J.I. Siepmann Chem 8561 9First way to determine β• differentiate eqn. 19 with respect to V∂E∂VN,β= −p +"βPjpjEje−βEjQ#−"βPjEje−βEjQ·Pjpje−βEjQ#(21)• differentiate eqn. 20 with respect to β∂p∂βN,V="PjEje−βEjQ·Pjpje−βEjQ#−"PjpjEje−βEjQ#(22)• adding eqns. 21 and β × 22∂E∂VN,β+ β∂p∂βN,V= −p + βpE − βpE + βEp − pE = −p (23)• comparing to known thermodynamic relation∂E∂VN,T− T∂p∂TN,V= −pwe can deduce that β = const/T , or in its more customary formβ =1kBT(24)J.I. Siepmann Chem 8561 10Connection to Entropy (Second way to determine β)• differentiateE holding N constant (closed system) using eqn. 18 and recognizingthat Ej(N, V ) can vary only with V if N is fixeddE =XjEjdPj+XjPjdEj= −1βXj(ln Pj+ ln Q)dPj+XjPj∂Ej∂VNdV• usingPjPj= 1 ,PjdPj= 0 and relationPjln PjdPj= d(PjPjln Pj)dE = −1βdXjPjln Pj− pdV (25)• comparing to known thermodynamic relationdE = T dS − pdVit follows thatS = −kBXjPjln Pj(26)• substituting eqn. 18 for Pjand using the definition ofE (eqn. 17) yieldsS = −kBXje−βEjQlne−βEjQ= −kBQXje−βEj(−βEj− ln Q)=ET+ kBln Q ≡ET−AT(27)• i.e. the Helmholtz free energy A is directly related to QA(N, V, T ) = −kBT ln Q(N, V, T ) (28)J.I. Siepmann Chem 8561 11• A is t he characteristic thermodynamic function of the canonical ensemble (havingthe three independent variables N, V , and T ) and Q is called the canonical ensemblepartition function• hence, if the partition function Q (and thus A) is known, we can differentiate A toobtain th e complete set of thermodynamic functionsdA = −S dT − p dV +XaµadNa(29)S = −∂A∂TV,N= kBT∂ ln Q∂TV,N+ kBln Q (30)p = −∂A∂VT,N= kBT∂ ln Q∂VT,N(31)E = −T2∂A/T∂TV,N= kBT2∂ ln Q∂TV,N(32)µi= −∂A∂NiT,V,Na6=i= −kBT∂ ln Q∂NiT,V,Na6=i(33)J.I. Siepmann Chem 8561 12Connection to 2ndlaw of thermodynamics• often it is more convenient to define Q with respect to energy levels (i.e., grouptogether all energy states belonging to the same energy level and define Ω(N, V, E),the microcanonical partition function, as the number of microscopic states(degeneracy) for a given system)Q(N, V, T ) =Xj(states)e−Ej(N,V )/kBT=Xi(levels)Ω(N, V, E)e−Ei(N,V )/kBT(34)• during a spontaneous process (e.q. expansion of gas into vacuum) a restraint isremoved and thereby the number of accessible quantum states of each energy levelwill either increase or remain constantΩ2(N, V′′, E) ≥ Ω1(N, V′, E)∆A = A2− A1= −kBT lnQ2Q1≤ 0 (35)Implications of T → 0 and T → ∞• if T → 0 and the lowest level E1is nond egenerate, thenQ = e−βE1h1 + Ω2e−β(E2−E1)+ · · ·i→ e−βE1(36)• hence P1→ 1 an d using eqn. 26limT →0S = 0 (37)• if T → ∞ the distribution over states becomes uniform, and since there is anindefinite number of stateslimT →∞S = ∞ (38)J.I. Siepmann Chem 8561 13THE GRAND CANONICAL ENSEMBLE• consists of a collection of open systems whose walls are both heat conductingand permeable to the passage of molecules. Again we can construct the ensembleby immersing the collection of such systems in a reservoir that acts both as a heatbath of temperature T and as a source/sink of particles with the flow of particlescontrolled by the chemical potential µ. After equilibration the reservoir is removedand the entire ensemble isolated from its surroundings.• since the particle number in each system of the grand canonical ensemble is allowedto fluctuate, we must now give the number of molecules N in addition to thequantum state to specify a given system• for each value of N, there is a different set of quantum states EN,j, and nN,jis thenumber of systems in the grand canonical ensemble that contain N molecules andare in state j• in addit ion to the two constraints on the number of systems N and the total energyEt(analoguous to the canonical ensemble) the possible distributions in the grandcanonical ensemble must satisfy a third constraint that the total number of particlesin the ensemble must be constant and equal to NtXNXjnN,j= N (39)XNXjnN,jEN,j= Et(40)XNXjnN,jN = Nt(41)• for any distribution n ≡ {nN,j}, th e number of states (degeneracy) is given byΩt(n) =N !QNQjnN,j!(42)J.I. Siepmann Chem 8561 14• maximize Ωt(following the same path as used for the canonical ensemble)∂∂nN,j"ln Ωt(n) − αXMXinM,i−βXMXinM,iEi− γXMXinM,iM#= 0 (43)PN,j(V, β, γ) ≡n∗N,jN= e−αe−βEN,j(V )e−γN(44)• again, eαcan be determined by summing eqn. 44 over occupation numbers (con-straint given in eqn. 39), and the symbol Ξ is given to the grand canonical partitionfunctionΞ(V, β, γ) ≡ eα=XNXje−βEN,j(V )e−γN(44)• thus we can now calculate the averages of the mechanical properties E, p, and N(which can be associated with their thermodynamic equivalents following Gibbs’ensemble postulate)E(V, β, γ) =1ΞXNXjEN,j(V ) e−βEN,j(V )e−γN= −∂ ln Ξ∂βV,γ(46)p(V, β, γ) =1ΞXNXj−∂EN,j∂Ve−βEN,j(V )e−γN=1β∂ ln Ξ∂Vβ,γ(47)N(V, β, γ) =1ΞXNXjN e−βEN,j(V )e−γN= −∂ ln Ξ∂γV,β(48)J.I. Siepmann Chem 8561 15Connection to thermal properties• to evaluate β and γ, we can follow the same line of argument as for the canonicalensemble (see eqns. 25 to 28)• differentiateE (substitute for EN,jfrom eqn. 44 and recognizing that EN,j(V ) canvary only with V )dE =XNXjEN,j(V )dPN,j+XNXjPN,jdEN,j(V )= −1βXNXj[γN + ln PN,j+ ln Ξ] dPN,j+XNXjPN,j∂EN,j(V )∂VdV (49)• using dN =PNPjN dPN,j(see eqn. 48)dE = −β−1dXNXjPN,jln PN,j−pdV −γβdN (50)• comparing to known thermodynamic relation for open systems dE = T dS −p dV +µ dN we can conclude thatT dS ≡ −β−1dXNXjPN,jln PN,j(51)µ ≡ −γβ(52)• given that the grand canonical ensemble must reduce to the canonical one if N isa constant (since the grand canonical ensemble might be viewed as a collection ofcanonical ensembles when the composition “freezes” by sudden insertion of wallswhich are impermeable to molecules), we can conclude that (again) β = 1/kBT andγ = −µkBT(53)S = −kBXNXjPN,jln PN,j(54)J.I. Siepmann Chem 8561 16• substituting eqn. 44 for PN,jyieldsS =ET−µNT+ kBln Ξ (55)• by comparison with the Gibbs free energy G = µN = E + pV − T S we