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Investigation #8 Part I 1 INTROD Name: _________________________ Partner(s): _________________________ Investigation #8 Part I _________________________ INTRODUCTION TO QUANTUM PHENONEMA Near the end of the 19th century, many practitioners of what was then known as “natural philosophy” (now called “physics”) believed that the widespread success of Newtonian mechanics and Maxwell’s theory of electromagnetism essentially completed our understanding of the operation and interactions of the physical world in terms of particles and waves. (O, human arrogance—do you know no bounds?) While there were a few discoveries at the end of 19th century, many scientists believed they would soon be explained in terms of the two “well understood” theories of the day. These discoveries included: • the discovery of x-rays by Wilhelm K. Roentgen in 1895 • the discovery of nuclear radioactivity by Alexandre-Edmond Becquerel in 1896 • the discovery of the electron by Joseph J. Thompson in 1898. At that time, there were also some well-known phenomena whose explanation in terms of Newton and Maxwell remained a mystery. These included: • the spectral distribution of wavelengths from hot glowing objects (first measured by Josef Stefan in 1879) • the ejection of electrons from a metal surface by ultraviolet light (discovered on accident by Heinrich Hertz in 1887). Attempts to explain these observations in terms of the “classical theories” of Newton and Maxwell were either inadequate or led to predictions that contradicted experimental results. What appeared to be minor cracks in the foundation of physics, eventually led to the “quantum revolution” which completely altered our perception of nature. The purpose of this investigation and the next is to observe some of the phenomena that essentially led to the quantum revolution. First, you will examine the spectral distribution of wavelengths of a “blackbody emitter” and determine the relationship between absolute temperature and wavelength for such an emitter. Next, you will observe and measure the quantization of energy levels by bombarding gas atoms with electrons. In particular, you will determine the excitation energies of mercury and neon. Blackbody Radiation Spectrum (Background) One of the unsolved mysteries at the end of the 19th century was the spectral distribution of wavelengths of blackbody radiation. A “blackbody” is the name given to describe a perfect absorber—one that absorbs 100% of the radiation incident upon it. In the latter half of the 19th century, Gustav Kirchoff was able to show that the most efficient absorbers of electromagnetic waves were also the most efficient radiators of this energy 1 as well. Thus, a blackbody absorber would also be a perfect emitter. The startling discovery made by Stefan in 1879 was that the total intensity (over all wavelengths) radiated from the interior of such a body depends only on temperature and is independent of the material. In other words, the spectral distribution for a blackbody emitter at a given temperature is the same regardless of the material. This implies that the emission of blackbody radiation must be a fundamental phenomenon. A blackbody (or a good approximation to one) can be constructed by making a small hole in a material that leads to a cavity with rough walls in the interior. Light that enters the hole has little chance of escaping. It is 100% absorbed. When the material is heated, the material begins to radiate until thermal equilibrium is achieved. This occurs when the rate of absorption equals the rate of emission. The radiation that does escape the hole can then be measured and analyzed. Early attempts to explain the observed spectrum met with failure. Ludwig Boltzmann tried to use a thermodynamic approach in terms of Carnot cycles. Wilhelm Wien built upon this and derived an expression that worked well at short wavelength, but failed at long wavelengths. At the turn of the century, Lord Rayleigh assumed the source of the radiation was the electric charges in the material. Behaving like little harmonic oscillators, a given temperature would set allow the oscillators to set up standing waves of radiation. Combining this with the equipartition theorem, Rayleigh and J. Jeans found an expression that worked well at very long wavelengths, but led to more severe discrepancies as the wavelengths got shorter. As λ → 0 , the expression predicts infinite energy within the cavity. This result became known as the “ultraviolet catastrophe.” In 1900, Max Planck eventually solved the problem. He found the solution by pure mathematical reasoning. Contrary to the known laws of “classical physics,” Planck found that by assuming the energy of a harmonic oscillator could only take on integer multiples of the fundamental frequency of the oscillator he could bridge the gap between the Wien expression and the Rayleigh-Jean expression and remove the ultraviolet catastrophe. While Planck’s results fit the data perfectly at all wavelengths, it is said that even he did not want to accept the implications of his own expression. Although Wien did not derive the correct expression for the spectral distribution of blackbody radiation, he did correctly identify a relationship between the wavelength of maximum intensity and the absolute temperature. This expression known as Wien’s displacement law is given by λmaxT = (constant). In the homework, you will use the wavelengths of maximum light intensity and the corresponding temperatures that occurred in your measurements to calculate the Wien constant for each of your three datasets. 