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Bohr Model/ Spectroscopy

Purpose: To understand and make use of the relationship between Bohr’s model of the atom and the observation that heated gasses emit light at discrete wavelengths rather than in continuous spectra. This lab will use a diffraction grating to determine the specific wavelengths of light emitted by various light sources.

Background for the Bohr Model of the Hydrogen Atom:

In today's lab, you will have the opportunity to explore the basic principles used by Niels Bohr in his development of a model to explain the hydrogen atom. Although a more complete and complicated quantum mechanical model has been developed to explain atomic structure, the clarity and relative simplicity of Bohr's model remain essential to our understanding of the inner workings of the atom. You will be able to see the striking similarities between an electron orbiting a proton and the motion of the planets under the Sun's gravitational influence. Despite a difference of 1020 (yes, 100 billion-billion) in scale between planetary orbits and atomic radii, you will be able to use the same basic principles of uniform circular motion and conservation of energy to explain their motions. Following Bohr's footsteps, you too will be able to predict the hydrogen spectrum.

Bohr postulated that the electrons orbiting an atom could only occupy certain states that had a “quantized” amount of angular momentum. This so-called “magic assumption” required that for an electron to be in a stable orbit, it would need to have an angular momentum equal to nh/2, where h is Planck’s constant and n is an integer. When an electron moves from one of the higher energy orbits to a lower energy orbit, energy is released in the form of a photon of light. The energy of the photon is inversely proportional to its wavelength. The derivation of an equation, primarily using equations that you have already worked with in other applications, relating the energy and wavelength of the photon emitted as an electron shifts orbits, is included in appendix I. Thus, using Bohr’s model of the atom, theoretical values for the different wavelengths of the light emitted by hydrogen could be found and compared to experimentally measured results.

Introduction: Diffraction, rather loosely defined, is the flaring of light as it passes through a narrow slit or past an obstacle. The flaring of the light passing through multiple adjacent narrow slits produces constructive interference patterns that are determined by the relationship between the wavelength of the light, the width of the slits and the distance between slits (d). A “diffraction grating” is a piece of transparent plastic that has a set of very fine parallel grooves or lines etched in it. Similar to waves in a pond, if the path length difference between light rays coming from adjacent slits landing on the screen is equal to an integer number of wavelengths, then a bright spot will appear. A diffraction grating produces a pattern similar to that of a prism, although the mechanism is different. (diffraction vs refraction) We will use a diffraction grating of [initially] unknown characteristics to determine the wavelength of various colors of light produced by various light sources, starting with hydrogen.

Procedure: First, we must determine the slit spacing in your diffraction grating. Your tools for this mission will be a laser light source of known wavelength, a diffraction grating and a meter-stick “crossbow” to help measure the positions of the lines. The layout is shown in Figure 1 below.

Determining Diffraction Grating Spacing

When light of one particular wavelength, λ, is incident on a diffraction grating, the light constructively interferes at positions that are related by an angle θ from the diffraction grating given by:

d sin θ=mλ

where d is the distance between the grooves or lines that are etched into the grating and m is an integer: 0, 1, 2, etc. This equation can be derived using some simple geometry. The diffracted light paths are shown in Figure 1.

Lasers are light sources that can be used to calibrate your diffraction grating because they produce a single, known wavelength. You can use the known wavelength of your laser and the diffraction pattern it produces to calculate the spacing of the slits in your diffraction grating(s). Use an Excel graph to find the spacing between slits on your diffraction grating. What will you plot on the x-axis? What will you plot on the y-axis?

Next, we will predict the visible wavelengths that hydrogen should produce, based upon Bohr’s model. The human eye can see photons with wavelengths in the range ~400 nm to ~700nm. Use Equation 8 from the appendix to predict the four transitions of hydrogen that fall within our visible range (many people will only be able to see 3 of the wavelengths) Try various values of n and n’ to determine which transitions will produce λ in the visible range. Make a table that clearly shows n, n’ and λ for the four possible transitions.

And finally, we can get down to observing the actual wavelengths for comparison: Set up your equipment as shown in the diagram on the next page, ensuring that the meter sticks are at right angles to each other and the light source and slit are positioned to center the light on the diffraction grating.

Two meter stick

One meter stick

Light source

Light slit

Diffraction Grating

Partner 1

Partner 2

Determining wavelength of light emissions

Partner 2 will look through the diffraction grating and guide partner 1 to the apparent location of the light bar(s) on the one meter stick. The distance from the intersection of the two sticks to the diffraction grating and the apparent position of the light bar(s) will define the tangent of the angle of diffraction for each wavelength. where λ is the wavelength of the light line seen, and d is the spacing between slits in the diffraction grating determined with the use of a laser. (We will only be working with the first order maxima, m=1.) After measuring the visible wavelengths of hydrogen, follow the same procedures for at least one other light source.

