MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.5.62 Lecture #5: Molecular Partition Function: Replace E(assembly) by ε(molecule) Readings: Hill, pp. 59-70; Maczek, pp. 16-19; Metiu pp. 49-55 Overview: We've learned to calculate thermodynamic (macroscopic) properties of a system from the partition function. However, the partition function, as it is presentlywritten, depends on the energy levels available to the entire many-particle system. We still need to input an understanding of the energy levels of a single molecule(microscopic) into a description of the energy levels of the entire many-particle system(= assembly). Today we will do this, using statistics (combinatorics). Goal: Reformulate Q as a function of the energies, εi, of states of individual molecules rather than the energies Ej of an assembly of molecules. Procedure: Change labeling of states from "α-type" (assembly centered) description to an occupation number, ni, (molecule centered) n-type description. α-type assembly description (list of the state of each molecule in assembly) m1xm1ym1z m2xm2ym2z m3xm3ym3z m4xm4ym4z m5xm5ym5z 1 1 1 2 1 1 1 1 1 1 2 2 2 1 1 molecule 1 molecule 2 molecule 3 molecule 4 molecule 5 state #1 state #2 state #1 state #3 state #2 energy ε1 energy ε2 energy ε1 energy ε3 energy ε2 To construct an n-type description (list of number of systems in each allowed moleculestate: less information) of the same assembly state: Define ni ≡ occupation number = number of molecules in ith molecular state. For example (2, 2, 1) means: n1 = 2 molecules in state #1 with energy ε1 n2 = 2 molecules in state #2 with energy ε2 n3 = 1 molecule in state #3 with energy ε3   5.62 Spring 2008 Lecture 5, Page 2 Thus, an α−type state could be re-expressed in terms of a set of individual particle energylevel occupation numbers (called a “configuration”): {ni} = {n1, n2, n3, ··· } where N = ∑ ni and E = ∑ ni εi shorthand i levels i levels energy ofenergy of ithtotal # of assembly molecular statemolecules This is a change of focus from labels that identify sum over molecularstates ≈1023 individual molecules to labels that identify molecular states and the number of molecules in each of those states. Note that different (α-type) assembly states can have the same (n-type) occupationnumbers. For example, switch the occupied energy states between molecules 1 and 2. Expand definition of degeneracy to include occupation numbers: Ω({ni}) ≡ degeneracy = number of (α-type) assembly states with the same set {ni} ofoccupation numbers (or total E) Rewrite Q ... Q(N,V,T) = e–E j /kT∑ j sum over possible (α-type) states of assembly = ∑ Ω({n i}) e–E({ni})/kT{n i } sum over all sets of occupation numbers {ni} such that ni∑ = N i revised 1/9/08 9:35 AM 5.62 Spring 2008 Lecture 5, Page 3 COMBINATORICS Determining Ω for a given set of {ni}: How many ways are there to arrange molecules such that occupation numbers aregiven by {ni}? This is Ω. Another way to ask the question ... How many ways are thereto put N molecules into a set of molecular states with n1 in state #1, n2 in state #2, etc. state # energy # molecules 1 ε1 n1 2 ε2 n2 3 ε3 n3   i εi ni   This is a simple combinatorial problem. We put all N molecules in sequence and put thefirst n1 in state #1, the next n2 in state #2, etc. Then the number of ways of arrangingmolecules into states is just the number of sequences, which is just N! =N(N–1)(N–2)(N–3) ··· (2)(1), because there are N places in the sequence to put the firstmolecule, (N–1) places to put the second, etc. However, this overcounts because the order of molecules chosen for molecular state #1 isnot important. That is, all ways of renumbering molecules in state #1 are equivalent.There are (n1!) of them. A similar factor of (ni!) needs to be used to correct forovercounting in each state. For distinguishable molecules, the # of ways of putting N molecules in a set of statessuch that the first state gets n1 molecules, the second state gets n2 molecules, etc. is ... N! N! =Ω({ni}) = n1!n2!n3!⋅⋅⋅ ni!⋅⋅⋅ ∏ni! i a multinomial coefficient What is a multinomial coefficient? In the expansion of (a + b + c)N N! the term an1bn2cn3 is multiplied by the coefficient n1!n 2!n3! revised 1/9/08 9:35 AM  5.62 Spring 2008 Lecture 5, Page 4 where ∑ ni= N i (a + b + c + d)3 = a3 + 3a2b + 6abc + ··· N! 3! 3! 3! n1!n 2!n3!n 4! 3!0!0!0! 2 !1!0!0! 1!1!1!0! Note: 0! = 1 Rewrite Q in Terms of the New Expression for Ω ({ }Q N,V, T( )= ∑Ω({ }ni )e−E ni ) kT { } niThis is a sum over all sets of occupation N! −∑ njε j kT numbers, and E ni )= ∑ .({ }njεj= ∑ jni∏ nj!e j { } j revised 1/9/08 9:35 AM5.62 Spring 2008 Lecture 5, Page 5 Reformulate the expression for Q energy of i-th molecular state Q N,V, T( )= {ni} ∏ N! nj!e– ∑i niεi /kT where E ({ })= ∑∑ ni niεii j # of molecules in state i Reduce this to a function of the sum over states of a single molecule: = ∑⎛ N! ⎞ ⎡∏ (e−εi kT )ni ⎤ {ni} ⎝⎜⎜∏ j nj! ⎠⎟⎟ ⎣⎢ i ⎦⎥ note that e− εi kT is raised to the ni -th power Now impose multinomial trick =(e−ε1 kT + e−ε2 kT + e−ε3 kT )N … N ⎛∑ e−εi kT ⎞ = ⎠⎟⎝⎜ i sum over states of a single molecule Define: q = i ∑ e−εi /kT sum over states of MOLECULAR PARTITION FUNCTION εi = molecular energy ofstate i a single molecule Q(N,V,T) = [q(V,T)]N N-molecule SingleCanonical Molecule Partition Canonical Function Partition Function FOR INDEPENDENT, DISTINGUISHABLE PARTICLES! revised 1/9/08 9:35