Letter pubs.acs.org/jchemeduc
A Specific Mathematical Form for Wien’s Displacement Law as vmax/T = constant Brian Wesley Williams* Department of Chemistry, Bucknell University, Lewisburg, Pensylvania 17837, United States ABSTRACT: For Wien’s displacement law in terms of frequency v and temperature T (vmax/T = constant) a specific mathematical form for the equality is derived. KEYWORDS: Upper Division Undergraduate, Physical Chemistry, Mathematics/Symbolic Mathematics, Spectroscopy
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n a recent issue of the Journal,1 David Ball considered Wien’s displacement law as a function of frequency vmax and temperature T: vmax/T = constant. Here, vmax represents the frequency associated with the maximum value of the Planck’s distribution law, also considered in terms of frequency. Determining a numerical value for vmax/T involved taking the first derivative of the distribution with respect to frequency, setting this equal to zero, and defining a variable x = hv/kT in terms of Planck’s constant h and Boltzmann’s constant k. This resulted in the equation ⎛ 1 ⎞ ⎟ = 0 3 − x(e x)⎜ x ⎝ e − 1⎠
AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Ball, D. W. Wien’s Displacement Law as a Function of Frequency. J. Chem. Educ. 2013, 90 (9), 1250−1252. (2) Valluri, S. R.; Jeffery, D. J.; Corless, R. M. Some Applications of the Lambert W Function to Physics. Can. J. Phys. 2000, 78 (9), 823− 831.
(1)
for the value of x corresponding to vmax/T. The author commented that “this equation is not solvable analytically” and obtained a numerical solution for x.1 The point of this letter is to note that eq 1 has a specific solution in terms of the Lambert W function (defined such that for yey = z, W(z) = y).2 For eq 1, multiplication by (ex − 1) gives 3(ex − 1) − xex = 0, leading to (x − 3)ex = −3. Multiplication by e−3 then gives (x − 3)e(x−3) = −3e−3 and ultimately
x = 3 + W ( −3e−3)
(2)
For negative real arguments z such that −1e−1 ≤ z ≤ 0, the Lambert function W(z) has two real valued branches, W(0,z) and W(−1,z) . This gives two possible solutions to eq 2: x = 3 + W (0, −3e−3) = 3 + ( −0.17856063) = 2.82143937 and x = 3 + W ( −1, −3e−3) = 3 + (− 3) = 0
(3)
As vmax is positive, the first solution is acceptable, and matches the numerical solution previously presented for x. So, the final form for the displacement law becomes vmax/T = (3 + W(0,− 3e−3))k/h. This example demonstrates the value of the Lambert W function in representing solutions to equalities involving the exponential function. © 2014 American Chemical Society and Division of Chemical Education, Inc.
Published: April 7, 2014 623
dx.doi.org/10.1021/ed400827f | J. Chem. Educ. 2014, 91, 623−623
