Recognizing the formula for the constant-volume heat capacity Cv=(a(Eav/aT)Ny allows the fractional fluctuation in the energy around the mean energy eave=E/M to be expressed as (E-Eave )ave Eave =kT Cv/ave What does this fractional fluctuation formula tell us? On its left-hand side it gives a measure of the fractional spread of energies over which each of the containers ranges about its mean energy Eave. On the right side, it contains a ratio of two quantities that are extensive properties, the heat capacity and the mean energy. That is, both Cy and eave will be proportional to the number n of molecules in the container as long as N is reasonably large. However, because the right-hand side involves Cv/Eav it is proportional to N-and thus will be very small for large N as long as Cy does not become large. As a result, except near so-called critical points where the heat capacity does indeed become extremely large, the fractional fluctuation in the energy of a given container ofN molecules will be very small (i.e, proportional to n). It is this fact that causes the narrow distribution in energies that we discussed earlier in this section PAGE 16
PAGE 16 Recognizing the formula for the constant-volume heat capacity CV = (¶(Eave)/¶T)N,V allows the fractional fluctuation in the energy around the mean energy Eave = E/M to be expressed as: (E-Eave)) 2 ave/Eave 2 = kT2 CV/Eave 2 . What does this fractional fluctuation formula tell us? On its left-hand side it gives a measure of the fractional spread of energies over which each of the containers ranges about its mean energy Eave. On the right side, it contains a ratio of two quantities that are extensive properties, the heat capacity and the mean energy. That is, both CV and Eave will be proportional to the number N of molecules in the container as long as N is reasonably large. However, because the right-hand side involves CV/Eave 2 , it is proportional to N-1 and thus will be very small for large N as long as CV does not become large. As a result, except near so-called critical points where the heat capacity does indeed become extremely large, the fractional fluctuation in the energy of a given container of N molecules will be very small (i.e., proportional to N-1). It is this fact that causes the narrow distribution in energies that we discussed earlier in this section
B Partition Functions and Thermodynamic Properties Let us now examine how this idea of the most probable energy distribution being dominant gives rise to equations that offer molecular-level expressions of thermodynamic properties. The first equation is the fundamental Boltzmann population formula that we already examined P,Q2; exp(-E, /kT)Q, which expresses the probability for finding the N-molecule system in its J quantum state having energy E, and degeneracy s Using this result, it is possible to compute the average energy <E> of the system <E>=∑:P:E and, as we saw earlier in this Section, to show that this quantity can be recast as <E>=kT a(InQ/aT)NV To review how this proof is carried out, we substitute the expressions for Pi and for Q into the expression for <E> <E>=2,E,2 exp(E/kT)H212, exp(-EykT)) PAGE 1
PAGE 17 B. Partition Functions and Thermodynamic Properties Let us now examine how this idea of the most probable energy distribution being dominant gives rise to equations that offer molecular-level expressions of thermodynamic properties. The first equation is the fundamental Boltzmann population formula that we already examined: Pj = Wj exp(- Ej /kT)/Q, which expresses the probability for finding the N-molecule system in its Jth quantum state having energy Ej and degeneracy Wj . Using this result, it is possible to compute the average energy <E> of the system <E> = Sj Pj Ej , and, as we saw earlier in this Section, to show that this quantity can be recast as <E> = kT2 ¶(lnQ/¶T)N,V . To review how this proof is carried out, we substitute the expressions for Pj and for Q into the expression for <E>: <E> = {Sj Ej Wj exp(-Ej /kT)}/{Sl Wl exp(-El /kT)}
By noting that a(exp(-E/kT)aT=(1/kT)E, exp(E/kT), we can then rewrite <E>as <E>=kt(2,,a(exp(E/kT)OT)(Q exp(-E/kT); And then recalling that aX/aT/X=aInX/aT, we finally obtain <E>=kT(aIn(Q)OT)NV All other equilibrium properties can also be expressed in terms of the partition function Q. For example, if the average pressure <p> is defined as the pressure of each antum state P,=(OE, /OV)N multiplied by the probability P, for accessing that quantum state, summed over all such states, one can show, realizing that only E, (not T or S2)depend on the volume V,that <p>=E(OE, /OVN Q exp(-E, /kT)Q =kt(aInQ/OVNNT PAGE 18
