5.1: Ensembles of N-molecule Systems
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When we begin our discussion of Boltzmann statistics in Chapter 20, we note that there exists, in principle, a Schrödinger equation for an N-molecule system. For any particular set of boundary conditions, the solutions of this equation are a set of infinitely many wavefunctions, Ψi,j, for the N-molecule system. For every such wavefunction, there is a corresponding system energy, Ei. The wavefunctions reflect all of the attractive and repulsive interactions among the molecules of the system. Likewise, the energy levels of the system reflect all of these interactions.
In Section 20.12, we introduce the symbol ΩE to denote the degeneracy of the energy, E, of an N-molecule system. Because the constituent molecules are assumed to be distinguishable and non-interacting, we have
ΩE=∑{Ni},EW(Ni,gi)
In the solution of the Schrödinger equation for a system of N interacting molecules, each system-energy level, Ei, can be degenerate. We again let Ω denote the degeneracy of an energy level of the system. We use Ωi (rather than ΩEi) to represent the degeneracy of Ei. It is important to recognize that the symbol “Ωi” now denotes an intrinsic quantum-mechanical property of the N-particle system.
In Chapters 21 and 22, we denote the parallel properties of an individual molecule by ψi,j for the molecular wavefunctions, ϵi for the corresponding energy levels, and gi for the degeneracy of the ith energy level. We imagine creating an N-molecule system by collecting N non-interacting molecules in a fixed volume and at a fixed temperature.
In exactly the same way, we now imagine collecting ˆN of these N-molecule, constant-volume, constant-temperature systems. An aggregate of many multi-molecule systems is called an ensemble. Just as we assume that no forces act among the non-interacting molecules we consider earlier, we assume that no forces act among the systems of the ensemble. However, as we emphasize above, our model for the systems of an ensemble recognizes that intermolecular forces among the molecules of an individual system can be important. We can imagine specifying the properties of the individual systems in a variety of ways. A collection is called a canonical ensemble if each of the systems in the ensemble has the same values of N, V, and T. (The sense of this name is that by specifying constant N, V, and T, we create the ensemble that can be described most simply.)
The canonical ensemble is a collection of ˆN identical systems, just as the N-molecule system is a collection of N identical molecules. We imagine piling the systems that comprise the ensemble into a gigantic three-dimensional stack. We then immerse the entire stack—the ensemble—in a constant temperature bath. The ensemble and its constituent systems are at the constant temperature T. The volume of the ensemble is ˆNV. Because we can specify the location of any system in the ensemble by specifying its x-, y-, and z-coordinates in the stack, the individual systems that comprise the ensemble are distinguishable from one another. Thus the ensemble is analogous to a crystalline N-molecule system, in which the individual molecules are distinguishable from one another because each occupies a particular location in the crystal lattice, the entire crystal is at the constant temperature, T, and the crystal volume is NVmolecule.
Since the ensemble is a conceptual construct, we can make the number of systems in the ensemble, ˆN, as large as we please. Each system in the ensemble will have one of the quantum-mechanically allowed energies, Ei. We let the number of systems that have energy E1 be ˆN1. Similarly, we let the number with energy E2 be ˆN2, and the number with energy Ei be ˆNi. Thus at any given instant, the ensemble is characterized by a population set, {ˆN1, ˆN2, …, ˆNi,…}, in exactly the same way that an N-molecule system is characterized by a population set, {N1, N2,…, Ni,…}. We have
ˆN=∞∑i=1ˆNi
While all of the systems in the ensemble are immersed in the same constant-temperature bath, the energy of any one system in the ensemble is completely independent of the energy of any other system. This means that the total energy of the ensemble, ˆE, is given by
ˆE=∞∑i=1ˆNiEi
Property | System | Ensemble |
---|---|---|
Quantum entity | Molecule at fixed volume and temperature | System comprising a collection of N molecules at fixed volume and temperature |
Aggregate of quantum entities | System comprising a collection of N molecules at fixed volume and temperature | Ensemble comprising ˆN systems each of which contains N molecules |
Number of quantum entities in aggregate | N | ˆN |
Wave functions/quantum states | ψi | Ψi |
Energy levels | ϵi | Ei |
Energy level degeneracies | gi | Ωi |
Probability that an energy level is occupied | Pi | ˆPi |
Number of quantum entities in the ith energy level | Ni | ˆNi |
Probability that a quantum state is occupied | ρ(ϵi) | ˆρ(Ei) |
Energy of the aggregate’s kth population set | Ek=∑Nk,iϵi | ˆEk=∑ˆNk,iϵi |
Expected value of the energy of the aggregate | ⟨E⟩=N∑Piϵi | ⟨ˆE⟩=ˆN∑ˆPiEi |
The population set, {ˆN1, ˆN2, …, ˆNi,…}, that characterizes the ensemble is not constant in time. However, by the same arguments that we apply to the N-molecule system, there is a population set
{ˆN⦁1, ˆN⦁2,…, ˆN⦁i,…}
which characterizes the ensemble when it is at equilibrium in the constant-temperature bath.
