In the Laboratory
An Experimental Determination of the Second Radiation Constant
W
Paul Coppens Department of Physical and Colloid Chemistry, Vrije Universiteit Brussel, Pleinlaan 2, B-1050 Brussels, Belgium;
[email protected] In most physical chemistry textbooks (1) black body radiation is dealt with in connection with the failures of classical physics. Although a number of articles on radiation laws have been published in this Journal (2–6), to my knowledge no experimental determination of the second radiation constant c2 has been described. Planck’s distribution law for a “black body” can be written as, E λ dλ =
c1 λ−5 dλ c exp 2 − 1 λT
(1)
where E λ is the emissive power,1 c1 the first radiation constant (= 2πhc 2), c2 the second radiation constant (= hc兾k), and λ is the wavelength of the black body radiation. For a “grey body” with spectral emissivity ελ and an area S, eq 1 becomes E λ dλ =
ε λ S c1 λ−5 dλ c exp 2 − 1 λT
(2)
For reasonable temperatures, eq 2 can be replaced by the Planck–Wien approximation, E λ dλ = ε λ S c1 λ−5 exp −
c2 dλ λT
(3)
which means that a plot of ln E λ versus 1兾T at a constant λ should give a straight line with slope ᎑c2兾 λ. This statement is not completely correct since ελ is a function of temperature. However, under the experimental conditions used here the variation of the spectral emissivity of the black body radiation (tungsten) with temperature is rather small, about 3%, so that for the present purpose it is taken as constant.
Experimental The experimental setup is shown in Figure 1. As “grey body” a projector lamp, L (Osram Xenophot HLX, 24 V, 250 W), with tungsten filament, K (area S), is used. Applying a potential difference, V, at the extremities of this filament causes a current, I, to flow. The product VI represents the electric power fed to the filament. As losses by conduction and convection can be ignored at normal operation temperatures of this sort of lamp, all the electric power is converted into radiation. This polychromatic radiation passes through an interference filter, F (365 nm in this work).2 The resulting monochromatic radiation is detected by a photomultiplier tube, PM (RCA 1P28), and the outcoming signal, i, is displayed on a X-t recorder (or measured by any other means). The potential difference V on the filament is varied between 4 and 17 V in steps of 0.5 V. This causes the current I and the temperature T to vary. Both V and I are measured by multimeters. The temperature is calculated as shown below. The signal, i, from the X-t recorder along with the temperature are used to calculate the second radiation constant. Hazards The high voltage applied onto the photocathode of the photomultiplier is of the order of 1 kV dc and normal precautions have to be taken when handling the equipment. Under normal working conditions there is no possibility for touching this voltage. Calculations The emissive power j (equivalent to E λ in eqs 1–3) falling onto the photocathode of the photomultiplier tube, is given by, j = A ε λ Sc1 λ−5 exp −
F X-t recorder
L K
PM I high voltage PM
V
Figure 1. Experimental setup: L–projector lamp, K–tungsten filament, V–voltage difference, I–current, F–interference filter, PM–photomultiplier tube.
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c2 λT
(4)
where A is the constant transmitted fraction of radiation. Equation 4 takes into account the absorption of radiation by the glass wall of the bulb of the projector lamp L; by the interference filter, F; and by the envelope of the photomultiplier tube, PM. The quantum efficiency of the photocathode at a particular wavelength is constant and so is the gain of the photomultiplier tube as long as the high voltage on this device is not changed. The signal, i, on the X-t recorder is proportional to the emissive power j, i = GA ε λ Sc1 λ−5 exp −
c2 λT
Journal of Chemical Education • Vol. 80 No. 11 November 2003 • JChemEd.chem.wisc.edu
(5)
In the Laboratory
Table 1. Overview of the Experimental Data, Measured and Calculated V/Va
a
I/Aa
VI/W
T/K
T ᎑1/(10᎑4 K᎑1)
i a/arb.unit
ln i
14.50
4.46
120.07
1869
5.3505
7.5
2.0149
15.00
4.72
123.60
1932
5.1760
15.5
2.7408
15.50
4.96
127.28
1990
5.0251
28.5
3.3499
16.00
5.20
131.20
2045
4.8900
50.5
3.9220
16.56
5.45
135.75
2104
4.7529
89
4.4886
17.00
5.65
139.55
2148
4.6555
135
4.9053
17.54
5.88
144.34
2200
4.5455
211
5.3519
17.97
6.06
148.30
2240
4.4643
290
5.6699
18.50
6.28
153.38
2288
4.3706
418
6.0355
18.98
6.46
158.01
2328
4.2955
567.5
6.3412
19.50
6.66
163.27
2371
4.2176
780
6.6593
10.05
6.87
169.04
2416
4.1391
1037.5
6.9446
10.56
7.06
174.55
2456
4.0717
1400
7.2442
11.00
7.21
179.31
2489
4.0177
1710
7.4442
11.50
7.39
184.99
2526
3.9588
2140
7.6686
11.98
7.56
190.57
2561
3.9047
2800
7.9374
12.50
7.74
196.75
2598
3.8491
3460
8.1490
13.02
7.93
103.25
2635
3.7951
3950
8.2815
13.50
8.07
108.95
2666
3.7509
4600
8.4338
14.03
8.25
115.75
2702
3.7010
5675
8.6438
14.50
8.40
121.80
2732
3.6603
7300
8.8956
14.98
8.55
128.08
2763
3.6193
8300
9.0240
15.50
8.72
135.16
2796
3.5765
9100
9.1160
16.00
8.84
141.44
2824
3.5411
10650
9.2733
16.50
9.02
148.83
2857
3.5002
11900
9.3843
These data points were obtained from the experiment.
