1. Connection between QM and Thermodynamics

Transcription

1 Calculaton of hermodynamc and Knetc Propertes 1. Connecton between QM and hermodynamcs We have focused to ths pont on the many approaches and detals of calculatng the nternal tronc energy of a sngle molecule, that s, the energy assocated wth takng nfntely separated consttuent nucle and trons at rest and formng a molecule: H + + O e H O E We typcally calculate E wthn the orn-oppenhemer approxmaton,.e. wthn the approxmaton that the nucle of the molecule are all fxed n space. We know that even at 0 K molecules have a zeropont vbratonal energy (ZPE), and the 0 K nternal energy of a molecule s thus: E 0 = E + ZPE ZPE can be calculated pretty relably wthn the harmonc approxmaton, accordng to 3n 6 1 ZPE = h ν = 1 where ν are the harmonc vbratonal frequences, obtaned from a vbratonal frequency analyss. E s not drectly observable; U 0 n prncple s. U 0 s the mnmum energy a molecule can have. Hgher energes are possble by tronc exctaton, or vbratonal exctaton, or due to rotatonal moton, or translatonal moton. If we assume that they are separable: E = E 0 + E + E vb + E rot + E trans o fully descrbe mcroscopc state of a molecule, we would have to specfy all these quanttes. Macroscopcally, though, what we are typcally nterested n s the equlbrum value of nternal energy, U (or some other thermodynamc quantty, lke S or H) at some specfed, externally mposed thermodynamc condtons, lke temperature or pressure P. hese equlbrum values are obtaned by averagng over all the possble states of an ensemble of molecules. he way ths average s constructed s the realm of statstcal thermodynamcs. Most mportant for us s the canoncal ensemble, n whch the free varables are,, and. We ll generally choose to be 1 mole, so we can talk about molar values of quanttes. he relatve probablty of a molecule beng n a partcular state E above E 0 n the canoncal ensemble s gven by the oltzmann factor, E E/ k P e = e he sum over all the states s the canoncal partton functon: E Q (,, ) = e Good news: all thermodynamc quanttes can be defned n terms of ths functon (see table). ad news: not so easy to calculate for some general collecton of molecules, lke a lqud, where E runs over all the states of all the molecules. Prof. W. F. Schneder CE Computatonal Chemstry 1/8 1:17:59 PM 11/8/007 Unversty of otre Dame

2 Calculaton of hermodynamc and Knetc Propertes he Equatons of the Canoncal Ensemble (,, ndependent varables) OE: All energes are referenced to E 0. = 1 k Sngle partcle partton functon Full Canoncal Ensemble Dstngushable Partcles (e.g., atoms n a crystal lattce E (,, ) = e q (, ) = e ε ( ) Q Indstngushable Partcles (e.g., molecules n a flud), q = e ε Full partton functon E (,, ) = e Q= q(, ) q (, ) Q Q =! Log partton functon ln Q ln q ln q ln! ( ln ln 1) q + Internal Energy (U) ln Q ln q ln q Pressure (P) 1 lnq ln q ln q Helmholtz Energy (A = U S) ln Q ( ln q + 1 ) ln q Entropy (S) k ( U + ln Q) k ( U ln q) k + { U + ( ln q + 1 )} Enthalpy (H = U + P) ln Q U + ln q U + ln q U + Gbbs Energy (G = A + P) ln Q ln Q + ln q ln q+ ln q + 1 ln q + Chemcal Potental A μ = 1 lnq ln q ln q Prof. W. F. Schneder CE Computatonal Chemstry /8 1:17:59 PM 11/8/007 Unversty of otre Dame

3 Calculaton of hermodynamc and Knetc Propertes. Ideal Gas hermodynamcs For a gas of dentcal molecules, though, the partton functon can be (nearly exactly) factored: (, ) = = ε q Q (,, ), q (, ) e! q(,) s the molecular partton functon, and s calculated by summng over the nternal energy states ε of an ndvdual molecule (startng at E 0 ). Further smplfed by factorng nto contrbutons from varous (3) degrees of freedom: trans rot vb q (, ) e ε e ε e ε e ε = trans rot vb = q q q q trans rot vb = U E Utrans Urot Uvb U Smlarly for other thermodynamc quanttes. For example, U C = = C + C + C + C v v, trans v, rot v, vb v, Have to separately calculate contrbutons from each DOF: ranslatonal 3 degrees of freedom. Assume partcle-n-a-box. Δε trans 0 q 10 trans 30 Imples P = nr. ote we have to choose a standard pressure/concentraton here! Gaussan chooses by default 1 atm as pressure. Rotatonal construct sum for each of 3 rotatonal degrees of freedom. Assume rgd rotor, means we need moments of nerta from molecular geometry. Δε rot e q 100' s rot bratonal construct sum for each of 3 6 normal modes wthn harmonc approxmaton. Means we need vbratonal spectrum. Δε vb 0. 1 e q 1 Electronc sum over excted tronc states. At usual temperatures, only ground state s mportant. Δ ε 1 e q = degeneracy of ground state vb Prof. W. F. Schneder CE Computatonal Chemstry 3/8 1:17:59 PM 11/8/007 Unversty of otre Dame

