PHYS272_Lab10

=Physics for Scientists and Engineers - Lab 10​ PreLab =  This assignment is due at the start of the laboratory period.

1. You use an open spectrometer to take measurements of the blue hydrogen line. You find that the distance from the central slit to the blue line on the left is s 1 = 12.4 ± 0.2 cm and the distance to the blue line on the right is s 2 = 12.0 ± 0.2 cm. Find the average distance s = (s 1 + s 2 )/2 and the uncertainty in the average distance. Use the fact that if y=f(x 1, x 2 ) then

math \displaystyle \Delta y = \sqrt{(\partial f / \partial x_1)^2 \Delta x_1^2 + (\partial f / \partial x_2)^2 \Delta x_2^2}. math

Next, find the angle θ between the center of the diffraction pattern and the blue line using the fact that the distance between the diffraction grating and the screen is D = 50.0 ± 0.5 cm and θ = s/D. Find the uncertainty in θ also. Finally, find the wavelength of the blue line using λ = d sin( θ ) where d = 1/(500,000 m -1 ) and the uncertainty in the wavelength. Does the uncertainty interval overlap with the accepted value of 486.3 nm?

2. The wavelengths of light emitted by Hydrogen result from an electron in a more excited m state falling down to a less excited n state and are given by 1/ λ = R H (1/n 2 - 1/m 2 ) and R H = 1.09678 × 10 7 m -1. Calculate the possible wavelengths emitted by an electron jumping down to n = 1 from m = 2, 3, 4, 5, 6, 7, 8, 9, 10. Then, calculate the wavelengths emitted by an electron jumping down to n = 2 from m = 3, 4, 5, 6, 7, 8, 9, 10. Repeat this process up to n = 5. Finally, circle all of the wavelengths that are in the visible part of the spectrum. How many visible wavelengths are there total?

3. Repeat problem number 2 for singly ionized Helium, for which R He = 4.387 × 10 7 m -1. = = =Physics for Scientists and Engineers – Lab 10 =

Objectives:

 * To determine wave lengths of the bright emission lines of hydrogen.
 * To test the relationship between wavelength and energy as implied by the Bohr model.
 * To determine the value of the Rydberg constant.
 * To observe and describe the more complex spectra of other atoms including the doublet structure of sodium lines and to identify the atomic species of three unknown sources.

Equipment:

 * Spectroscope platform and lab jack
 * Grating post and diffraction grating
 * Hydrogen geissler tube light source
 * Other geissler tube light sources
 * Mercury light source
 * Sodium light source
 * Unknown sources
 * Ocean Optics Spectrometer
 * Winsco Spectrum Analysis chart
 * Graphical Analysis software

Physical Principles:
 The relation between the velocity c of a wave, the wavelength λ, period T and frequency f is given by

math \displaystyle (1)\ \ \ \lambda\ =\ c\, T\ =\ \frac{c}{f} \ \ \ \ \ \mbox {where} \ \ \ \ c\ =\ 2.998\ \times \ 10^8\, \ {\frac{\mbox{m}}{\mbox{s}}} math

When two waves (assume equal amplitudes) arrive at a point such that both waves have their maximum disturbances at the same time, the resulting disturbance has an amplitude which is the sum of the two amplitudes (twice the amplitude) and we say the waves are in phase. The energy of the disturbance increases as the square of the amplitude (4 times the energy in one wave). When the two waves arrive such that one wave has a maximum disturbance (crest) when the other has its disturbance in the opposite direction (trough), the resulting amplitude is the difference between the amplitudes of the first and second wave (zero, for example), and we say that the waves are 180° out of phase.

The diffraction grating consists of a large number of very narrow slits (500 slits per mm for our gratings). Light from a source passes through a slit and after traveling a distance D arrives at the grating. The waves reaching different points of the grating are in phase. Light rays which leave the grating at an angle θ will have phase differences which are the bases of the triangles illustrated in figure 1. If the base of the smallest triangle is equal to the wave length λ, the waves will be in phase and a bright line (image of the slit) will be seen at the angle θ from the slit. If this base is much different than the wavelength light from many pairs of slits will be nearly out of phase and will cancel and no light will be seen in this direction. Thus the condition that the slit will be imaged by a color with a wavelength λ at an angle θ is

math (2)\ \ \ d\, \sin \theta\ =\ n\, \lambda\ math

where the angle θ (in radians) is given by

math \displaystyle (3)\ \ \ \theta\ =\ \frac {x} {D} math

with //x// the distance along the arc from the slit to the direction of the bright image of the slit in that color and D = 0.50 m is the radius of the scale. This follows from the definition of angle and the fact that the grating is placed at the center of curvature of the scale arc. The grating has 500,000 slits per meter so that d = 1/(500,000 m -1 ) = 2×10 -6 m. Thus an observation of the distance, x, of the first order (n=1) virtual image of the slit in a particular color with equations (2) and (3) gives the value of the wavelength.

