In the Classroom
A Simple Derivation of the Boltzmann Distribution Sean A. C. McDowell Department of Biological and Chemical Sciences, University of the West Indies, Cave Hill Campus, P.O. Box 64, Bridgetown, Barbados;
[email protected] The Boltzmann distribution is a very important concept in chemistry, occupying a central position in the science because it governs the equilibrium distribution of molecules at a particular temperature. The concept of equilibrium is fundamental to understanding physical and chemical processes and the theories that govern them. The importance of the Boltzmann distribution cannot be understated. It is used in a wide range of applications, examples of which are determining the molecular mass of macromolecules (e.g., polymers) by sedimentation techniques; estimating the temperature of celestial bodies, high-temperature flames, and low-pressure gases; determining the distribution of gas molecules in the atmosphere; and calculating the intensity of spectral lines. The derivation of the Boltzmann distribution is usually taught as part of a statistical mechanics course in physical chemistry at the undergraduate university level and is covered in detail in most standard textbooks (1–4). The equation is usually first encountered by students at the introductory university level, the derivation being relegated to the final year of the degree program—where, in my experience, it is usually found to be a particularly formidable aspect of the physical chemistry course. As was pointed out in a fairly recent paper in this Journal (5), the problem usually stems from conceptual difficulties and the unfamiliarity of mathematical devices such as Stirling’s approximation and the Lagrange method of undetermined multipliers, which are used in most physical chemistry textbooks (1–4). In this paper a simplified derivation of the Boltzmann distribution is proposed as an alternative to the more or less standard approaches given in the textbooks (1–4), which, it is hoped, may actually enhance students’ understanding of the more rigorous treatments. This method should be within the grasp of final-year students, who should have had some experience with concepts from macroscopic thermodynamics. It is also an alternative method to the one proposed in ref 5, where the author focuses on the point of minimum Helmholtz energy in a closed system. In the present work a closed system (constant N, V, T ) is studied but here the fundamental thermodynamic property of entropy is linked to the probabilistic molecular distributions in a straightforward manner and it is believed that this method may be conceptually simpler than those given in refs 1–5. It should be noted that this method does not use the entropy criterion for equilibrium (in a thermally isolated system), since energy transfer between the system and its surroundings is allowed.
the two lowest (nondegenerate) energy levels e0 and e1 are occupied and contain n0 and n 1 molecules, respectively. Traditionally the ground-state energy e0 is set to zero for simplicity because only the relative energies and not their absolute values are relevant here. The number of ways in which this configuration can be realized (i.e., the number of microstates W ) is given by
Derivation of the Boltzmann Distribution
assuming that temperature remains effectively constant and with the integral over S taken from S to S ′ and the integral over U from U to U ′. Since the change in energy, U ′ – U, is simply the energy e added to the original system, then it follows from the previous equation that
Let us assume that we have a closed system containing a large number N of distinguishable molecules in thermal equilibrium with its surroundings at a temperature T and under constant volume conditions. We also suppose that only
W=
N! n 0! n 1!
(1)
with N = n0 + n1. The connection between the statistical mechanical description of the system (embodied in the concept of the number of microstates) and the bulk thermodynamic description given by the entropy (S) is expressed in Boltzmann’s famous equation S = k lnW
(2)
k being, of course, Boltzmann’s constant. This equation can be shown to be reasonable, since students should already be familiar with the fact that the approach to equilibrium corresponds to maximizing entropy, and it can be easily shown that this also corresponds to maximizing W (or ln W ), thereby implying a link between these two concepts. Substitution of eq 1 into eq 2 gives S = k {ln(N !) – ln(n0!) – ln(n1!)}
(3)
Now suppose that a small amount of energy e (corresponding to the separation between the energy levels e0 and e1) is added to the system and promotes one molecule from the lower to upper energy level (i.e., n0 → n0 – 1, n1 → n1 + 1). Hence, S ′ = k {ln(N !) – ln[(n0 – 1)!] – ln[(n1 + 1)!]}
(4)
The change in entropy, ∆S, is thus ∆S = S ′ – S = k ln{n0/(n1 + 1)} = k ln{n0/n1}, n1 >>> 1 (5) From the laws of thermodynamics we obtain a connection between the entropy and total energy (U ) of the system: dS = dU/T
(6)
since under constant volume conditions dU = dq. If we choose a reversible path, dqreversible = dU, and by definition dqreversible = T dS, so that dU = T dS. Since U and S are state functions, the latter equation is always true regardless of whether we choose a reversible path or not. Consequently, ∆S = ∫dS = ∫dU/T = 1/T ∫dU
JChemEd.chem.wisc.edu • Vol. 76 No. 10 October 1999 • Journal of Chemical Education
(7)
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In the Classroom
∆S = e/T
(8)
k ln(n0/n1) = e/T
(9)
n1/n0 = exp({ e/kT)
(10)
Hence, from eqs 5 and 8, and This, of course, is the Boltzmann distribution law, which may be generalized for any two arbitrary energy levels. It may also be extended to the case of indistinguishable molecules and to degenerate energy levels. Conclusion A straightforward derivation of the Boltzmann distribution law has been presented that may be a useful alternative to the more standard treatments used in the teaching of statistical mechanics at the senior undergraduate university level. Conceptually and mathematically simpler, the development given here builds on students’ knowledge of thermodynamics and may serve as a conceptual aid in enhancing understanding of one of the most fundamental topics in chemistry. Literature Cited 1. Barrow, G. M. Physical Chemistry, 6th ed.; McGraw-Hill: New York, 1996. 2. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 2nd ed.; Houghton Mifflin: Boston, 1995. 3. Atkins, P. W. Physical Chemistry, 5th ed.; Oxford University Press: Oxford, 1994. 4. Vemulapalli, G. K. Physical Chemistry; Prentice Hall: Englewood Cliffs, NJ, 1993. 5. Russell, D. K. J. Chem. Educ. 1996, 73, 299.
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Journal of Chemical Education • Vol. 76 No. 10 October 1999 • JChemEd.chem.wisc.edu