Analysis: Determine the “d” value of your diffraction grating using a laser. Using equation 8 from appendix I, construct a table that shows the values of n and n’ that will theoretically produce λ in the visible range. Use the Diffraction grating’s value of “d” you found using the laser to determine the visible wavelengths of hydrogen and at least one other light source for comparison with “expected” values. (Error analysis)

Appendix I :

The Theory

You will be combining some familiar equations (and perhaps some new ones) to derive the expressions that will be used in today’s experiment. Finally, you will be able to predict the visible wavelengths of hydrogen.

1.1 Electrons Orbit Protons in Uniform Circular Motion

Electrons orbit protons under the influence of an attractive Coulomb force. The magnitude of the Coulombic force is given by:

Equation 1

where e is the magnitude of the charge on a proton and or an electron, r is the radius of the electron's orbit and k is Coulomb's constant [ 1/(4o) ].

Assume that the electron is in uniform circular motion about a proton at rest. Using Newton's Second Law and all you know about uniform circular motion, show that

Equation 2

where m is the mass of the electron and v is the speed of the electron.

1.2 Angular Momentum is Quantized

Bohr's idea that the electron's angular momentum must be quantized was an important key to an understanding of the hydrogen atom. From 141/171, you should remember that angular momentum (L) is a quantity similar to linear momentum, but can be used to describe an object that is undergoing uniform circular motion (such as an electron that is orbiting a nucleus). One definition of the angular momentum of the object electron is defined as the linear momentum of the particle (p) times the radius at which the object is rotating (r). Bohr postulated that electrons could only occupy orbits where their angular momentum was an integral number, n, of some quantized units.

Equation 3

The integer n is known as a “quantum number”. The constant h is called Planck’s constant, the same constant you may have seen in Planck’s theory of black body radiation. (h = 6.63 x 10-34 Js) Use this postulate and your result from Equation 2 to show that the radius of an allowed electron's orbit is given by

Equation 4

1.3 Electrons Bound to Protons

The total energy of the system is equal to the sum of the electron's kinetic energy and the potential energy between the proton and the electron. We found that when the electron and proton are infinitely far apart, the PE = 0. When a proton and electron are a distance r apart, we find that the potential energy was given by:

Equation 5

With the help of equation 2, show that the sum of the kinetic and potential energies of the electron can be given by:

Equation 6

1.4 Electron's Energy is Quantized

Now, since the electrons can only orbit the proton at discrete radii (this is what equation 4 states), this will require that the energy of the electron-proton system will be quantized as well. This means that only certain energy values are allowed (i.e. specific states of allowed energy). Use your results from Equation 4 and Equation 6, show that

Equation 7

Therefore, the energy levels of the hydrogen atom are quantized!

1.5 Light Emitted by the Hydrogen Atom

You are nearly able to predict what specific wavelengths of light that hydrogen will emit. As you can tell from Equation 7, the energy of the system depends only on a constant times (1/n)2. By specifying the quantum number n, you can specify the total energy of the system. A hydrogen atom will emit a photon of light when it jumps from one allowed energy orbit to another (in order to conserve energy). The energy E of a photon is related to its wavelength λ by

where c is the speed of light. Now, if the electron jumps from some initial (upper) state to some final (lower) state n ( > n), the energy lost by the electron is

When the electron loses energy, it is given off in the form of a photon with

Equation 8

Verify that the constant in the second expression of Equation 8 is given by 91.2 nm. (Verify the units as well as the value, and don’t let your calculator tell you the value is zero… You might have to be “tricky” when entering it into your calculator.)

Appendix II:

Date Last Modified April 11, 2014 Page Number 1

AIR (about 80% N2, 20% O2 gases). Strong spectrum is effectively the same as that of pure N2. See NITROGEN.

ARGON (Ar gas). Weak multiple lines, most intense in violet, least intense in red.

Color	Wavelength Å
Violet	4200 (Hazy)
Violet	4400 (Hazy)
Violet	4600
Green	4950
Green	5250
Green	5500
Green	5600
Green	5700
Yellow	5950
Red	6100
Red	6250
Red	6300
Red	6400
Red	6500
Red	6600
Red	6700
Red	6800
Red	7100
Red	7200

BROMINE (Br2 gas). Strong, multiple line spectrum from violet to red, with about 7 prominent lines.

