Electron transitions

Electron transitions are the technical name for the phenomenon of electrons either gaining or losing energy to change the radius of their orbit around an atom.

The Main Idea

Electron transition to greater orbits and higher energy levels most commonly occurs when an orbiting electron is struck by a high energy photon. Photons, the quanta of light and all electromagnetic radiation, have no mass but carry energy. This means when a photon collides with an electron, the electron can completely absorb the photon and thus all its energy with no change in mass. The transition of an electron to a higher energy level is only permitted when the electron absorbs energy greater than or equal to the difference between the two energy levels. A photon carrying insufficient energy to trigger a transition will either not be absorbed or will be immediately ejected by the electron, either way the result is no change for the electron. If an electron has excess energy after transitioning to a higher energy level, that is if the electron has more energy than is allowed by the energy formula for its particular level, but not enough energy to transition to a yet higher level, this energy will be ejected in the form of a photon (this is permitted by conservation of mass as photons have no mass and thus their creation results in no net change).

Transition to a lower energy level and smaller radius of orbit or a "downward" transition (corresponding to a decrease in the electron's principle quantum number [math]\displaystyle{ n }[/math]) is in many ways the inverse of transition to a higher energy level and greater radius of orbit or "upward" transition (corresponding to a decrease in the electron's principle quantum number [math]\displaystyle{ n }[/math]). As a result of the first and second laws of thermodynamics, electrons prefer to be in the lowest energy level possible (the smallest value of [math]\displaystyle{ n }[/math]). Thus, upward transition as a result of energy input (i.e. collision with a photon) is rarely long lasting as the electron desires to rid itself of its new energy and transition back to its lowest available energy level and smallest available orbit orbit. In order to achieve this downward transition to an orbit of smaller radius the electron must have the lower energy characteristic of this smaller orbit determined by the energy equation. The excess energy the electron carries which keeps it in its larger orbit cannot disappear due to conservation of energy, so something must be done to shed the undesired energy. In order to achieve this an electron wishing to transition downwards ejects a photon with an energy equal to the exact difference in the electron's energies before and after the transition. Consequently, this energy is also equal to the difference between the allowed energy of the initial and final value of [math]\displaystyle{ n }[/math]. Thus, the photon energy is equivalent to ∆[math]\displaystyle{ E_{n} }[/math].

The negative energies of the electrons are what keeps them in orbit around the nucleus, but it is possible to free the electrons from this orbit by bringing this energy to 0 or any positive energy. Because the electron's energy while in orbit around the nucleus is negative a positive energy of equal magnitude is required to free it from its orbit. Hence why the electron's energy found using [math]\displaystyle{ E_{n} }[/math] is referred to as its binding energy, because it represents the hurdle that must be surmounted in order to remove the electron from orbit.

The increasing ease with which an electron may be freed as it transitions to larger orbits and thus higher energy levels is an interesting and intuitive consequence of this phenomenon. As the principle quantum number [math]\displaystyle{ n }[/math] increases, energy increases in value approaching 0 as [math]\displaystyle{ n }[/math] (and therefore the orbital radius) approach infinity. Thus, the closer the energy level of the electron is to 0 the less energy is required to bring it fully to 0 and thereby free the electron. This is primarily a consequence of negatively charged electrons in larger radii being farther from the positively charged nucleus and thus feeling a weaker electric attraction.

Mathematical Model

where [math]\displaystyle{ n }[/math] = 1,2,3...

• Radius of orbit: [math]\displaystyle{ r_{n} = a_{0}n^2 }[/math]
  • Where [math]\displaystyle{ a_{0} = {\frac{4π ε_{0}ħ^2}{m_{e}e^2}} = 0.0529 }[/math] nm
• Electron energy level or Binding Energy: [math]\displaystyle{ E_{n} = {\frac{13.6 eV}{n^2}} }[/math]
• Photon frequency: [math]\displaystyle{ λ = \frac{hc}{E_{photon}} }[/math]
  • Where h is Planck's constant and c is the speed of light.

Derivations found here.

A Computational Model

Pictured above is a simulation of the Franck-Hertz experiment first performed in 1914, which proved the quantization of energy predicted by the Bohr model. For an in depth explanation of this experiment see here. The blue sphere pictured moving from left to right is an electron accelerated by the voltage difference between the two plates, and the stationary white sphere is an atom containing orbiting electrons. When the voltage difference between the two plates is low, the accelerating electron does not gather enough kinetic energy before striking the atom to cause its orbiting electron to transition to an excited state ([math]\displaystyle{ n \gt 1 }[/math]. However, when the voltage is high enough the electron gathers enough kinetic energy to cause an upward electron transition upon collision. This effect is denoted by the sphere representing the atom

This was observed in practice (as obviously in 1914 the technology did not exist to observe such interactions) by measuring the current caused by the electrons striking the opposite plate. Franck and Hertz were essentially measuring the final energy shown on the graph for different voltage differences. Initially, for low voltage differences the current would increase as increasing the voltage difference resulted in higher final energies for the accelerating electrons and thus larger currents. However, when the voltage was increased just enough to give the accelerating electron enough energy to cause a transition in the atom's electron, the accelerating electron loses most of its energy in the process. This resulted in lower measured currents, from which the Franck and Hertz concluded that the atom would only absorb the accelerating electron's energy if it was sufficient to cause transition. This meant that the atom's electron's could only exist at very specific energy levels that were thus quantized, confirming the Bohr Model's predictions.

