Chapter 27
The Giancoli onLine Tutor

You should see a lowercase alpha here: a. If not you need to get a browser that supports greek fonts such as NetScape 4.5.

Early Quantum Theory and Models of the Atom

I. The Spectrum of light

The approximate spectrum of light from the Sun is shown at the left. The spectrum is described by a complicated formula developed by Max Planck, the peak in its spectrum is given by l = 2.90x10^-3/T, where l is the wavelength in meters and T is the temperature in kelvin. Notice that all the colors of visible light are present (along with UV and IR radiation). This can be contrasted with light from atoms which contains only light of specific wavelengths.

The visible spectrum from the Hydrogen atom is shown at the right. Only for specific colors are present with wavelengths of 656 nm, 486 nm, 434 nm and 410 nm. If you where given the set of numbers [1 4 9 16...] and asked to predict the next number you would quickly and correctly guess at 25 (5²). Similarly the wavelengths of the light from hydrogen form a numerical pattern:

1/l = 1.097x10⁷(1/2²-1/n²), n=3, 4, 5 and 6 known as the Balmer Series

Higher values of n give rise to specific UV 'colors'. Other UV lines follow the pattern given by

1/l = 1.097x10⁷(1/1²-1/n²), n=2, 3, 4... known as the Lyman Series

And IR 'lines' are found to satisfy the relationship

1/l = 1.097x10⁷(1/3²-1/n²), n=4, 5, 6... known as the Paschen Series

Light of wavelength l has a frequency f=c/l. It is now known to be composed of energy packets called photons, each with Energy E = hf, where h is planck's constant (6.63x10^-34 Js or 4.14x10^-15 eVs).

Interactive Example
What is the wavelength to the nearest nm of the photon corresponding to the case n=7 in the same group that has the four visible lines? Enter your results below and click outside.

What is the frequency to the nearest THz? A Tera is 10¹².

What is the energy of the associated photon to the nearest eV?

If the photon was from a star rather than an atom and it represented the peak wavelength to the nearest kelvin what is the stars temperature?

II. The Bohr Atom

If you don't have the Quicktime plugin there is probaly a broken image at the left. You can view a GIF animation version of the movie here.

If you have the movie plugin it is probably best to step back and forth through the movie using the buttons at the right. The Animation at the left models the Bohr atom.

Frames 1(abc) shows an electron making a transition from the n=3 to the n=2 state (first line of the Balmer Series) emitting the red photon. It is most useful to use Planck's constant in the form h=4.14x10^-15 eVs ,
hf= DE= -1.51-(-3.4)= 1.89
f= 1.89/4.14x10^-15= 4.57x10¹⁴
l= c/f= 3x10⁸/4.57x10¹⁴= 6.56x10-7 m= 656 nm
Frames 2(abc) shows a transition from the n=4 to the n= 2 state which produces the photon in the blue-greenregion of the spectrum.

Frames 3(abc) shows a transition from n=5 to n=2 and the blue photon that results.

Frames 4(abc) shows a transition from n=6 to n=2 with the photon at the violet end of the spectrum. Transitions from higher levels to the n=2 level but the resulting photons are all in the UV region and are not visible . Similarly transitions to the n=1 level will also be in the UV. Transitions, also, can occur from upper levels to the n=3 (or higher) levels. These will be in the infrared.

Frames 5(abc) shows light incident on the atom, if the light contains photons of the right frequency (E₂ = E₁ + hf), as is the case here, it can cause the electron to make a transition to a higher state.

Frames 6(abc) show an electron in the ground state being struck by an UV photon (represented here in black). If the protons energy is greater than 13.6 the electron can be ejected from the atom. Any excess energy will be carried off as kinetic energy of the electron. This is an example of the PhotoElectric Effect as applied to an atom. In this case
KE(in eV) = hf - 13.6
where h is as given above.

III. The Interaction of Photons with Matter

The Photoelectric Effect

The photoelectric effect is similar with metals . In a metal the conduction electrons are free to move throughout. However, the electrons are held within the surface of the metal by the 'Work Function' (W_o), and the least tightly held (those thought of as being at the surface) are held with the energy W_o. Thus the maximum kinetic energy of an electron ejected from a metal is
KE_max(in eV) = hf - W_o(in eV)
Since Kinetic Energy must be positive, there is a minimum frequency called the cutoff frequency, f_o, for an electron to be ejected at all given by
f_o = W_o/h
W_o and h must be expressed in consistent units, either J and Js or eV and eVs.

The Compton Effect

High energy photons such as X-ray photons interact with'free' electrons by scattering off them with billiard ball like collisions. In such a collision the photon looses energy and hence its frequency must drop and its wavelength lengthen, correspondingly.
If the wavelength l of the incident photon, then the wavelength, l', of the scattered photon is given by
l = l' + (h/mc)(1-cos(f))
where m is the electron mass and h/mc = 2.4x10^-12 m. Note that if f =0 there is no loss of energy because the photon misses the electron. At 180 degrees we have a direct head-on collision and a maximum loss of energy. All other angles are grazing collisions.

Pair Production

At still higher photon energies greater than 1.022 MeV (l<0.00122 nm, a gamma ray), the photon can create electron-positron(antielectrons) pairs. 0.511 MeV goes into creating the mass of each particle, any surplus goes equally into the kinetic energy of each particle. In order to conserve momentum this must occur near a nucleus.

Interactive Example
A 3 eV photon strikes a metal with work function of 2.0 eV. What is the maximum kinectic energy of the emitted electrons to the nearest eV? Enter your results below and click outside.

What is the speed of the emitted electron?

What is the cutoff frequency, f_o, to the nearest THz for the metal target?

A 0.15 nm X-ray photon 'scatters' off an electron at 0 degrees. What is its new wavelength in nm?

A 0.15 nm X-ray photon 'scatters' off an electron at 90 degrees. What is its new wavelength in nm?

What is the energy of the scattered electron in the above example?

A gamma ray with a wavelength of 0.001 nm creates an electron positron pair. What is the energy of the electron to the nearest keV ?

III. The Wave Nature of Matter

One other topic covered here is the de Broglie wavelength of a particle. This is very much a precursor to Quantum Mechanics to be covered later. For every free particle there is a wavelength, l, given by

l = h/p where p is mv or mv/sqrt(1 - v²/c²). Use the latter if v is greater than 3x10⁷ m/s.
Here you need to use h= 6.63x10^-34 Js.

Interactive Example
What is the wavelength of a 0.1 g travelling at 1 m/s?
Enter your answer and click outside .

What is the wavelength to the nearest mm of an electron (m= 9.1x10^-31 kg) travelling at 1 m/s? (enter as a decimal number)
Enter your answer and click outside.