PAGE 18 By noting that ¶ (exp(-Ej /kT))/¶T = (1/kT2 ) Ej exp(-Ej /kT), we can then rewrite <E> as <E> = kT2 {Sj Wj¶ (exp(-Ej /kT))/¶T }/{Sl Wl exp(-El /kT)}. And then recalling that {¶X/¶T}/X = ¶lnX/¶T, we finally obtain <E> = kT2 (¶ln(Q)/¶T)N,V. All other equilibrium properties can also be expressed in terms of the partition function Q. For example, if the average pressure <p> is defined as the pressure of each quantum state pj = (¶Ej /¶V)N multiplied by the probability Pj for accessing that quantum state, summed over all such states, one can show, realizing that only Ej (not T or W) depend on the volume V, that <p> = Sj (¶Ej /¶V)N Wj exp(- Ej /kT)/Q = kT(¶lnQ/¶V)N,T
Without belaboring the point further, it is possible to express all of the usual thermodynamic quantities in terms of the partition function Q. The average energy and average pressure are given above; the average entropy is given <S>=kInQ + kT(aInQ/aN)VT the helmholtz free energy a A=-kT InO and the chemical potential u is expressed as follows H=-kT(aInQ/aN) As we saw earlier, it is also possible to express fluctuations in thermodynamic properties in terms of derivatives of partition functions and, thus, as derivatives of other properties. For example, the fluctuation in the energy <(E-E>)> was shown above to be gIven <(E-<F>)2>=kTCy The Statistical Mechanics text by McQuarrie has an excellent treatment of these topic and shows how all of these expressions are derived PAGE 1
PAGE 19 Without belaboring the point further, it is possible to express all of the usual thermodynamic quantities in terms of the partition function Q. The average energy and average pressure are given above; the average entropy is given as <S> = k lnQ + kT(¶lnQ/¶N)V,T the Helmholtz free energy A is A = -kT lnQ and the chemical potential m is expressed as follows: m = -kT (¶lnQ/¶N)T,V. As we saw earlier, it is also possible to express fluctuations in thermodynamic properties in terms of derivatives of partition functions and, thus, as derivatives of other properties. For example, the fluctuation in the energy <(E-<E>)2> was shown above to be given by <(E-<E>)2> = kT2 CV. The Statistical Mechanics text by McQuarrie has an excellent treatment of these topics and shows how all of these expressions are derived
So, if one were able to evaluate the partition function Q for N molecules in a volume V at a temperature T, either by summing the quantum-state degeneracy and exp(E, /kT)factor Q=2Q exp(E,/kT) or by carrying out the phase-Space integral over all M of the coordinates and momenta of the system Q=hMJexp( -H(q, p)kt) dq dp one could then use the above formulas to evaluate any thermodynamic properties as derivatives of InQ What do these partition functions mean? They represent the thermal-average number of quantum states that are accessible to the system this can be seen best by again noting that, in the quantum expression Q=2, Q2, exp(-E,/kT) the partition function is equal to a sum of (i) the number of quantum states in the j energy level multiplied by (ii)the Boltzmann population factor exp(-E /kT)of that level So, Q is dimensionless and is a measure of how many states the system can access at temperature T. Another way to think of Q is suggested by rewriting the Helmholtz free PAGE 20
PAGE 20 So, if one were able to evaluate the partition function Q for N molecules in a volume V at a temperature T, either by summing the quantum-state degeneracy and exp(-Ej /kT) factors Q = Sj Wj exp(- Ej /kT), or by carrying out the phase-space integral over all M of the coordinates and momenta of the system Q = h-M ò exp (- H(q, p)/kT) dq dp , one could then use the above formulas to evaluate any thermodynamic properties as derivatives of lnQ. What do these partition functions mean? They represent the thermal-average number of quantum states that are accessible to the system. This can be seen best by again noting that, in the quantum expression, Q = Sj Wj exp(- Ej /kT) the partition function is equal to a sum of (i) the number of quantum states in the jth energy level multiplied by (ii) the Boltzmann population factor exp(-Ej /kT) of that level. So, Q is dimensionless and is a measure of how many states the system can access at temperature T. Another way to think of Q is suggested by rewriting the Helmholtz free