4 Calculaton of hermodynamc and Knetc Propertes he Statstcal hermodynamcs of an Ideal Gas All energes referenced to ther 0 K values All thermodynamc quanttes expressed on a per mole bass ranslatonal DOFs 3-D partcle n a box model 1 hermal wavelength Λ= h π m For Λ << L, k 1 qtrans = 3 = (essentally always true) 3 Λ P Λ e e k Utrans = R Cv,trans = R Strans = Rln Rln 3 = 3 Λ PΛ Rotatonal DOFs rgd rotor model Lnear molecule rotatonal temperatureθ R = hc k ( ) ( ) 1 θr 1 J J+ 1 1,unsymmetrc qrot = J 1 e, θr σ σ + = σ θr,symmetrc J = 0 ( ( σθr )) Urot = R Cv,rot = R Srot = R 1 ln on-lnear molecule rotatonal temperatures θ = hc 1 1 π 3 qrot, θac,, σ = rotatonalsymmetry number σ θaθθc 3 3 R σθaθθ C Urot = R Cv,rot = R Srot = 3 ln 3 π bratonal DOFs harmonc oscllator model Sngle harmonc mode vbratonal temperature θ = hc ν k 1 qvb =, θ (generally must use full form) 1 e θ θ θ e θ v θ θ θ θ θ θ Uvb = R C,vb = R Svb = R ln ( 1 e ) e 1 e 1 e 1 Multple harmonc modes vbratonal temperatures θv = hc ν v k 1 qvb = e θ v 1 v θ v v v e v θ v v θv θv v e v e v e θ θ θ θv Uvb = R C,vb = R Svb = R ln ( 1 e ) Electronc DOFs q spn multplcty Prof. W. F. Schneder CE Computatonal Chemstry 4/8 1:17:59 PM 11/8/007 Unversty of otre Dame α α k

5 Calculaton of hermodynamc and Knetc Propertes 3. Gaussan thermochemstry output bratonal frequency calculaton wthn Gaussan calculates all these quanttes for you. See Secton 3 of Ochtersk, hermochemstry n Gaussan, for descrpton. 4. Reacton thermodynamcs Often nterested n changes n thermodynamc quanttes accompanyng a chemcal reacton. Have to calculate dfferences: Δ U (, ) =Δ E +Δ ZPE +Δ Utrans (, ) +Δ Urot ( ) +ΔUvb ( ) Δ S (, ) =Δ S (, ) +Δ S ( ) +ΔS ( ) trans rot vb Can do ths for any condton you want. If you want a partcular standard state (e.g. 73 K, 1 bar), have to choose densty/ approprately. A {R} E PES ΔE ΔU(0) Equlbrum constants, determned by. K ( ) e ΔG (, ) =. Correspondng concentraton unts and standard state See Secton 4 of Ochtersk, hermochemstry n Gaussan, for descrpton. 5. Mxng tronc structure and expermental thermochemstry Sometmes want to relate to some expermental standard, lke heat (enthalpy) of formaton. Recall that ths s an enthalpy relatve to elemental speces, and these are not always easy to calculate wth a partcular method. Use a thermodynamc cycle, and combne expermental and calculated quanttes as necessary. 0 ote that Δ H f = 0 for the elements n ther most stable state at all temperatures by defnton, but that does not mean that the enthalpy of an element s not temperature dependent. Can calculate or measure ths quantty dependng on convenence. Same approach can be used for S, G. Prof. W. F. Schneder CE Computatonal Chemstry 5/8 1:17:59 PM 11/8/007 Unversty of otre Dame