We define the probability, ˆPi, that a system of the ensemble has energy Ei to be the fraction of the systems in the ensemble with this energy, when the ensemble is at equilibrium at the specified temperature. Thus, by definition,
ˆPi=ˆN⦁iˆN.
We define the probability that a system is in one of the states, Ψi,j, with energy Ei, as
ˆρ(Ei)=ˆPiΩi
The method we have used to construct the canonical ensemble insures that the entire ensemble is always at the specified temperature. If the component systems are at equilibrium, the ensemble is at equilibrium. The expected value of the ensemble energy is
⟨ˆE⟩=ˆN∞∑i=1ˆPiEi=∞∑i=1ˆN⦁iEi
Because the number of systems in the ensemble, ˆN, is very large, we know from the central limit theorem that any observed value for the ensemble energy will be indistinguishable from the expected value. To an excellent approximation, we have at any time,
ˆE=⟨ˆE⟩
and
ˆN⦁i=ˆNi
The table above summarizes the terminology that we have developed to characterize molecules, N-molecule systems, and ˆN-system ensembles of N-molecule systems.
We can now apply to an ensemble of ˆN, distinguishable, non-interacting systems the same logic that we applied to a system of N, distinguishable, non-interacting molecules. The probability that a system is in one of the energy levels is
1=ˆP1+ˆP2+⋯+ˆPi+…
The total probability sum for the constant-temperature ensemble is
1=(ˆP1+ˆP2+⋯+ˆPi+…)ˆN=∑{ˆNi}ˆW(ˆNi,Ωi)ˆρ(E1)ˆN1ˆρ(E2)ˆN2…ˆρ(Ei)ˆNi…
where
ˆW(ˆNi,Ωi)=ˆN!∞∏i=1ΩˆNiiˆNi!
Moreover, we can imagine instantaneously isolating the ensemble from the temperature bath in which it is immersed. This is a wholly conceptual change, which we effect by replacing the fluid of the constant-temperature bath with a solid blanket of insulation. The ensemble is then an isolated system whose energy, ˆE, is constant. Every system of the isolated ensemble is immersed in a constant-temperature bath, where the constant-temperature bath consists of the ˆN−1 systems that make up the rest of the ensemble. This is an important feature of the ensemble treatment. It means that any conclusion we reach about the systems of the constant-energy ensemble is also a conclusion about each of the ˆN identical, constant-temperature systems that comprise the isolated, constant-energy ensemble.
Only certain population sets, {ˆN1, ˆN2, …, ˆNi,…}, are consistent with the fixed value, ˆE, of the isolated ensemble. For each of these population sets, there are ˆW(ˆNi,Ωi) system states. The probability of each of these system states is proportional to ˆρ(E1)ˆN1ˆρ(E2)ˆN2…ˆρ(Ei)ˆNi…. By the principle of equal a priori probability, every system state of the fixed-energy ensemble occurs with equal probability. We again conclude that the population set that characterizes the equilibrium state of the constant-energy ensemble, {ˆN⦁1, ˆN⦁2,…, ˆN⦁i,…}, is the one for which ˆW or lnˆW is a maximum, subject to the constraints
ˆN=∞∑i=1ˆNi
and
ˆE=∞∑i=1ˆNiEi
The fact that we can make ˆN arbitrarily large ensures that any term, ˆN⦁i, in the equilibrium-characterizing population set can be very large, so that ˆN⦁i can be found using Stirling’s approximation and Lagrange’s method of undetermined multipliers. We have the mnemonic function Fmn=ˆNlnˆN−ˆN+∞∑i=1(ˆNilnΩi −ˆNilnˆNi +ˆNi) +α(ˆN−∞∑i=1ˆNi)+β(ˆE−∞∑i=1ˆNiEi) so that
(∂Fmn∂ˆN⦁i)j≠i=lnΩi −ˆN⦁iˆN⦁i−lnˆN⦁i +1−α−βEi=0
and
lnˆN⦁i =lnΩi −α−βEi
or
ˆN⦁i=Ωiexp(−α)exp−βEi
When we make use of the constraint on the total number of systems in the ensemble, we have
ˆN=∞∑i=1ˆN⦁i=exp(−α)∞∑i=1Ωiexp(−βEi)
so that
exp(−α)=ˆNZ−1
where the partition function for a system of N possibly-interacting molecules is
Z=∞∑i=1Ωiexp(−βEi)
The probability that a system has energy Ei is equal to the equilibrium fraction of systems in the ensemble that have energy Ei, so that
ˆPi=ˆN⦁iˆN=Ωiexp(−βEi)Z