1 EXPERIMENT 12 THE GRATING SPECTROMETER AND ATOMIC SPECTRA OBJECTIVES • Learn the theory of the grating spectrometer • Observe the spectrum of mercury and hydrogen • Measure the grating constant of a diffraction grating • Measure the Rydberg Constant INTRODUCTION: In the previous experiment diffraction and interference were discussed and at the end a diffraction grating was introduced. In this lab you will apply what you learned to measure the wavelengths of light in the visible bright line spectrum of some elements. You will use a diffraction grating mounted on a spectrometer to measure the angular position of the interference maxima. Knowing the grating spacing and the angular position of an interference maxima, the wavelength of the light can be calculated. In the first part of the lab, you will use the known wavelengths of the mercury spectrum and the angles measured using the grating spectrometer to determine the grating spacing of the diffraction grating. In the second part of the lab you will use the calculated grating spacing and the spectrometer to determine the wavelengths of light in the visible hydrogen spectrum. White light is made up of all the colors of the rainbow - red, orange, yellow, green, blue, and violet. Different colors correspond to different wavelengths. Human eyes are sensitive to light with wavelengths in the range 390 nm (violet) to 750 nm (red) (1 nm = one nanometer = 10-9 m). 300nm 400 500 600 700violet indigo blue green yellow redwavelength(nm)ultraviolet infraredRange of human vision Figure 1The Grating Spectrometer Physics 252, v1.0) 2 Recall from the previous lab: mλ = d sin θm (1) Here, d = distance between slits m = order number of interference maximum λ = wavelength of light θm = angle from the center line to the mth interference maximum Unlike the previous lab, the grating spectrometer allows you to directly measure the angle θm in degrees and minutes of arc (60 arc minutes = 1 degree). If a plot of mλ vs. sin θm is made, it has the form of a straight line with mλ playing the role of y and sin θm playing the role of x. The equation of the line should have an intercept of zero and a slope of d, the spacing between adjacent slits in the diffraction grating. When the wavelengths of light emitted by excited atomic gases were first studied with precision spectrometers, scientists observed that most of the light was emitted at a few discrete wavelengths. In this experiment, you will observe the discrete emission lines of hydrogen and mercury gases. These discrete lines originate from the deexcitation of atoms. That the lines are discrete was initially a great surprise because it implied that the valid energies of electrons in atoms were also discrete. If any energies were possible, a continuous spectrum would be expected not a discrete line spectrum. This discovery led to the development of quantum mechanics, which is one of the foundations of modern physics. Figure 2 To put this discovery in perspective, consider what one might have expected if Newton’s laws governed the energies of atomic electrons. Atomic hydrogen consists of a proton and an electron, which is bound to the proton by the attractive Coulomb force: 22rkeFC= (2)The Prism Spectrometer (Version 4.0, 6/18/2008) 3 where k=8.99x10922CNm; e=1.6x10-19 C is the magnitude of the electron and proton charges; and r is the distance between the electron and proton. If the electron is in a circular orbit about the proton, as shown above, Newton’s second law states Fc = ma, where a is the centripetal acceleration of the electron rv2 (here, v is the speed of the electron). Combining Newton’s second law with equation (2): 222rkermv= (3) (The proton is so much heavier than the electron that one can assume that it remains stationary while the electron orbits about it.) The potential energy U(r) of the electron is given by: rkerU2)( −= (4) The total energy of the electron is the sum of the kinetic (T) and potential (U) energies E=T+U. Rearranging equation (3) yields: 22222 rkemv= (5) The left hand side of equation (5) is the kinetic energy, therefore the total energy is: rkerkeE222−= or rkeE22−= (6) It is negative because the electron is bound in the attractive Coulomb potential of the proton’s electric field. This derivation, which follows from Newton’s laws, implies that the energy of the electron is negative and has an arbitrary magnitude that depends only on the radius r of the electron’s orbit. This turns out to be incorrect for small systems like an atom. In fact, the electron in a hydrogen atom cannot be in any arbitrary orbit, but only those orbits with energies given by the Bohr formula: 20nEE −= (7)The Grating Spectrometer Physics 252, v1.0) 4 where, Eo = 2.2·10-18 J = 13.6 eV (eV is an electron volt and 1 eV = 1.6 x 10-19 J) and n = 1,2,3,4… The negative sign in the equation for the En means that the system has less energy than a free electron and proton or that the electron is bound to the proton, and it takes 13.6 eV of energy to remove the electron from the atom. Bohr's Model of the Hydrogen Atom Bohr proposed to modify this planetary model of the atom by making the following assumptions: 1. Electrons can exist only in certain special orbits about the central nucleus. The orbits are called stationary orbits or stationary states. The possible orbits are those with angular momenta that are multiples of a number called Planck’s constant (denoted by h). 2. The dynamic equilibrium of the system in the stationary states is governed by Newtonian mechanics. 3. Transitions between different stationary states are accompanied by the emission or absorption of radiation the frequency of which is given by the Planck formula, EhfΔ= (8) where f is the frequency and EΔis the difference in the energy between the two stationary states. The first assumption was made to eliminate the problem of the electrons radiating due to the acceleration in their circular orbits. This assumption was made ad hoc with no justification other than the resulting model reproduced the results of experiments. The second assumption stated that the features of the model could be calculated with classical mechanics. The third assumption was the most controversial because