Bohr's Theory of the Hydrogen Atom

Background:

While Rutherford was doing his pioneering work leading to the discovery of the atomic nucleus, chemists were completing spectra analysis of different elements.  They found that each element, when excited in its gaseous state, produced a unique spectral fingerprint of brightly colored lines which could be seen when viewed through closely separated slits.  The element most studied was hydrogen which had three distinctively observable lines in the visible spectrum - - red, blue/cyan, and violet. 

The first person to propose a mathematical relationship for these lines was J. J. Balmer and we now call hydrogen's visible spectrum the Balmer series.  Another pattern in hydrogen's spectral lines was noted by J. Rydberg and W. Ritz.  When you added together the frequencies of any two lines, you produced the frequency of a third line.  There had to be an atomic model that would predict these patterns.  That model was conceived by Neils Bohr (1913).

Standing Waves

1) Open http://www.cbu.edu/~jvarrian/applets/waves2/simwav_g.htm

2) In the “Control” window, click “Continuous” and “free end”.

3) What do the following represent?

a) Red Line

b) Green Line

c) Blue Dots

4) Click “Play” to watch the animation for a while and then click “Pause”. To view what is happening in more detail, click “Forward” until the red and green waves overlap crest to crest and trough to trough. The “crest” is the high point of the wave and the “trough” the low point. When the initiated and reflected waves overlap each other in this manner, you have constructive interference. Describe the resulting wave in this case.

5) Continue to click “Forward” until the red crests overlap the green troughs and vice verse. This is called total destructive interference. Describe the resulting wave in this case.

The result of the interference of the two waves in this simulation is called a standing wave. Standing waves are produced whenever two waves of identical frequency interfere with one another while traveling opposite directions along the same medium. Standing wave patterns are characterized by certain fixed points along the medium which undergo no displacement (nodes). The nodes are always located at the same location along the medium, giving the entire pattern an appearance of standing still (thus the name "standing waves").

6) Get a long rope, spring, or Slinky. Hold one end and have a partner hold another end. Try shaking the rope back and forth with different frequencies to form standing waves of at least three different wavelengths. Draw your standing waves below:

a) slow

b) medium

c) fast

7) Do all shaking frequencies produce a standing wave? If not, what do you observe? Why do you think this happens?

The Wave Nature of Matter

8) Go to http://www.whatthebleep.com/trailer/doubleslit.wm.low.html and watch “Double Slit Experiment from Down The Rabbit Hole”. You will need a computer with a speaker.

9) How do scientists know that electrons behave like a wave as well as a particle?

Bohr Model of the Atom

The Bohr model that explained the spectrum of the hydrogen atom was based on the following assumptions.

• The electron in a hydrogen atom travels around the nucleus in a circular orbit.
• The energy of the electron in an orbit is proportional to its distance from the nucleus. The further the electron is from the nucleus, the more energy it has.
• Only a limited number of orbits with certain energies are allowed. In other words, the orbits are quantized.
• The quantized electron energy states correspond to electron orbitals of specific radii.
• Light is absorbed when an electron jumps to a higher energy orbit and emitted when an electron falls into a lower energy orbit.
• The energy of the light emitted or absorbed is exactly equal to the difference between the energies of the orbits.

10) Go to http://www.walter-fendt.de/ph11e/bohrh.htm and read the introduction.

a) What condition must be met for an electron to be in a stable orbit?

b) How does the circumference of the orbit compare to the wavelength of the electron?

11) Click “Wave model”.

a) What does the green wavy line represent (read the intro)?

b) What does the blue line represent?

12) Choose principal quantum numbers 1-6 from the menu and record the radii and energies in the data table. Count how many total wavelengths are in the orbit for “circumference of orbit in terms of wavelength “.

Data Table – Bohr Model of Hydrogen

principle quantum number	radius of orbit (m)	energy (j)	energy (eV)	circumference of orbit in terms of wavelength
n=1				l1
n=2				l2
n=3				l3
n=4				l4
n=5				l5
n=6				l6

13) Choose principal quantum number n=1. Left click on the blue circle and drag it outwards.

a) What do you notice about the green wavy line when it is between energy levels?

b) Why can’t electrons be found between energy levels?

constructive interference destructive interference

DeBroglie’s equation relates the wavelength of matter (in this case, an electron) to its momentum (mass X velocity).

l_DeB = h/(mv) where h = 6.63 x 10 ^{- 34} J s

It was found that an orbital radius would be stable if an integer multiple of the electron's deBroglie wavelength (l_DeB) equaled the orbital’s circumference (2 p r):

nl_DeB = 2 p r

Let’s find out how fast an electron in the first energy level is moving.

