Communication pubs.acs.org/jchemeduc
Wien’s Displacement Law as a Function of Frequency David W. Ball* Department of Chemistry, Cleveland State University, Cleveland, Ohio 44115, United States ABSTRACT: Wien’s displacement law has an important part in the development of modern quantum theory, and predicting it was one of the triumphs of Planck’s distribution law. It is usually expressed in terms of wavelength. Less known, however, is its expression in terms of frequency. Here, we derive Wien’s law as a function of frequency and point out its major predictive difference from the more common version. This version of Wien’s law would be a useful demonstration of a change in variables for a physical chemistry lecture course and the differences in predictions upon variable change.
KEYWORDS: Upper Division Undergraduate, Physical Chemistry, Textbooks/Reference Books, Mathematics/Symbolic Mathematics, Nomenclature/Units/Symbol, Spectroscopy ien’s displacement law (hereafter, simply “Wien’s law”) is explicitly mentioned in a minority of physical chemistry textbooks.1 This omission seems a shame for two reasons. First, historically the announcement of Wien’s law in 1893 was one of the unexplained phenomena of classical mechanics, ultimately acting as one test of the new quantum theory of light. Second, even today Wien’s law is used spectroscopically to estimate the temperature of objects from a distance, either close (for example, using a hand-held device to measure the temperature of molten metal) or distant (for example, in estimating the temperature of the surface of the sun or other celestial object). Wien’s law usually states that for the (blackbody) radiation emitted from an object having nonzero absolute temperature, the wavelength maximum of the radiation intensity curve, λmax, is inversely related to absolute temperature T:
W
λmax T = constant
(1)
Figure 1. A plot of Planck’s distribution law versus wavelength, showing how the wavelength maximum decreases as the temperature is increased. The curve just barely above the x axis is for T = 3000 K.
This constant is generally given as 2898 μm K.1a (The current CODATA-recommended value is 2.8977685(51) × 10−3 m K.2) Figure 1 shows plots of Planck’s distribution law versus temperature, demonstrating that as the temperature increases, λmax decreases. One of the successes of Planck’s distribution law (hereafter, Planck’s law) for light is that it could predict Wien’s law. Because Planck’s law expresses some property of light (spectral radiance, energy density, power flux) versus the light’s wavelength, the derivative of the expression for Planck’s law can be taken, set equal to zero, and solved for the maximum wavelength, yielding an expression that verifies Wien’s law. However, Planck’s law can also be expressed in terms of light’s frequency, begging the following question: what is the form of Wien’s law in terms of frequency, rather than wavelength? This form of Wien’s law was mentioned by © 2013 American Chemical Society and Division of Chemical Education, Inc.
Bluestone,3 but not explicitly derived nor discussed. The process is similar to the derivation of the wavelength version, but the result is a bit different. Performing the derivation and considering the results provides additional insight into the energy distribution of blackbody radiation, and also illustrates how a change in variableeven a related variablecan impose a significant difference in predicted trends.
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THE DERIVATION We start with Planck’s law written in terms of frequency: Published: July 31, 2013 1250
dx.doi.org/10.1021/ed400113z | J. Chem. Educ. 2013, 90, 1250−1252
Journal of Chemical Education U=
8πν 3h ⎛ 1 ⎜ c 3 ⎝ ehν / kT −
Communication
⎞ ⎟ 1⎠
⎛ 1 ⎞ ⎟ = 0 = 3 − x(e x)⎜ x ⎝ e − 1⎠
(2)
where U is the energy density of the light (in units of J/m3s), ν is the frequency of the light, h is Planck’s constant, c is the speed of light, and k is Boltzmann’s constant. Plots of this function versus the same temperatures as Figure 1 are shown in
=3−
xex =0 ex − 1
(6)
This equation is not solvable analytically. However, it is easy to solve numerically. Table 1 shows a portion of an Excel Table 1. Author’s Systematic Solution of the Expression xe x 3 − ex − 1 = 0 3−
x=
→ 2a 1.418023 0.686965 −0.15719 0.276436 0.191395 0.105472 0.018707 −0.02498 −0.00748 0.001257 0.000384 3.44 × 10−5 −5.3 × 10−5 8.19 × 10−6 −5.5 × 10−7 3.25 × 10−7 −2.4 × 10−8 1.85 × 10−9
0 1 2 3 2.5 2.6 2.7 2.8 2.85 2.83 2.82 2.821 2.8214 2.8215 2.82143 2.82144 2.821439 2.8214394 2.82143937
Figure 2. A plot of Planck’s distribution law versus frequency, which is a less-common representation. The frequency maximum increases with temperature. The curve just above the x axis is for T = 2000 K, and the curve for T = 1000 K is almost completely coincident with the x axis on this scale.
xex = e −1 x
a
The fraction in the expression is indeterminate at x = 0, but analysis shows that the function converges to 2 at x = 0.
