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Cite This: J. Chem. Educ. XXXX, XXX, XXX−XXX
Alternative Derivation of the Maxwell Distribution of Speeds Francisco Rivadulla*
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Centro Singular de Investigación en Química Biolóxica e Materiais Moleculares (CiQUS), Departamento de Química-Física, Universidade de Santiago de Compostela, 17582 Santiago de Compostela, Spain ABSTRACT: The Maxwell distribution of speeds, f(v), is the starting point for the calculation of the transport coefficients in kinetic-molecular theory. Most physical chemistry textbooks follow a path to derive f(v) similar to that used by Maxwell, which makes it difficult for students to understand its relationship with the equilibrium state of the system, and its probabilistic character. The mathematical implications of the distribution, such as its relationship with other probability distribution functions, are also not sufficiently developed. Herein, we discuss how deriving the Maxwell distribution of speeds from the Boltzmann distribution function allows students to connect with the basic concepts of statistical thermodynamics, such as those treated in an introductory undergraduate course. We also present the relationship between Maxwell’s distribution and other probability distribution functions, which can be helpful for deriving simple proportionality relationships between f(v) and v, under different conditions. KEYWORDS: Statistical Mechanics, Physical Chemistry, Kinetic-Molecular Theory
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INTRODUCTION In an effort to provide a tool for the objective assessment of student learning, the American Chemical Society Examination Institute (ACS-EI) developed an Anchoring Concepts Content Map (ACCM) as a proposal of the most relevant content that should be included in a four-year chemistry degree.1 For the case of physical chemistry, its ACCM includes a number of specific content areas on statistical thermodynamics, which articulate several enduring concepts.2 These include the fundamentals of kinetic-molecular theory and statistical thermodynamics, needed for understanding the connection between microscopic composition and macroscopic properties of a chemical system.3 In this regard, the Maxwell distribution of molecular speeds, f(v), constitutes the basis of the kinetic theory of gases, as it is the starting point for the calculation of the most probable, average, and root-mean-square speeds of gas molecules.4−7 However, following the classical derivation of f(v) as it is done in most physical chemistry textbooks does not sufficiently highlight the probabilistic character of the function, nor its connection to the state of equilibrium of the system. Deriving general mathematical relationships with other statistical distribution functions is also important. This helps the students derive intuitive proportionality relationships between f(v) and v under different conditions (i.e., restrictions in the dimensions of movement) relevant in chemical problems.
derivation, particularly for students facing this topic for the first time. Even more important, this oversight could prevent a full appreciation of the essence of transport phenomena: the deviation of the distribution function from its value at equilibrium. Therefore, understanding the process of transport requires first that the connection between equilibrium and the Maxwell distribution function is fully understood. A few years after Maxwell, Boltzmann provided a derivation of f(v) on the basis of his probability distribution function for the energy states of a system (see refs 12 and 13 for a discussion of Boltzmann’s contribution to this subject). What I propose here is a simple version of this derivation as a practical classroom example of a calculation of the distribution of the kinetic energy over the particles of the system. For an isolated system at equilibrium, the population of the energy levels representing the (quantized) kinetic energy along each of the three different Cartesian directions can be calculated from the Boltzmann distribution: nvi(x) N
e−mvi (x)/2kBT 2
∑i e−mvi (x)/2kBT
(1)
where nEi(x)/N represents the fraction of molecules at the (nondegenerate) ith kinetic energy state along the x direction. Chemistry students are familiar with the quantization of kinetic energy from introductory courses of quantum chemistry; thus, the result of the energy difference between consecutive kinetic energy levels, at least in the onedimensional case, is well-known to them.4 For molecules moving along macroscopic dimensions, this calculation provides an energy spacing sufficiently small that the kinetic
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DERIVATION OF f(v) FROM THE BOLTZMANN DISTRIBUTION LAW In the famous 4th Proposition of his 1860 paper “On the Motion and Collisions of Perfectly Elastic Spheres”,8 Maxwell derived an analytical expression for the velocity distribution of the particles of an ideal gas that is still used by many physical chemistry textbooks.4,5,9−11 However, the probabilistic nature of the velocity distribution and its relationship with thermodynamic equilibrium is not obvious from this © XXXX American Chemical Society and Division of Chemical Education, Inc.
