Ideal KMI± Molecules freely rotating, vibrating, Separate : E = Eln

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1 Ideal 1 KMI± Molecules freely rotating vibrating Separate : E EhanstEwttEvibEEeeazeEmaopE@EarPEeiEuibptae Q Eln!

2 Tetrode Show equations of otnonicsl ensemble 2 trance 1 qtt Emms Yns ^ah#a UH U n ) ZNKBT PV NKBT srhktf] Initiate Sacku EE sh1r enlejfsvofenlejfsct ] lnkf5k ' You choose the standard state! molecule Thy m#he via With these tools cos get at the ideal heat apseity of Any molecule cjge gas

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6 nrt[ nrt[lnv+3zlnt 3ha+1 lnv ] 6 ' coast ]

7 mcoh What ' Asides : 7 Using Stott so kle+eu4sti ) can show that where m is the molecular MASS And h Planck 's constant Saekur Tetrode e u n)cnrl~ +NRln v + Ns S klnsl { position DOFS D8 Fs momentum Are the " microstates to court in on ideal gas? Eton molecules specified by } Pwr position ( wlu scalp And momentum Gestes w/t ) from yt distribution p " ) The two EOS 's wild stove have been inferred kinetic t theory the Boltzmann And S from stat meds

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12 KBT qi %3 µ KI KBT lnfth ) luta %) KBT ln (hs RIP ) ln ( k3 Fo) + kpilnplp ugp)m GP )+kbtl~pf pilnqtp ' ) KBT 1 h% '

13 @!%ifer µ

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15 Dme 140 Fermi verily strthb Was apply

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17 Advanced Thermodynamics ND CBE Table 1: Statistical Thermodynamics of an Ideal Gas Translational DOFs 3D particle in a box model trans 2 h 2 1/2 2mL 2 h k B 2 m For T>> trans << L q trans V/ 3 (essentially always true)! U trans 3 2 RT C vtrans 3 2 R S trans R ln e5/2 V N 3 R ln e5/2 k B T P 3 Rotational DOFs Rigid rotor model Linear molecule rot hcb/k B! q rot 1 X 1 (2l + 1)e l(l+1) rot/t 1 T 1 unsymmetric T >> rot rot 2 symmetric l0 U rot RT C vrot R S rot R(1 ln( rot /T )) Nonlinear molecule rot hcb /k B q rot 1 T 3 rot rot rot U rot 3 2 RT C vrot 3 2 R S rot R 2 Vibrational DOFs Harmonic oscillator model Single harmonic mode vib h /k B 1/2 T >> rot rotational symmetry number 3 ln rot rot rot T 3 q vib 1 1 e vib/t T vib T >> vib U vib C vvib S vibi! 2 vib R e R vib e vib/2t vib /T vib/t 1 T e R vib/t 1 e ln(1 e vib/t ) vib/t 1 Multiple harmonic modes vibi h i /k B q vib Y i 1 1 e vibi/t R X i U vib C vvib S vibi vibi e R X! 2 vibi e vibi/2t vibi /T vibi/t 1 T e R vibi/t 1 e ln(1 e vibi/t ) vibi/t 1 i Electronic DOFs q elec spinmultiplicity January c 2012 W F Schneider 1

18 Pt +Ekx ' 170 Note that in high T limit each contributes some factor of RT to total DOF energy In dsssicsl limit that energies Are continuous on uee E P Eee p q position p CAR show if write Kite momentum tpittsdpdg gettpiefdpdq energy depends quadrabdcslly on p And( or each such DOF contributes YZRT to U translational : Paines 'ERT " KRT uibmbonsl ' g RT Called law of known pre QM equiptrtiwon was very well

19 Each ( 180 Apply ideas to a solid a a e No µ scopic trans a ' On only vibrations pot n 2 a g / Assume each Atom is A harmonic oscillator w/ same spring 3RT to energy constant ( 3 uib Dots ) Known to law of Dulong t Petitt contributes p 3R 3rd fifo high T ftdls at low Einstein realized that if these Are QM oscillators then must use of :b Qto+ ( qis )3N 1+ ish Assuming Note these Are distinguishable Ouib same for All

20 Why Think 190 really should model As bosons Euis a 3R(a ye&i R 54 L ( 1 e Qist ) ' U 3 Nevis Mu± closer to observation Properly gives Cpo As Too Not quite right functions form At low T though? Shouldn't think of these ss individual oscillators of as waves of different wavelength / frequency MM need to cslculsbe density of these qcv ) gwsdv idu Consider stl frequencies such that ST gfo 0 ) du 3N Voebxe o h p a EEIS

21 ktb( ( Other ensembles TT 1 PjYr(uµn ) ucanonnal VN TIN pje4%eun TEETH pjluj u ) EYPE "f/dtpn ) Gibbs isobaric / GLTPn) gpnj) isothermal LKTP n)fqhµn)ep%dv pjlu; n)eypemnp/( TV ) µ grand canonical ktµ HIV µ) QHun)enH

22 grand Langmuir adsorption How many adsorbate Adsorbate at a surface at a given T tµ? occupy sites + An sinaru site{ internal does of A QCT AN ) labor q? Acwnitrous A! Ale?_ n )! " Often want to know N(µ D canonical LT A µ ) n ) enmp AEOQHA ( A E ee%nt aau l + qsie EMP ) " fbiwmiaf Ak U[ G Am) tl ( 1 tqae# )

23 HE N ) Xesiee gas tak6ti@uep ideal N o F nesertir Langmuir isotherm eheup Esietheup :3: x: E n to some external reservoir? we want to relate µ ( Tip) µ (tp ) + ktln%o Itg// ' attendees puts two systems osnagaemeenogy

24 " EH eoep EMOP ( % ) q!tt ( % ) eaep 230 Define KG ) Eqsle E lkm x O KCT ) % * T ) % Equilibrium constant is ratio of partition functions times a factor to account for energy offset " standard