math (4)\ \ \ \ \lambda \ = \ d\, \sin \theta \ math

Light energy is absorbed by atoms in bundles called photons. The energy of a photon is related to the frequency and wavelength of the photon by the Einstein relation

math \displaystyle (5)\ \ \ \ \ E\ =\ \frac {h \, c} {\lambda} math

where Planck's constant h = 6.624×10 -34 J s. Atoms exist in certain discrete energy levels and in the case of the simple atom of hydrogen the energies of these levels are given by

math \displaystyle (6)\ \ \ E_n\ =\ \frac{-E_0\,}{n^2}\ \ \ \ \ \ \ \ \ \ \ n\ =\ 1,\ 2,\ 3,\ ... math

where n is some positive integer, 1, 2, 3, …, and

math E_0 = 13.6 \, \mathrm{eV} = 2.18\times 10^-18 \, \mathrm{J}. math

 When atoms are in excited states (n values greater than 1) they can emit a photon and enter a state with a lower (more negative) energy and smaller value of n (see figure 2). If m is the integer for the higher state and n is the integer for the lower state the energy difference between the states is equal to the energy of the emitted photon. Thus

math \displaystyle (7)\ \ \ E_{photon}\ = \ E_{m \to n} \ = E_0 \,\left (\frac{1}{n^2}\ -\ \frac{1}{m^2} \right) math

 and from equation (5)

math \displaystyle (8)\ \ \ \frac{1}{\lambda}\ = \ \frac{E_0 }{h\,c}\,\left (\frac{1}{n^2} - \frac{1}{m^2} \right) =R_H\left (\frac{1}{n^2}\ -\ \frac{1}{m^2} \right) math

 where

math \displaystyle R_H = \frac{E_0} {hc} = 1.09678 \times 10^7 \, \mathrm{m}^{-1} math

is called the Rydberg constant. For the hydrogen lines in the visible spectrum, (.4×10 -6 m < λ < .7×10 -6 m), n = 2 and m = 3, 4, 5, 6,….

For an interesting simulation of a hydrogen atom, visit []. In the upper left, change the experiment to “Bohr Model” and observe how the various colors of light interact with the electron and affect which energy levels it occupies.

= = =Procedure: =

Patterns caused by diffraction gratings
 Insert the hydrogen tube in the power supply shown in figure 3 and turn on the electric power. Set the spectroscope platform on the lab jack in front of the light source and adjust the height so that the slit is at the level of the center of the Geisler tube. Set the platform slit close to the source and align it so that the light source is in line with the grating and slit. Place the grating on the grating post and place the post in the hole on the platform. Rotate the grating about a horizontal axis while looking through the grating toward the slit until the colored lines are in a horizontal row on both sides of the slit. Measure and record the distance D between the grating and slit.

Read and record the position x of the red hydrogen line (m = 3) on the right and left side of the scale, in a table. Compute and record the average value of θ and its uncertainty in the table and from there calculate the wavelength (λ) and the uncertainty in the wavelength. Compare this with the accepted value. Repeat the process for the blue (m = 4) and violet lines (m = 5).

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Hydrogen spectrum
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Use the Ocean Optics Spectrometer and SpectraSuite software to record a hydrogen spectrum to record the hydrogen spectrum and read the values of the wavelengths of transitions from the m 6, 7 and 8 levels into the n = 2 level.

Complete the following table and compare your measured values with the accepted ones. Construct a plot of y = 1/λ versus x = 1/m 2 using your experimental values. Include error bars in your plot. From the graph determine the value of the Rydberg constant ( R = -slope) and determine the percentage of error from the accepted value of 1.09678×10 7 m -1. From the y intercept show that in equation (8) n = 2.


 * Table: Hydrogen spectra**
 * ~ Energy level ||~ Accepted Wavelength ||~ Measured Wavelength ||~ Uncertainty in Wavelength ||~ Percent Error ||
 * m = 3 || 656.5 nm ||  ||   ||   ||
 * m = 4 || 486.3 nm ||  ||   ||   ||
 * m = 5 || 434.2 nm ||  ||   ||   ||
 * m = 6 || 410.3 nm ||  ||   ||   ||
 * m = 7 || 397.1 nm ||  ||   ||   ||
 * m = 8 || 389.0 nm ||  ||   ||   ||

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Helium Spectrum
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> With the Star Spectroscope observe the visible lines of Helium and record the color, energy and wavelength of each of these lines. Could any of these result from transitions in singly ionized Helium? You calculated the wavelengths for ionized Helium transitions in your prelab. Note that <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">E 0 = Z 2 E 0H which is 54.4 eV and the Rydberg constant for helium is R He = 4 R H or 4.387×10 7 m -1. Which wavelength seen with the Ocean Optics spectrometer is closest to one of the lines in the visible region that you calculated in your prelab? What are the initial and final states of this transition?.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Spectrum of other light sources
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> View two other Geisler tubes, fluorescent lights and tungsten filaments through the spectroscope and compare qualitatively your observations of the spectra with that of hydrogen. Do you think these spectra can be fit to equation (8). Explain your response to this question. Identify the elements in the two other Geisler tubes by reference to the Winsco Spectrum Analysis Chart.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Sodium and Mercury light sources
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Use the Project Star Spectrometer shown in figure 4 to observe the sodium light source and the mercury light source and record the color and wave lengths of the bright lines. Note that when viewed with the Ocean Optics Spectrometer all the sodium lines are double. Can you suggest a reason why this might be so? Compare your observations with the values shown below.

The wavelengths of the brightest lines of sodium and mercury are listed below in nanometers.
 * ~ Sodium: || 615.4, 616.1, 589.0, 598.6, 568.3, 568.8, 514.9, 515.4, 497.9, 498.3, 474.8, 475.2, 466.5, 466.9, 449.4, 449.8 ||
 * ~ Mercury: || 579.1, 579.0, 546.1, 435.8 ||

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Fraunhofer absorption lines
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Direct the Project Star Spectrometer out of the window in HYH219 with the room darkened. Record the wavelengths of four of the darkest of the Fraunhofer absorption lines in a table. Can you identify any of these lines?