Homework on Quantized Energy Levels

Feel free to work on the homework in groups. The work you hand in, however, should reflect your understanding of the material and be in your own words.Students who turn in identical (or close to identical) homework assignments will be asked to explain their answers orally to the TA or prof. A student who cannot explain how he or she arrived at a given answer will be charged with academic dishonesty.

You should show all of your calculations (neatly) and justify all of your answers for full credit.

1. In your own words, explain why elements can be identified by their emission spectra. ("In your own words" implies that each student's answer should be distincive and not merely a paraphrase of someone else's answer.)
The unique configuration of electrons for an element results in unique values of allowed energies in atoms of that element. Emission spectra are produced when electrons transition between energy levels and emit the extra energy in the form of light. Since the allowed energy levels are unique to an element, so are the energies corresponding to transitions between those levels. Thus the light emitted by an atom will have only one of those unique energies. The energy of light determines its color, so the colors of the emission spectrum will uniquely identify an element.

The energy levels in a Hydrogen atom are given by E_n = -(13.6 eV)/n², where n is any positive integer. The energy of light is related to its wavelength by E = hc/l = (1240 eV nm)/l Answer the next few questions about Hydrogen.

2. What are the lowest 3 energies a hydrogen atom can have? What values of n do they correspond to?
-13.6 eV, n = 1; -3.4 eV, n = 2; -1.51 eV, n = 3

3. What are the energies and wavelengths of light emitted for each of the 3 transitions among those lowest three states?
n=2 to n=1, 10.2 eV, 122 nm;
n=3 to n=1, 12.1 eV, 102 nm;
n=3 to n=2, 1.9 eV, 653 nm

4. Which (if any) of those wavelengths are in the visible spectrum?
653 nm is visible as red light.

5. Will light having a wavelength of 121 nm ever be emitted by a hydrogen atom? How do you know this?
No. There is no pair of energy levels in hydrogen whose difference corresponds to a wavelength of 121 nm. (If you got 121 nm for the n=2 to n=1 transition in the previous problem, then obviously a yes answer is correct here.)

6. Is there a limit to the number of wavelengths emitted by a certain element? If so, what is it due to? If not, why not?
No. As n approaches infinity (E = 0), the discrete energy levels will get closer and closer, eventually appearing more or less continuous.

1.	In your own words, explain why elements can be identified by their emission spectra. ("In your own words" implies that each student's answer should be distincive and not merely a paraphrase of someone else's answer.) The unique configuration of electrons for an element results in unique values of allowed energies in atoms of that element. Emission spectra are produced when electrons transition between energy levels and emit the extra energy in the form of light. Since the allowed energy levels are unique to an element, so are the energies corresponding to transitions between those levels. Thus the light emitted by an atom will have only one of those unique energies. The energy of light determines its color, so the colors of the emission spectrum will uniquely identify an element.
	The energy levels in a Hydrogen atom are given by E_n = -(13.6 eV)/n², where n is any positive integer. The energy of light is related to its wavelength by E = hc/l = (1240 eV nm)/l Answer the next few questions about Hydrogen.
2.	What are the lowest 3 energies a hydrogen atom can have? What values of n do they correspond to? *-13.6 eV, n = 1; -3.4 eV, n = 2; -1.51 eV, n = 3*
3.	What are the energies and wavelengths of light emitted for each of the 3 transitions among those lowest three states? n=2 to n=1, 10.2 eV, 122 nm; n=3 to n=1, 12.1 eV, 102 nm; n=3 to n=2, 1.9 eV, 653 nm
4.	Which (if any) of those wavelengths are in the visible spectrum? 653 nm is visible as red light.
5.	Will light having a wavelength of 121 nm ever be emitted by a hydrogen atom? How do you know this? No. There is no pair of energy levels in hydrogen whose difference corresponds to a wavelength of 121 nm. (If you got 121 nm for the n=2 to n=1 transition in the previous problem, then obviously a yes answer is correct here.)
6.	Is there a limit to the number of wavelengths emitted by a certain element? If so, what is it due to? If not, why not? No. As n approaches infinity (E = 0), the discrete energy levels will get closer and closer, eventually appearing more or less continuous.