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 #3: Canonical Partition Function: Replace {Pi} by Q Canonical Distribution Function Denominator of canonical distribution function has a special name ... Q(N,V,T) = ∑e−Ej kT j CANONICAL PARTITION FUNCTION Sum of "Boltzmann factor", e–Ej/kT, over states of the assembly originally called "zustandsumme" ≡ Z ≡ sum over states Q is a very very important quantity. We will use Q to calculate macroscopic properties from microscopic properties Rewrite Canonical Distribution Function in terms of Q ... e−EjkT = Q m FEEL THE POWER OF Pj — can now calculate macroscopic properties from ensemble average ... but more convenient to use Q. REPLACING Pj IN ENSEMBLE AVERAGE BY Q example: E = ∑ Pj Ej = f(Q) j Define β ≡ 1/kT Pj = e−E j kT e−Em kT m ∑ Pj = e− E j kT e−Em kT∑5.62 Spring 2008 Lecture 3, Page 2 (Q N,V,T)= ∑ e−Ej kT ∑ e−βEj= j j ∂Q ∑ E je−βEj= − ∂β j e−βEjNow Pj = Q so e−βEj = QPj Therefore ∂∂β Q = −∑ j PjE jQ = −Q∑ j PjE j But E = ∑ PjEj j So ∂∂β Q = −E Q or E = − Q1 ∂∂β Q E = − 1 ∂Q ∂lnQ = −∂lnQ ∂lnQ Q ∂β = −∂β ∂(1 kT)= −∂T ∂T ∂ 1 kT( ) E = kT2 ∂lnQ ∂T ⎛ ⎝⎜ ⎞ ⎠⎟ N,V = kT2 ∂lnQ ∂ln T ∂ln T ∂T = kT ∂lnQ ∂ln T This is the ensemble average for E written in terms of Q instead of PjWriting S in terms of Q instead of Pj S = −k∑ j Pj ln Pj = −k∑ j Pj ln ⎜⎛⎝ e−EQj /kT ⎞⎠⎟ S = −k∑ Pj ⎣⎢ kT − lnQ ⎦⎥ = T ⎡−Ej ⎤∑ j PjE j + k lnQ j S = k lnQ + E T = k ln Q + k ∂lnQ ∂ln T ⎛⎝⎜⎞⎠⎟N,V revised 1/9/08 9:35 AM5.62 Spring 2008 Lecture 3, Page 3 WRITING ALL THERMODYNAMIC FUNCTIONS OR MACROSCOPIC PROPERTIES IN TERMS OF Q From thermo ... (A = E − TS = E − T k lnQ + E / T )= E − kTln Q − TET A = –kT ln Q Helmholtz free energy Note that both A and Q have natural variables N, V, T. From thermo ... ⎛ ∂A ∂V ⎞ p = –⎜⎝ ⎟⎠ pressure T,N p = kT ∂lnQ ∂V ⎛⎝⎜⎞⎠⎟T,N from thermo ... ⎛ ∂A ∂N ⎞ ⎟⎠T,V (For µ, always naturalchemical potential variables held constant.)µ = ⎜⎝ µ = −kT ∂lnQ ∂N ⎛ ⎝⎜ ⎞ ⎠⎟ T,V H ≡ E + pV⎫ ⎬ write in terms of Q in homeworkG ≡ A + pV⎭ Now we have a rudimentary structure or framework for relating the microscopic properties as given by Q, the sum over states of assemblies present in the canonical ensemble, to macroscopic or thermodynamic properties. Note that Q (or Pj) tells us the distribution of assembly states present in the ensemble. We see that it is the energy of the state of the assembly that determines its probability of being in the ensemble. So now we need to know what are the energies of the assemblies, Ej, so that Q for specific systems may be calculated. Once Q is known, we can calculate all macroscopic thermodynamic properties from the above expressions!! revised 1/9/08 9:35 AM5.62 Spring 2008 Lecture 3, Page 4 A LOOSE END: DEGENERACY — BACK TO Pj Sometimes a more useful form of Pj is P(E). GOAL: Derive P(E) P(E) ≡ probability of finding an assembly state with energy E. Each j in Q stands for a distinguishable state of the assembly. kT + e−Eγ kT + … = ∑e−Ej kTkT + e−EβQ = …+ e−Eα j But many distinguishable assembly states are degenerate (i.e. have the same energy) Eα = Eβ = Eγ = E Q = …+ 3e−E /kT +… = ∑Ω(N,V,E)e−E /kT E ↑ Ω(N,V,E) ≡ degeneracy = no. of distinguishableassembly states with energy E. So Q(N, V, T) = e−E j kT j ∑ = E ∑ Ω(N,V,E)e−E kT sum over energy levelspresent in ensemble sum over states of assemblies ( ) = ( )P E∑ Pj = ∑ e−E j kT Q N,V, Tj∍ Ej =E j∍ Ej =E Sum over those assembly statesthat belong to the set of assembly states whose Ej = E P(E) = Ω(N,V,E)e−E kT Q(N,V, T) probability of finding an assembly state with energy E in ensemble revised 1/9/08 9:35