Color	Wavelength Å
Violet	4200
Violet	4250
Violet	4500
Blue	4750
Blue	4800

CARBON DIOXIDE (CO2 gas). About 6 intense lines from carbon (C ), superimposed on the spectrum from oxygen (O). See CARBONIC ACID.

CARBONIC ACID (H2CO3 vapor). Spectrum resembles that of carbon dioxide, plus conspicuous red line from hydrogen.

Color	Wavelength Å
Violet	4150
Violet	4250
Violet	4450
Violet	4550
Blue	4900
Green	5100
Green	5200
Green	5300
Green	5400
Green	5650
Red	6100
Red	6200
Red	6300
Red	6600

CHLORINE (Cl2 gas). Medium intensity multiline spectrum from violet to orange, with 3 stronger lines in the blue/green region.

Color	Wavelength Å
Violet	4450
Violet	4550
Blue	4850
Green	5100
Green	5200
Green	5400
Green	5450
Green	5700
Yellow	5900
Red	6000
Red	6250
Red	6350
Red	6550
Red	6650

HELIUM (He gas). Strong spectrum with 2 violet, 2 green, 1 yellow & 2 red lines being prominent.

Color	Wavelength Å
Violet	4000
Blue	4500
Blue	4550
Blue	4800
Green	5000
Green	5100
Yellow	5850
Red	6500
Red	6800
Red	7200

IODINE (I2 vapor). Strong spectrum with lines so closely spaced that appearance is “blurry”, especially in orange/red region.

KRYPTON (Kr gas). Strong spectral lines in violet, green, orange & red portions.

Color	Wavelength Å
Violet	4300 (Hazy)
Violet	4400 (Hazy)
Violet	4500
Violet	4550
Blue	4900
Green	5600
Green	5650
Green	5700
Yellow	5900
Red	6100
Red	6300
Red	6500
Red	6650

MERCURY (Hg vapor). Strong spectrum composed of 3 violet, 1 green, 1 yellow & 1 orange lines. Mercury lamps are used as light sources for these wavelengths.

Color	Wavelength Å
Violet	4500
Violet	4600
Green	5000
Green	5050
Green	5600
Yellow	5900
Red	6100
Red	6250
Red	6600
Red	6800
Red	7200

NITROGEN (N2 gas). Strong spectrum of many lines from violet to red.

Color	Wavelength Å
Violet	4000
Violet	4050
Violet	4100
Violet	4150
Violet	4200
Violet	4250
Violet	4400
Violet	4450
Blue	5000
Blue	5050
Blue	5200
Green	5300
Green	5400
Green	5500
Green	5600
Yellow	5800
Yellow	5850
Yellow	5900
Red	6000
Red	6150
Red	6200
Red	6250
Red	6300
Red	6350
Red	6400
Red	6450
Red	6500
Red	6600
Red	6700
Red	6750
Red	6800
Red	6850

NEON (Ne gas). Strong spectrum of multiple lines in green, yellow, orange, red. Note absence of violet lines. Used in “neon lights”.

Color	Wavelength Å
Blue	4750
Blue	4900
Green	5100
Green	5250
Green	5600
Green	5700
Yellow	5800
Yellow	5900
Yellow	6000
Red	6050
Red	6100
Red	6150
Red	6200
Red	6600
Red	6650
Red	6700
Red	6850
Red	7050
Red	7150

OXYGEN (O2 gas). Very weak spectrum covering violet, blue/violet, green & red (2 lines) regions.

Color	Wavelength Å
Violet	4400
Blue	4900
Green	5250
Green	5400
Green	5500
Green	5650
Red	6150
Red	6250
Red	6600
Red	6650

WATER (H2O vapor). Three strong hydrogen lines & weak spectrum from oxygen.

Color	Wavelength Å
Violet	4300
Violet	4400
Blue	4900
Green	5200
Green	5400
Green	5500
Green	5600
Red	6050
Red	6100
Red	6650

XENON (Xe gas). Weak spectrum of 2 violet & 2 green lines.

Color	Wavelength Å
Blue	4700
Green	4850
Green	5000
Red	6250
Red	6400

Lab Evaluation – Please provide feedback on the following areas, comparing this lab to your previous labs: How much fun you had completing this lab; How well the lap prep period explained this lab; The amount of work required compared to the time allotted; Your understanding of this lab; The difficulty of this lab; How well this lab tied in with the lecture. Please assign each of the listed categories with a value from 1-5, with 5 being the best, 1 the worst. Comments supporting or elaborating on your assessment can also be very helpful in improving the future labs.

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