Grating Spectrometer.doc

phys 2211llab 19the grating spectrometer and the rydberg constantpurposelight carries an extraordinary amount of information. beyond its ability to form images, the mixture of wavelengths the spectrum of which light is usually composed can tell us a great deal about distant or inaccessible objects. it is by examining spectra that we have been able to determine such things as the structure of atoms, the composition of stars, the velocities of galaxies, even the age of the universe.the tool for doing this is the spectrometer. in this lab we will use a grating spectrometer to measure the wavelengths of light emitted by several elements and to identify an unknown gas. we will also use it to take a peak inside the hydrogen atom and “weigh” its electron.principlesthe grating spectrometer consists of a diffraction grating set behind a narrow slit. light passing through the slit and the grating is diffracted at an angle that depends on its wavelength. red light (long wavelengths) will emerge from the grating at larger angles than blue light (shorter wavelengths). the observer sees a separate image of the slit for each wavelength (color) in the light.the set of wavelengths in the light from an object is called its spectrum. there are two general types of spectra: continuous and discrete. in a continuous spectrum, there is a complete range of wavelengths so that the colors gradually pass from blue to red. a rainbow is a continuous spectrum created by water droplets in the air. (the drops act as small prisms prisms will also diffract light according to wavelength.)in a discrete spectrum, only selected wavelengths are represented. the light from a neon sign is a selection of discrete wavelengths. our eyes average these out into a single color, but a spectrometer displays each wavelength individually.how light is generatedit will be helpful to consider how light is generated. matter is made of charged particles electrons and protons, which together with neutrons form atoms and molecules. when charged particles vibrate, they emit electromagnetic radiation light. (we will call any em radiation “light”, even though it may not be in the visible range of frequencies.)the light waves generated by charged particles will have the same frequency as their frequency of vibration. the wavelength of the light is inversely related to the frequency. (recall that the speed of a wave is its wavelength times its frequency: v = f. since the speed of light is fixed, the wavelength is determined by =v/f.)one mechanism of light generation is called thermal radiation. all atoms and molecules are continually vibrating or colliding with each other, at frequencies determined by their temperature. because of the random nature of their motions, there is a complete range of frequencies involved, and so a continuous spectrum of light is generated.a second mechanism involves the shifting of electrons between energy levels inside an atom. each atom in nature has a unique configuration of energy levels in which the electrons of the atom reside. when an atom receives a kick of energy for instance by a collision with another atom one or more of its electrons will shift to a higher energy level. to return to its “ground” state, the electron must give up this extra energy, and it does so by emitting light. the wavelength of the light depends on the difference in energy between the two levels.if we excite an elemental gas for instance neon at any given time there will be many electrons undergoing all possible energy level shifts. since there is a discrete (i. e., countable) number of possible shifts, a discrete spectrum of light will be given off by the gas. this is what the neon in a neon sign is doing, and this is what we will observe with the spectrometers.the hydrogen spectrum and the rydberg constantsince the beginning of spectroscopy in the 19th century, scientists have noted patterns in the spectral lines of the elements. (the image of the spectroscopes slit looks like a line of light, so different wavelengths are commonly called “lines”.) the four lines in the visible spectrum of hydrogen can be quantified by the empirical formula(n = 3, 4, 5, 6)rh = 1.097 x 107 m-1rh is called the rydberg constant (for hydrogen) the above is called the balmer formula and refers to the visible lines in the spectrum. if we include infrared and ultraviolet lines, the formula can be generalized to(1)these patterns were mysterious at first, but we now know they are clues to the internal structure of atoms. the ns refer to the energy levels of the atom, with the lowest energy level (the ground state) being labeled n = 1. as an electron drops from a high energy level (labeled by ni to a lower state, labeled by nf, it will emit light of wavelength .the bohr model of the hydrogen atommaking the connection between the above empirical formula and the internal structure of an atom took all the tricks in the classical physicists bag from mechanics through electrodynamics and optics as well as some new ideas that came to be known as quantum mechanics.the story can be outlined briefly. first, in studying thermal radiation, the physicist max planck found that its spectrum could be understood only if it were postulated that the energy of radiation was proportional to its frequency, e = hf(h = 6.626 x 10-34 j-sec)and that the energy stored at a given frequency in an electromagnetic field had to be an integer multiple of a fundamental energy: efield = nhf, with n integer. the constant h is known as plancks constant. this was the origin of the idea of the quantum that energy comes in discrete dollops rather than continuously.einstein carried the story further by proposing that light, although it travels as a wave, is emitted and absorbed by electrons in discrete packets, now called photons. by identifying these photons with plancks quanta, he was able to successfully explain the photoelectric effect.finally, niels bohr applied these ideas to the hydrogen atom, and added a few ideas of his own. it was known that the hydrogen atom contained a negative charge in the form of an electron, and a positive charge in the form of a nucleus (a proton), but it was a mystery how the two could form a stable configuration without falling into each other. if the electron simply orbited the nucleus like a planet around the sun, it would quickly radiate away its energy and fall into the nucleus. (it is well established that all accelerating charges radiate, and the electron is no exception. the glow of a light bulb comes from the acceleration of electrons.)bohr proposed that the electron did indeed orbit the nucleus, but that its angular momentum is quantized, so that the electron is fundamentally restricted to certain discrete values of angular momentum. he showed that these allowed values had to be proportional to plancks constant h, so bohr postulate wasl = n (n = 1, 2, 3, . . .)the symbol is a shorthand for h/2 and is read “h-bar”.thus, bohr said, the electron couldnt radiate, because if it did it would lose energy continuously, which would change its speed, which would change its angular momentum to values that are not allowed.instead, the electron would only radiate when it jumped from one allowed orbit to another. each orbit must have a definite energy associated with it, so to switch orbits the electron must radiate or absorb a photon of energy equal to the difference in energies between the orbits. this fixed the frequency, consequently the wavelength, of radiation the atom could emit. hence, the atom would emit a discrete spectrum.energy levelsbohr derived the allowed energy levels by applying a few basic principles from mechanics and electrodynamics:1.