2 In this simulation activity, you will examine the radiation emitted from an (ideal) blackbody. Go to the following URL: https://phet.colorado.edu/sims/html/blackbody-spectrum/latest/blackbody-spectrum_en.html When the simulation opens, you will see: • A plot of “Spectral Power Density” (SPD )as a function of wavelength. (Note the units of each axis.) The scale of the axes can be adjusted independently using the Zoom In or Zoom Out buttons at the end of each axis. • There is a thermometer on the right with some reference values. You can change the temperature of the blackbody with the slider. • The box to the left of the thermometer, will allow you to toggle visibility of the data points (Graph Values), the range different parts of the electromagnetic spectrum (Labels), and the total intensity emitted by the blackbody (Intensity). The camera button will freeze the spectrum currently displayed so that you can change the temperature to observe a second spectrum. You can view up to three spectra simultaneously. The eraser will remove all previous display spectra. Questions: You should recognize that the quantity (SPD)(dλ) = to the intensity of the emitted radiation that having wavelengths between λ and λ + dλ.) If you wished to calculate the total intensity of radiation emitted in just the visible region of the spectrum (say, 400 nm ≤ λ ≤ 700 nm), what mathematical operation will you have to perform? Write the mathematical expression that will allow you to do this. Question: If you wished to calculate the total intensity of radiation emitted over all wavelengths what is the mathematical expression that will allow you to do this? Procedure 1. Explore the simulation, by playing with the various parameters described above. 2. Set the thermometer to 1000 K. Adjust the axes so that the peak of the spectrum is approximately in the center of the graph. 3 3. Record the Spectral Power Density, Wavelength of the Peak Intensity, and the Total Intensity of the radiation in the first row of Table 1. 4. Increase the temperature in increments of 1000K and repeat Step 3 for at least 9 more temperatures. Adjust the scale of the axes as needed so that peak of the spectrum is approximately in the center of the graph. Table 1: Blackbody Spectral Power Peak Intensity of Temperature Density Wavelength Radiation 2 2 T (K) (MW/m /µm) λmax (µm) Ι (MW/m ) 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Questions: Qualitatively, what change do you observe for the location of the peak of the spectra as the temperature of the blackbody is increased? Question: Are your results consistent with higher temperatures producing higher intensities? Explain any discrepancies. 4 INTRODUCTION TO QUANTUM PHENOMONA Homework (For all questions, you must show your work to for maximum credit.) Planck’s expression for the intensity of radiation per wavelength is given by 2πc2h ⎛ 1 ⎞ IPlanck (λ,T) = ⎜ ⎟, λ 5 ⎝ e(hc λkBT) −1⎠ where c is the speed of light, h is Planck’s constant, and kB is the Boltzmann constant. 1. Recall that the Rayleigh-Jeans radiation law worked well at long wavelengths, but failed to accurately reproduce the experimental findings at short wavelengths. Show that in the limit of long wavelengths (low energy) Planck’s radiation law (shown above) reduces to the Rayleigh-Jeans expression: 2πck T I (λ,T) = B . Rayleigh−Jeans λ 4 (Hint: Express the exponential function in a power series expansion and neglect the quadratic and higher order terms.) 2. Wien’s radiation law worked well at short wavelengths, but failed to accurately reproduce the experimental findings at long wavelengths. Wien’s expression is given by C I (λ,T) = 1 e−C2 λT , Wein λ 5 where C1 and C2 are constants. Show that in the limit of short wavelengths (high energy) Planck’s expression reduces to the Wien expression and determine the values of C1 and C2 in terms of the constants given in Wein’s expression. 5 3. As stated in the lab packet, Wien did not derive the correct expression for the spectral distribution of blackbody radiation. However, he correctly identified the relationship between the wavelength of maximum intensity and the absolute temperature. Recall that this expression (known as Wien’s displacement law) is given by λmaxT = (constant). Using the peak wavelengths (λmax) of the maximum spectral power density and the corresponding temperatures that you measured in Table 1, make a plot of λmax as a function of 1/T.
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