Figure 2. We take the derivative of this equation with respect to ν, applying the chain rule: dU =
=
24πν 2h ⎛ 1 ⎜ hν / kT 3 ⎝ c − e ⎛ h ⎞ hν / kT ⎜ ⎟ (e ) ⎝ kT ⎠
24πν 2h ⎛ 1 ⎜ c 3 ⎝ ehν / kT −
⎞ 8πν 3h ⎛ 1 ⎟+ ⎜ 3 ⎝ hν / kT ⎠ c − 1 e
⎞2 ⎟ ( −1) 1⎠
⎞ 8πν 3h2 hν / kT ⎛ 1 ⎟− (e )⎜ hν / kT 3 ⎠ ⎝ kTc 1 e −
⎞ ⎟ 1⎠
spreadsheet that was used to determine the value of x systematically. Ultimately, we find that (7) x = 2.8214393... for the point at which eq 2 is a maximum. Thus, substituting into eq 5, we have
2
2.8214393... ≡
(3)
This expression is set to zero to find the curve maximum: =
=0
Quite a bit cancels from both terms. When this is done, what remains is
T = 1.70099... × 10−11 K s νmax
hν kT
(8b)
There are a few things to note about eq 8. First, the constant involved in the equation is not directly related to the constant in the wavelength version of Wien’s equation. That is, a simple substitution using ν = c/λ does not generate the frequency version of the constant. Second, a given temperature does not predict equivalent frequency/wavelength pairs using both versions of Wien’s law, as demonstrated in Table 2. If they did, then the last column would equal c, the speed of light.
(4)
At this point, we will define a unitless parameter x as hν/kT: x≡
(1.38065 × 10−23 J/K)T
Rearranging this allows us to rewrite Wien’s law in terms of frequency: νmax = 5.87893... × 1010 K−1 s−1 (8a) T or, reciprocally,
⎞ 8πν 3h2 hν / kT ⎛ ⎞2 24πν 2h ⎛ 1 1 ⎜ ⎟ ⎜ ⎟ − (e ) ⎝ ehν / kT − 1 ⎠ c 3 ⎝ ehν / kT − 1 ⎠ kTc 3
⎞ hν hν / kT ⎛ 1 ⎟=0 =3− (e )⎜ hν / kT ⎝e kT − 1⎠
(6.6260684 × 10−34 J s)νmax
(5)
Substituting this unitless parameter into eq 4, we get 1251
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Table 2. Values of λmax and νmax from the Two Versions of Wien’s Law Temp/K 1000 2000 3000 4000 5000 6000 7000 8000 a
λmax/nm
νmax/s−1
2898 1449 966.0 724.5 579.6 483.0 414.0 362.3
× × × × × × × ×
5.879 1.176 1.764 2.352 2.939 3.527 4.115 4.703
λmax × νmax/(m/s)a 13
10 1014 1014 1014 1014 1014 1014 1014
1.704 1.704 1.704 1.704 1.703 1.704 1.704 1.704
× × × × × × × ×
108 108 108 108 108 108 108 108
Note the conversion from nm to m in the last column.
Instead, the products of λmax × νmax equal 1.704 × 108 m/s, which can be easily demonstrated as λmax × νmax = (λmax T ) ×
⎛ νmax ⎞ ⎜ ⎟ ⎝ T ⎠
= (2898 μm K) × (5.879 × 1010 K−1 s−1) = 1.704 × 1014 μm/s = 1.704 × 108 m/s
which the last column in Table 2 shows. Why, finally, are these numbers related in an unexpected way? The answer is because we start with an expression for energy density and then maximize it versus wavelength for the classic expression of Wien’s law and then versus frequency for the version derived here. Energy density is different per unit frequency than it is per unit wavelength; the energy density per unit wavelength is the more familiar U=
8πhλ 3 1 3 hc / λkT c e −1
(9)
which numerically maximizes to a different expression. Because both forms of Planck’s law must be maximized numerically, the constant defined by λmax × νmax, 1.704 × 108 m/s, cannot be expressed exactly in terms of fundamental constants. As such, the maximum points in Figures 1 and 2 occur at seemingly mathematically unrelated points.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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REFERENCES
(1) Of the available physical chemistry textbooks, the only ones that mentioned Wien’s displacement law are: (a) Ball, D. W.; Physical Chemistry; Cengage/Brooks-Cole Publishing: Pacific Grove, CA, 2003. (b) McQuarrie, D. A.; Simon, J. D.; Physical Chemistry; University Science Books: Sausolito, CA, 1997. (c) Noggle, J. H.; Physical Chemistry; HarperCollins: New York, 1996. (d) Silbey, R. J.; Alberty, R. A.; Physical Chemistry; John Wiley & Sons: New York, 2001. (2) Mohr, P. J.; Taylor, B. N.; Newell, D. B. Rev. Mod. Phys. 2008, 80, 633−730. (3) Bluestone, S. J. Chem. Educ. 2001, 78, 215−8.
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