()the energy of the electron is the sum of its kinetic and potential energies.its potential energy is the electrostatic potential energy between the electron and the nucleus, both with magnitude of charge e.2.(n = 1, 2, 3, . . .)a particle in a circular orbit has angular momentum l = mvr. by bohrs postulate this must have one of the values n, with n an integer greater than zero.3.the centripetal force on the electron is provided by the electrostatic force between it and the nucleus.putting these three facts together leads to an expression for the energies of the allowed levels:(2)for n = 1, 2, 3 . . .the negative sign indicates the electron is in an energy “well”. it must receive energy from outside to escape from the nucleus. the lowest energy state is represented by n = 1. using the accepted values for the constants, this energy is -2.18 x 10-18 joules. a positive energy of this amount would be required for the electron to escape from the nucleus. this is indeed the ionization energy of hydrogen in its ground state.balmer formula revisited.when the electron makes a transition from a higher to a lower energy level, it must emit a photon of energy . by the planck-einstein relation, , where c is the speed of light. soor(we used h = 2.) this is the balmer formula, with(3)using accepted values of the constants, this give rh = 1.097 x 107 m-1, in excellent agreement with experiment.the bohr model was very successful in describing hydrogen and other one-electron atoms. however, it failed when applied to more complicated atoms. a new version of quantum mechanics had to be developed, and that has been very successful in describing all phenomena at the atomic level. though no longer used, the bohr model still serves as a stepping-stone from classical to modern physics.we will observe the four lines in the hydrogen spectrum and use the measured wavelengths to determine rh. we can then use this and expression (3) to calculate the mass of the electron.156aug 05the grating spectrometerlab 11procedures: elemental spectrain this first part, we will calibrate the spectrometers by measuring lines in the spectrum of helium. graphing the known values against the measured values gives us a calibration curve for adjusting the readings of the instrument.once we have our calibration curve in hand, we will use it to measure the spectrum of mercury, and to identify unknown element. finally, we will observe and describe the continuous spectra of a light bulb and of the fluorescent lighting.equipment spectroscopes spectrum tube power supply helium gas discharge tube mercury gas discharge tube unknown gas discharge tubes incandescent lamp graph papernote: -always turn the power supply off when removing or inserting a tube. keep your finger clear of the sockets on the power supply. the power supply applies a high voltage to the gas tubes-handle the tubes by their ends or by the small glass handle. do not touch the tube in the middle.-turn the power supply off when not observing. the tubes can get quite hot. use a paper towel to handle the tubes if they are hot.i. the helium spectrum place the helium tube in the power supply and turn it on. familiarize yourself with the use of the grating spectrometer. in particular, learn how to measure the wavelengths of the lines that appear. we will measure the wavelengths in nanometers rather than angstroms, since this is the practical limit of measurement for these spectroscopes. measure the wavelength of 8 lines in the helium spectrum. also note the color and relative intensity of the lines and record your observations. for relative intensity, use this scale:1 = the brightest lines2 = moderately bright3= faint4= edge of visibility calibrate the spectroscope by graphing the reference wavelengths for these lines against your measured values. (see appendix for reference values). the reference values go on the vertical axis; your experimental values go on the horizontal axis. use the same scale on each axis. draw the best smooth curve through your data.-do a rough-and-ready calibration for use in lab. in your report, do a more careful calibration: do a least-squares fit for the data, treating the data as linear, to find the best-fit line. (if your data are clearly not linear, use a quadratic or exponential fit.)use your graph to adjust your measured values for the wavelengths. the wavelength scales in the spectrometers may not be accurately placed and there are usually significant parallax errors in reading the values.ii. measure the mercury spectrum. place the mercury tube in the power supply and measure the wavelengths of 8 of the brightest lines. record the values, colors and relative intensities. adjust your measured values using your calibration graph. starting at a measured wavelength, read vertically to the curve, then horizontally to the reference value. tabulate these adjusted values along with your data. identify the experimental wavelengths with the reference values given. take the percent error of your values.iii. identify an unknown spectrum. using the technique above, measure several spectral lines from the “unknown” spectrum tube and identify the element by comparing your values with those in the reference tables.iv. continuous spectraa. observe the light bulb of a table lamp with the spectrometer. diagram the spectrum vs. wavelength, annotating with the observed colors.b. observe a fluorescent lamp. what element can you distinguish in the spectrum?the grating spectrometerlab 11procedures: the rydberg constant the hydrogen spectrum and the ryberg constant measure the spectral lines of hydrogen. there are four visible lines. one will be extremely faint, but it can be measured with determination. use your values to determine the rydberg constant, rh:the visible lines correspond to n = 3, 4, 5, 6. calculate rh for each line and take an average. compare your value with the accepted value. calculate the average deviation in the values and use this as the uncertainty, drh. use your value for rh and the bohr formula for rh to calculate the mass of the electron, me:(k = 1/40) calculate the uncertainty in the mass, dm from the uncertainty