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Received: March 4, 2019 Revised: June 22, 2019
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DOI: 10.1021/acs.jchemed.9b00188 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
meaning of f(v) and its relation to the equilibrium state of the system.
energy can be safely considered as a continuous function of the velocity,14 and the summation in eq 1 can be replaced by an integral: f (vx) ≈
e−mvx +∞
2
/2kBT 2
∫−∞ e−mvx /2kBT
ij m yz zz = jjj j 2πkBT zz k {
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RELATIONSHIP OF f(v) TO OTHER PROBABILITY DISTRIBUTION FUNCTIONS It is relatively easy to derive a general form of the Maxwell distribution of speeds from probability theory and statistics of continuously distributed variables. Presenting this type of derivation to chemistry students allows them to become familiar with mathematical methods used in statistics, which are not too complex and serve as the tools to derive other distribution functions of real variables. Consider for instance n independent continuous variables xi, which are normally distributed (Gaussian-like), with mean μi and variance σi. Then, a new variable is defined as
2
e−mvx /2kBT = (2πkBT /m)1/2 dvx
1/2
e−mvx
2
/2kBT
(2)
This is the density distribution function for the velocity along the x direction. It has the shape of a Gaussian function centered at μ = 0 and with σ = (kBT/m)1/2. Therefore, calculation of the population of the (continuously distributed) velocity states directly from the Boltzmann distribution (i.e., at the equilibrium) leads to the correct expression for the density distribution functions of vx, vy, and vz. Because each component of the velocity can be considered as independent of the other two at this level of approximation, the total probability density of the velocity can be calculated as the product of the independent density distributions for each component of the velocity vector: ij m yz zz f (vx)f (vy)f (vz) = jjj j 2πkBT zz k { ij m yz zz = jjj j 2πkBT zz k {
ji xi − μi zyz Y = ∑ jjj z j σi zz { i k n
−m(vx 2 + vy 2 + vz 2)/2kBT
f (Y ) =
2
/2kBT
(3)
This function depends only on the magnitude of the velocity vector, v⃗. For being a probability distribution function, eq 3 must be normalized (i.e. integrating between 0 and ∞). The resulting normalization factor, 4πv2 dv, represents a differential change in volume in the velocity space defined by vx, vy, and vz. This leads to the final expression for the velocity distribution function: i M yz zz f (v) dv = jjj k 2πRT {
3/2
e−Mv
2
/2RT
× 4πv 2 dv
Y (n /2) − 1 −Y /2 e 2n /2Γ(n/2)
(6)
where the gamma function is Γ(p + 1) = pΓ(p), and Γ(1/2) = π . Probability density functions provide the probability that the variable is contained within a specific range of values. In this case, f(Y) presents an asymmetric belllike shape for n > 1 and tends to the symmetric Gaussian distribution as n increases (the result of the central limit theorem in probability theory). Coming back to the problem of the distribution of molecular speeds, as discussed in the preceding section, each of the components of the velocity, vx, vy, and vz, are normally distributed, with μ = 0 and σ, according to eq 2. Therefore, we can define a variable similar to Y in eq 5 of the form
3/2
e−mv
(5)
which follows the so-called χ2 distribution, with n degrees of freedom.15 The probability density of the χ2 distribution takes the form
3/2
e
2
2 2 ij vy yz ij v yz ij v yz 1 v2 Y = jjj x zzz + jjjj zzzz + jjj z zzz = 2 (vx 2 + vy 2 + vz 2) = 2 j σx z j σz z j σy z σ σ k { k { k {
(4)
2
which is the Maxwell distribution of speeds. Equation 4 is expressed in molar quantities; NAkB = R and NAm = R, where NA and R are the Avogadro number and the gas constant, respectively. The integration of the function f(v) dv between two values of v, provides the fraction of molecules whose speeds are included in that interval. This derivation is more intuitive for chemistry students after having completed an introductory lesson on the Boltzmann distribution and the partition function. It establishes a direct connection between the Boltzmann and Maxwell distributions, which indicates the probabilistic nature of f(v), and its condition of distribution at the equilibrium. The approach of eq 1 and the transition from it to eq 2 requires prior knowledge of basic aspects of quantum chemistry. I have assumed that students are familiar with concepts such as quantization of energy or separation between consecutive levels of kinetic energy in an ideal gas. If that is not the case, the instructor can follow exactly the same derivation, considering each of the velocity states in eq 1 as points in a continuous distribution. Although not formally correct in this case (the derivation of the nonquantized f(v) is a little more complicated), it allows students to understand the physical
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Because Y = v /σ must follow a χ distribution, going from f(Y) to f(v) requires the following transformation: 2
f (Y )
2
2
∂Y (v) = f (v); ∂v