For questions #14-17, please show your work and the formulas used.

14) Find the circumference of the electron’s orbit at E₁. (use the radius in the table for n=1)

15) Find the wavelength of an electron at E₁.

16) Find the velocity of the electron in the first energy level by using your wavelength from #15 and deBroglie’s equation. The mass of an electron is m_e- = 9.11 X 10^-31kg

17) How does this velocity compare to the speed of light, c = 3.0 X 10⁸ m/s? Express this as a percent.

(velocity of electron)/(speed of light) X 100% = ____________________________

Hydrogen Emission Lines

18) Open http://phys.educ.ksu.edu/vqm/free/h2spec.html and read the instructions.

19) Add lines to represent the principle energy levels n=1 through n=4. Use the values of energy (in eV) from your Bohr Model of Hydrogen Data Table.

20) Draw emission lines for n=2 à n=1, n=3 à n=2, n=4 à n=3.

Max Planck discovered that electromagnetic energy is quantized according to the formula

E = hn where h = 6.63 x 10 ^{- 34} J sec

where n represents the electromagnetic wave's frequency calculated according to the wave formula

n = c / l where c = 3.0 X 10⁸ m/s

Planck’s equation in terms of wavelength (l) is then

E = ( h c) / l so l = (hc) / E

21) Which of these emission lines is in the visible part of the spectrum (i.e., which of the arrows you drew makes a colored line appear in the second black row near the top of the web page)?

22) Why aren't the other two lines visible? Are they off the visible scale because they are too high energy? too low energy? What type of electromagnetic radiation might each one be?

23) Remember that the energy of the light emitted is exactly equal to the difference between the energies of the orbits. Find the energy of each of the transition lines drawn in step #20 and then the wavelength using Planck’s equation. Use the diagram after the table to determine the type of electromagnetic radiation (if the light is in the visible region, give the color. Remember that 1 nm = 10^-9 m).

transition	diff. in energy , DE (j)	wavelength, l (m)	type of EMR/color
n=2 à n=1	E2-E1 =
n=3 à n=2	E3-E2 =
n=4 à n=3	E4-E3 =

Show one of each kind of calculation.

24) Do your answers to #21 and #22 agree with your calculations?

25) Click “Clear all” on the simulation. Look at the figure to the right. Try to create the "Balmer Series".

26) What portion of the spectrum do you expect the (invisible to our eyes) Paschen Series lines to be in?

27) In the introduction, we mentioned another pattern in hydrogen's spectral lines noted by J. Rydberg and W. Ritz.  When you add together the frequencies of any two lines, you produce the frequency of a third line.  Find the energies, wavelengths and frequencies of the photons that could be given off as an electron transitions from E₄ to E₂ .

transition	diff. in energy , DE (j)	wavelength, l (m)	frequency, n (hz)
n=4 à n=2	E4-E2 =
n=4 à n=3	E4-E3 =
n=3 à n=2	E3-E2 =

Show one of each kind of calculation.

Show that the frequency of light given off as an electron transitions from E₄ à E₃ plus that given off from E₃ à E₂ equals the frequency from E₄ à E₂.

Summary:

Write a paragraph describing why electrons must occupy certain energy levels in the Bohr model of the atom and how the Bohr model can explain the atomic spectrum of hydrogen.

Equations in this handout (you may use this on a test):

DeBroglie’s equation l_DeB = h/(mv)

orbital circumference nl_DeB = 2 p r

Planck’s equation E = hn = ( hc ) / l

wave equation c = nl

Variables (units):

deBroglie’s wavelength (m) l_DeB

mass (kg) m

velocity (m/s) v

principle quantum number n

orbital radius (m) r

energy (j) E

frequency (hz) n

wavelength (m) l

Constants:

Planck’s constant h = 6.63 x 10 ^{- 34} J sec

speed of light c = 3.0 X 10⁸ m/s

mass of an electron m_e- = 9.11 X 10^-31 kg

Equivalencies:

1 eV = 1.6 X 10^-19 j

1 nm = 10^-9 m