6 Calculaton of hermodynamc and Knetc Propertes Sample thermodynamc cycle for calculatng the heat of formaton of methane: C (s, 98 K, 1 bar) ΔH 0 f (98 K) + H (g, 98 K, 1 bar) CH 4 (g, 98 K, 1 bar) 0 0 H (98K) H (0 K) Exp t or calc. C (s, 0 K, 1 bar) 0 0 H (98K) H (0 K) Exp t or calc. 0 (0 K) ΔH f Exp t or calc. H (g, 0 K, 1 bar) CH 4 (g, 0 K, 1 bar) ZPE ZPE C (g, 0 K, 1 bar) + H (0 K, rgd) ΔE calc. CH 4 (0 K, rgd) 6. Isodesmc reactons Ultmately relablty depends on ablty to calculate tronc energy dfferences, and these can be dffcult, especally for reactons that nvolve sgnfcant changes n tronc structure. Isodesmc reacton s a lttle trck of the trade that takes advantage of fact that correlaton errors are often constant for a gven bond type. Wrte unknown compound n terms of reactons that conserve bond types, for example: CH 4 + CH Cl CHCl 3 Uses CH F + HCHO CH 4 + COF reacton and known nformaton about frst three molecules to calculate thermodynamcs of the fourth. Prof. W. F. Schneder CE Computatonal Chemstry 6/8 1:17:59 PM 11/8/007 Unversty of otre Dame

7 Calculaton of hermodynamc and Knetc Propertes 7. Reacton Knetcs Reacton rates more challengng to predct than thermodynamcs the latter s a unversal property of a reacton for gven condtons, the former depends on the pathway(s) from reactants to products. Conceve of a reacton as happenng n a sequence of elementary steps. In general dffcult to predct these elementary steps a pror. Gven them, though, our task s to calculate the rates of the ndvdual steps. Unts? A + products rate = kcc ( ) A Conceved of as passage from one well to another. Can be complex dynamcal process. ranston state theory s a very useful model that s straghtforward to apply from computatonal results. S assumptons (n order of degree of approxmaton): 1. Exstence of a PES. Exstence of a dvdng surface ( pont of no return ) 3. Exstence of a crtcal transton state pont on dvdng surface 4. Exstence of a quas-equlbrum between reactants and transton state A+ [A] C 5. Harmonc (quadratc) form of the PES near the S Equlbrum and harmonc approxmatons lead to: Can equvalently be expressed k q 0 / n ΔE k Δ k ( ) = e ( c) h qaq k : frequency factor, recprocal tme h q : partton functon of S, excludng reacton coordnate mode qa, q : partton functon of reactants ΔE0 : zero-pont corrected actvaton energy P P c = = : standard concentraton unts Ak R Δn : changes n number of speces from reactants to S k Δn S R H / R k ( ) e Δ e Δ = ( c) h Δ S : standard entropy of actvaton Δ H : standard enthalpy of actvaton Prof. W. F. Schneder CE Computatonal Chemstry 7/8 1:17:59 PM 11/8/007 Unversty of otre Dame

8 Calculaton of hermodynamc and Knetc Propertes Example: Dels-Alder reacton H C=CHF + H C=CH CH=CH S h 6.661E 34 J s A 6.01E+3 /mol R J/mol K k E 3 J/K K P 1.00 atm H C=CHF H C=CH CH=CH S Delta (kj/mol) E ZPE delta E oltzmann 3.09E 17 q 1.59E E+1.07E E 10 3 k /h 6.1E+1 s 1 4 R/P l/mol 4p k/p 4.06E 0 cm^3 1**3*4: k(98.15) 5.09E 5 l/mol s 1**3*4p: k(98.15) 5.5E 33 cm3/molecule s Slow. What would I have to do to calculate at a hgher temperature? Or pressure? 8. Specal knetcs topcs Knetc sotope effect partton functons calculated by default for most abundant sotope. Substtuton of D for H can sgnfcantly mpact rate. unnelng correcton especally for reacton paths nvolvng lght atoms, tunnelng through barrer can sgnfcantly mpact rates. Hard to treat quanttatvely; smple models avalable based on knowng curvature at S. aratonal transton state theory what f a reacton doesn t have an energy barrer? May stll have a free energy barrer, place where Δ G s maxmzed. Found by searchng along IRC. on-adabatc dynamcs ncorporate effect of multple tronc states on reacton rate Prof. W. F. Schneder CE Computatonal Chemstry 8/8 1:17:59 PM 11/8/007 Unversty of otre Dame