f (Y (v))
2v = f (v ) σ2
Therefore, the general form of the probability density function of molecular speeds is f (v ) =
2 2 (n /2) − 1 2 2 2v (v /σ ) e−v /2σ 2 n /2 σ 2 Γ(n/2)
(8)
which, after substitution of n = 3 and σ = (kBT/m)1/2, results in i M zy zz f (v) = jjj k 2πRT {
3/2
e−Mv
2
/2RT
× 4πv 2
(9)
which leads to the Maxwell distribution of speeds after multiplying by dv. Taking σ = (kBT/m)1/2 is not an assumption a priori: the equipartition theorem states that = kT/m along each direction, and given that the mean square velocity is B
DOI: 10.1021/acs.jchemed.9b00188 J. Chem. Educ. XXXX, XXX, XXX−XXX
Journal of Chemical Education
Communication
the second moment of the distribution function, σ has to be equal to (kBT/m)1/2. The general form of f(v) expressed in eq 8 is very useful, and it serves to obtain proportionality relationships between f(v) and v under different conditions. For example, if the movement is restricted to two dimensions, eq 8 with n = 2 and σ = (kBT/ m)1/2 shows that ij m yz −mv 2 /2k T B zze × 2πv dv f (v) dv = jjj j 2πkBT zz k {
(7) Laidler, K.; Meiser, J. H.; Sanctuary, B. C. Physical Chemistry; Houghton Mifflin Company: Wilmington, MA, 2012. (8) Maxwell, J. C. On the motions and collisions of perfectly elastic spheres. London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 1860, 19, 19. (9) Levine, I. N. Physical Chemistry; McGraw Hill: New York, NY, 2002. (10) Landau, L. D.; Lifshitz, E. M. Statistical Physics; Pergamon Press: Oxford, U.K., 1969. (11) Ben-Naim, A.; Casadei, D. Modern Thermodynamics; World Scientific: Singapore, 2016. (12) Uffink, J. Boltzmann’s Work in Statistical Physics. In The Stanford Encyclopedia of Philosophy, Spring 2017 ed.; Zalta, E. N., Ed.; Stanford University, 2017. (13) Bach, A. Boltzmann’s Probability Distribution of 1877. Arch. Hist. Exact Sci. 1990, 41, 1−40. (14) Davidson, N. Statistical Mechanics; McGraw-Hill Book Company, Inc.: New York, 1962. (15) Paszek, E. Introduction to Statistics; OpenStax CNX; Rice University: Houston, TX, 2007. (16) MacDonald, I. A relation between the Maxwell-Boltzmann and chi-squared distributions. J. Chem. Educ. 1986, 63, 575. (17) Bennett, C. A.; Franklin, N. L. Statistical Analysis in Chemistry and the Chemical Industry; Literary Licensing, LLC: New York, 2013. (18) Sanger, M. J. Nuts and Bolts of Chemical Education Research; ACS Symposium Series; American Chemical Society: Washington, DC, 2008; pp 101−133. (19) Lewis, S. E. Tools of Chemistry Education Research; ACS Symposium Series; American Chemical Society: Washington, DC, 2014; pp 115−133.
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which is the Rayleigh distribution. The connection between f(v) and χ2 was previously noted by other authors,16 although it continues to be absent from most physical chemistry textbooks. However, the χ2 test is very widespread in analytical and industrial chemistry,17 as well as in chemical education research,18,19 for the analysis of the correlation between variables or the consistency between data series. Introducing the χ2 distribution to physical chemistry students would not mean an increase in the mathematical load of the curriculum but would be beneficial for the analysis of laboratory data and establishing relationships with other statistical distributions, as described above.
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SUMMARY In summary, we have shown a simple and intuitive derivation of f(v), which highlights its probabilistic character and, more importantly, its relationship with the state of equilibrium of the system. We also discussed the relationship between Maxwell’s distribution and the χ2 distribution, which can be very useful in obtaining relations of proportionality between f(v) and v in different situations relevant in chemistry.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. ORCID
Francisco Rivadulla: 0000-0003-3099-0159 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS F.R. would like to acknowledge Carlos López-Bueno from USC for fruitful discussions and Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2016−2019, ED431G/09) and the European Regional Development Fund.
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REFERENCES
(1) Murphy, K.; Holme, T.; Zenisky, A.; Caruthers, H.; Knaus, K. Building the ACS Exams Anchoring Concept Content Map for Undergraduate Chemistry. J. Chem. Educ. 2012, 89, 715−720. (2) Holme, T. A.; Reed, J. J.; Raker, J. R.; Murphy, K. L. The ACS Exams Institute Undergraduate Chemistry Anchoring Concepts Content Map IV: Physical Chemistry. J. Chem. Educ. 2018, 95, 238−241. (3) Cartier, S. F. An Integrated, Statistical Molecular Approach to the Physical Chemistry Curriculum. J. Chem. Educ. 2009, 86, 1397. (4) Atkins, P.; de Paula, J. Physical Chemistry; Oxford University Press: Oxford, U.K., 2006. (5) Engel, T.; Reid, P. Physical Chemistry; Pearson Education Inc.: Glenview, IL, 2009. (6) Dunbar, R. C. Deriving the Maxwell distribution. J. Chem. Educ. 1982, 59, 22. C
DOI: 10.1021/acs.jchemed.9b00188 J. Chem. Educ. XXXX, XXX, XXX−XXX