where G is a constant proportionality factor (product of the quantum efficiency of the photocathode and the gain of the photomultiplier tube). So ln i = B −
c2 λT
(6)
where B is equivalent to ln(GAελSc1λ᎑5 ) and is a constant. This represents the equation of a straight line with 1兾T as the independent variable. The slope, ᎑c2兾λ, allows the calculation of the second radiation constant c2 since λ is fixed by the choice of the interference filter. The determination of the temperature T is made with the use of another radiation law, the Stefan–Boltzmann relation, that states that the total emissive power Et of a “grey body” with area S at temperature T is given by,
(
E t = εt σ S T 4 − T0 4
)
(7)
where εt stands for the total emissivity, σ for the Stefan–Boltzmann constant (= 2π5k4兾15c2h3 = 5.670400 × 10᎑8 W m᎑2 K᎑4; ref 7 ), and T0 for room temperature (1). As the electric power fed to the filament is completely converted into radiation, it follows that Et = VI and from eq 7
T =
VI + T04 εt σ S
14
(8)
The area S of the filament can be calculated by measuring the length l and diameter φ of the filament wire S = πφl
(9)
This is done by using a Precision Tool Instrument measuring microscope, model 15. The total emissivity εt, however, is a function of temperature (8). So, the determination of the temperature must be done in an iterative way. For practical purposes the relation between εt and temperature T for tungsten in the temperature range 1000–3600 K can be expressed to a good approximation by a quadratic equation 2 ε t = −3.370 × 10 −8 (T / K ) + 2.468 × 10 −4 (T / K ) − 0.102 (10)
Results The experimental data obtained by students during a four-hour laboratory session are given in Table 1. A plot of
JChemEd.chem.wisc.edu • Vol. 80 No. 11 November 2003 • Journal of Chemical Education
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In the Laboratory W 10
Supplemental Material
Notes for the instructor are available in this issue of JCE Online.
8
6
ln i
Notes
4
2
0 3.5
4.0
4.5
T
ⴚ1
ⴚ4
/ (10
5.0
5.5
ⴚ1
K
)
1. Emissive power is the radiant energy of wavelength λ per unit wavelength interval emitted per unit time by a unit area of a black body at an absolute temperature T . 2. All other wavelengths would be satisfactory as long as they lie in the working wavelength region of the photomultiplier tube used for the detection of the radiation.
Literature Cited Figure 2. Ln i versus 1/T at a wavelength of 365 nm. Slope of the regression line is ᎑3.99 x 10᎑2 K.
ln i versus 1兾T at a wavelength of 365 nm is shown in Figure 2. The slope of the regression line is ᎑(399.8 ± 1.8) × 102 K. With λ = 365 nm this gives a value for c2 = (1.459 ± 0.007) × 10᎑2 m K, in good agreement with the literature value, c2 = 1.4387752 × 10᎑2 m K (7). Conclusion In spite of the relative simplicity and the low cost of the experimental setup (most of the components are commonly present in a physical chemistry laboratory) a reliable result for the value of c2 can be obtained by use of both the Planck law for monochromatic radiation and the Stefan–Boltzmann law for the total radiation of a “grey body”.
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1. See textbooks on physical chemistry, e.g., Atkins, P.; de Paula, J. Atkins’ Physical Chemistry, 7th ed.; Oxford University Press: Oxford, England, 2002. Silbey, R. J.; Alberty, R. A. Physical Chemistry, 3rd ed.; John Wiley & Sons Inc.: New York, 2001. Berry, R. S.; Rice, S. A.; Ross, J. Physical Chemistry, 2nd ed.; Oxford University Press: New York, 2000. Laidler, K. J.; Meiser, J. H. Physical Chemistry, 3rd ed.; Houghton Mifflin Company: Boston, MA, 1999. Adamson, A. W. A Textbook of Physical Chemistry, 3rd ed.; Academic Press: Orlando, FL, 1986. 2. Bluestone, S. J. Chem. Educ. 2001, 78, 215. 3. Larsen, R. D. J. Chem. Educ. 1985, 62, 199. 4. Pyle, J. T. J. Chem. Educ. 1985, 62, 488. 5. Dence, J. B. J. Chem. Educ. 1983, 60, 645. 6. Lehman, T. A. J. Chem. Educ. 1972, 49, 832. 7. Mohr, P. J.; Taylor, B. N. CODATA Recommended Values of the Fundamental Physical Constants: 1998. In J. Phys. Chem. Ref. Data, 1999, Vol. 28, No. 6. 8. CRC Handbook of Chemistry and Physics; Weast, R. C., Ed.; CRC Press: Florida, 1986; pp E–395.
Journal of Chemical Education • Vol. 80 No. 11 November 2003 • JChemEd.chem.wisc.edu
