Wavelength questions
You are an engineer designing a switch that works by the photoelectric effect. The metal you wish to use in your device requires 6.7x10 (to the -19 power. J/atom to remove an electron. Will the switch work if the light falling on the metal has a wavlength of 540nm or greater? Why or why not?
Radiation in the ultraviolet region of the electromagnetic spectrum is quite entergetic. It is this radiation that causes dyes to fade and your skin to burn. If you are bombarded with 1.00 mol of photons with a wavelength of 375nm, what amount of energy (in kilojoules per mole of photons) are you being subjected to?
What is the shortest wavelength photon an excited H atom can emit? Explain briefly.
All of your questions revolve around the photon theory of light, originally discovered by Max Planck in 1900 and expanded upon by Einstein in 1905. The crux of the theory is the fact that light can be modeled as a dicrete set of packets, called photons. Each packet carries a finite amount of energy which is specifed by the Planck relationship, which states:
Energy of a photon = Planck's Constant * frequency of light.
Thus the energy carried by light is directly proportional to its frequency. Therefore, blue light is more energetic than red light, because its frequency if higher. This idea immediately answers your first two questions if we also recall that the frequency of light, like any wave, is related to its wavelength by the equation:
frequency x wavelength = speed of the wave
or specificially for light:
frequency x wavelength = speed of light (which is 3.00 E 8 m/s)
(note the 3.00 E8 is my notation for 3.0 x 10^8)
In your first question your device needs 6.7E-19 J/atom to remove an electron from the atom. Where does this energy come from? It comes from a photon of light hitting the device and imparting its energy to it. Thus the energy of each photon hitting the device must be at least 6.7 E -19 J. Therefore, the frequency of the photon hitting the device must be:
6.7E-19 = Planck's constant x frequency of photon
6.7E-19 = 6.63E-34 x f
f=1.02 E 15 Hertz
Since we know the frequency, we can obtain the wavelength of the light needed by:
1.02 E 15 x wavelength = 3.00 E8
wave = 2.97 E -7 meters = 297 nanometers (a nm is 10^-9 of a meter and is a common unit used in light wavelengths).
Thus we need a wavelength of 297 nm or LESS, since the smaller the wavelength, the greater the frequency and energy. Thus, 540 nm or larger would not provide enough energy.
Your second question is asking almost the same thing. Again we need to know the amount of energy each photon of the 375 nm light provides. Thus we must reverse the calculations we just performed, going from wavelength to frequency to energy. Specifically
375 E -9 m x f = 3.00 E8
f=8 E 14 Hz
E=6.63E-34 * 8E 14
E=5.30 E-19 J/photon
That is the energy for each photon. Since one mole would be 6.02 E23 photons, you can easily find the energy per mole of photons. Note again that this problem is just the reverse of the previous. Traditionally with wavelength, frequency, energy problems, you will always use the two equations above to interconvert among all three relationships.
Your third question deals with the Bohr model of the hydrogen atom. The main take home lesson of the Bohr model is that the hydrogen electron can populate specific energy levels, with the energy of each level given by the equation:
En = -2.178 E-18J/n^2
where n is an integer called a quantum number. Thus, for example, the lowest energy the hydrogen electron can have, E1, would equal
E1=-2.178E-18J/1^2 = -2.178 E-18 J
E5 would be:
E5= -2.178 E-18J/5^2 = -2.178E-18 J/25 = -8.71 E -20 J
Don't let the negative sign bother you. Its just a convention we chemists use to indicate that the electron is bound to the nucleus of the atom.
Now, Bohr rationalized that the only way an electron could move from one energy level to another would be through the absorption or emission of a photon of light. IF the electron absorbed the photon, it would "jump" from a low level (like n=1) to a high level (like n=10). If the electron went from 10---> 1 it would emit a photon of the same energy. From what we have learned above, the shortest wavelength of energy the atom could emit would be equiv. to the highest frequency photon the atom could emit (again since frequency and wavelength are inversely proprotional). Since energy and frequency are directly proportional, if we can find the MAXIMUM amount of energy the electron could lose, we could find the frequency and hence the wavelength of the photon.
We know the lowest energy level of the hydrogen atom occurs when n=1, with the energy being -2.178 E-18 J (as shown above). What would be the max. energy level of the electron. That would occur when the value of n is as large as possible. In fact, it would be infinity. Thus the energy of the elctron here would be
Einfinity = -2.178E-18J/(infinity)^2 = 0
Thus the electron has an energy of 0 when it is in its highest energy level (and would be removed from the atom) Thus the energy difference between the two levels would be:
E final - E initial = E1 - Einfinity = -2.178E-18J - 0 = -2.178E-18 J
Thus number is the energy of the photon, with the minus sign telling us the photon is emitted from the atom. Thus the frequency of the photon would be:
2.178E-18J=6.63E-34 x f
f= 3.29 E 15 Hertz
You can then calculate the wavelength of the photon using the techniques we have described in problems 1 and 2