configurational integral: Q [ βu r,..., r ] d r... d r Mean value of a static quantity observable: Q X r,..., r [ βu r,..., r ] d r... d r 9/38 do not confuse U int. energy and U r,... potential Consequences Mean velocity 8 8 8kB v v vdv σ v M Mean quadratic velocity v q v vdv 3 M 3kB [cd aple; xaple axwell.ws] 6/38 heral de Broglie wavelength liit of quantu/classical description Exaple 3/38 a Calculate Λ for heliu at K. b Copare to the typical distance of atos in liquid heliu density.5 g c 3. a 6. Å; b 3.8 Å Most probable velocity d dv v ax M Speed of sound κ C p /C V κ κkb v zvuk M kb credit: hight3ch.co/superfluid-liquid-heliu/ erodynaics finished α? S k B [α βe] k B α U then 7/38 Helholtz free energy α U S F F ln e βe [...] canonical partition function statistical su Q or Z Interpretation: nuber of accessible states low-energy states are easily accessible, high-energy states are not Fro F we can obtain all quantities df pdv Sd: p F U F + S V S F H U + pv G F + pv Seiclassical partition function Hailton foralis: positions of atos r i, oenta p i. E H E pot + E kin, E pot U r,..., r, E kin i Su over states replaced by integrals: F ln Z p i 8/38 Z e βe h 3 [ βh r, r,..., r, p,..., p ] d r d p where h h Planck constant. Why the factorial? Particles are indistinguishable... but appear in different quantu states Why Planck constant? Has the correct diension Z ust be diensionless We get the sae result for noninteracting quantu particles in a box vide infra Seiclassical onoatoic ideal gas Q [] d r... d r d r d r V V V Q Z Λ 3 V Λ 3 V e Λ 3, F p V µ And verification: U F + S F F k B ln,v F k B ln Z ln Ve Λ 3 k B n V V F 3k B V Λ 3 V ln pλ 3 G F + pv ln Λ3 Ve + k B µ 3/38 e Euler nuber e ele. charge Quantu onoatoic ideal gas + 3/38 Eigenvalues of energy of a point ass in a a b c box: E h n x 8 a + n y b + n z c Assuption: high enough teperature so that a few particles copete for the sae quantu state it does not atter whether we have ferions or bosons. Equivalently, Λ distance between particles. Partition function: Z βe βe dn x dn y dn z V n x n y n Λ 3 z E E i Z Z Yes, it is the sae! he choice of factor /h 3 in the seiclassical Z was correct.

5 Isobaric enseble: p const + 33/38 he sae arguent now applied to V as for E before: V + V + V V V ogether: α i βe γv γ is a universal property of a barostat to be deterined fro ideal gas Ve βe kine γv dvd r V... d p V + e γv dv e βe kine γv dvd r... d p V e γv dv + γ ricks used: d p... d p gives the sae factor, Λ 3 d r... d r V correctly: XdVd r... d r [ V... V Xd r... d r ]dv the order of integration is iportant dv is the last alternatively by substituting V /3 ξ i r i one gets d r... d r V dξ... dξ then the integration order is irrelevant V e γv dv /γ + recursively per partes Grandcanonical enseble: µ const contd. + 37/38 he ectation value of X in the grandcanonical enseble is a Xe βe d r... d p a e βe d r... d p, where a e βµ. he last eq. is usually integrated over oenta incl. internal degrees of freedo; for X X, r it holds: a Xe βe d r... d r, where a e βe d r... d r a aq Λ 3 V should be equal to /p at liit γ p/. Isobaric enseble: p const contd. + 34/38 oralization constant α fro the Boltzann eq. and e α /Z p : S k B [α βe γv] k B α U p V ln Z p U S + p V G where βp guarantees diension Z p βp h 3 e βe+pv d r... d p dv hen V /p ln Zp β V + G p p or V k B p p Exaple: Verify the last eq. by differentiating ln Z p by p directly he ectation value of X in the isobaric enseble is Cheical potential + 38/38 he quotient in series can be ressed as a Λ 3 eβµres ρ, ρ V, µ res µ µ id ρ where µ res is the residual cheical potential cheical potential wrt the standard state of ideal gas at given teperature and volue density, which can be copared with tables after pressure is recalculated fro p st isong the ideal gas equation of state Xe βe+pv dvd r... d p Z p Grandcanonical enseble: µ const + 35/38 he sae arguent now applied to as for E, V before: + + ogether: α i βe + δ δ is a universal property of a source of particles to be deterined fro ideal gas V h 3 e βe kine δ d r... d p h 3 e βe kine δ d r... d p ricks used: d p... d p Λ 3 d r... d r V id. Λ 3eδ V V Λ 3eδ Λ 3eδ x e x, where x V Λ 3eδ derivative by x: x e x x xe x On coparing with βµ id,point particle V/Λ 3 one gets δ βµ. Grandcanonical enseble: µ const contd. + 36/38 oralization constant α fro the Boltzann eq. and e α /Z µv : S k B [α βe + δ] k B α U + µ ln Z µv U S µ Ω where Z µv e βµ h 3 e βe d r d p Grandcanonical potential Ω F µ F G pv df Sd pdv + µd dω Sd pdv dµ