Examples

Simple

Will a photon with energy [math]\displaystyle{ E_{photon} = 10.4eV }[/math] be able to transition a hydrogen electron in the ground state to the n = 2 level?

1. [math]\displaystyle{ E_1 = \frac{-13.6}{1} = -13.6eV }[/math]
   
2. [math]\displaystyle{ E_2 = \frac{-13.6}{2^2} = \frac{-13.6}{4} = -3.4eV }[/math]
   
3. [math]\displaystyle{ E_2 - E_1 = -3.4eV - -13.6eV = 10.2eV }[/math], so it requires [math]\displaystyle{ 10.2eV }[/math] to transition an electron from the ground state to the second level.
   
4. [math]\displaystyle{ 10.4eV \ge 10.2eV }[/math]
   

Yes, the photon has enough energy to bump the electron from the ground state to the second energy level.

Middling

If an electron orbiting hydrogen starts in the n = 4 orbit and ends in the ground state, how many photons with different energies can the atom emit?

The different possible orbital transitions possible in the atom are (where the numbers indicated are possible values of n):

4 -> 3

4 -> 2

4 -> 1

3 -> 2

3 -> 1

2 -> 1

There are 6 different possible transitions, which correspond to 6 different energy levels the photons emitted from these transitions can have.

Difficult

What is the energy of a Hydrogen electron in an orbit of radius .4761 nm? What form of electromagnetic radiation is necessary to free this electron from its orbit?

1. Using the formula for the radius of the orbit, find the value of n for this electron:
   1. [math]\displaystyle{ r = a_{0}n^2 }[/math]
   2. where n =1,2,3... and the Bohr Radius [math]\displaystyle{ a_{0} = 0.0529*10^{-9} }[/math]
   3. [math]\displaystyle{ n^2 = \frac{r}{a_{0}} = \frac{.4761*10^{-9}}{0.0529*10^{-9}} = 9 }[/math]
2. Using the value of n, calculate the energy of the electron:
   1. [math]\displaystyle{ E_{n} = \frac{-13.6 eV}{n^2} = \frac{-13.6 eV}{9} = -1.51 eV }[/math]
3. Set the electron's ionization energy (the energy required to free it from its orbit) equal to the energy of a photon ([math]\displaystyle{ E_{photon} = \frac{hc}{λ} }[/math]) and solve for the wavelength λ:
   1. [math]\displaystyle{ E_{Ionization} = |E_{n}| = 1.51 eV }[/math]
   2. [math]\displaystyle{ E_{Ionization} = \frac{hc}{λ} }[/math]
   3. [math]\displaystyle{ λ = \frac{hc}{E_{Ionization}} }[/math]
   4. [math]\displaystyle{ λ = \frac{(4.14*10^{-15} eV*s)(2.998*10^8 m/s)}{1.51 eV} = .822 μm }[/math]
4. Examine the wavelengths of electromagnetic radiation and determine the form with an interval of wavelengths containing .822 μm:
   1. A wavelength of .822 μm is characteristic of ultraviolet light.

History

In 1914, German physicists James Franck and Gustav Hertz (nephew of the physicist after whom hertz are named) performed an experiment to test the existence of quantized energy levels predicted by the Bohr Model of the Hydrogen atom presented the prior year. The experiment consisted of firing high energy electrons through a tube filled with mercury gas and measuring the current of the electrons when they were reabsorbed on the other side.

A graph of the measured current vs the initial energy of the electrons

Given the massive number of gas particles contained within the tube, the likelihood of an electron passing through without collision is small. If the electron has exclusively elastic collisions, collisions in which there is no change in net kinetic energy, it will be make it to the other side and be reabsorbed and measured as part of the current. With exclusively elastic collision, increasing the initial energy the electrons ought to result in a higher current measured as a result.

When the experiment was performed, the graph of measured current vs initial energy did not show a strong positive correlation but rather seemed staggered and jagged. This is a result of the fired electrons having in elastic collisions. Were these jagged points on the graph extremely sporadic it could be considered a result of random inelastic collisions. Their extreme uniformity however is a result of the fired electrons having inelastic collisions with electrons orbiting the gas atoms, but only after gaining enough energy (in this case 4.9 electron volts) to allow the orbiting electrons to transition to more excited orbits for the mercury atom. When the fired electrons had less than 4.9 electron volts of initial energy their collisions with orbiting electrons was elastic because the orbiting electrons had no chance of absorbing the energy they needed to transition to a higher energy level. Thus, the energy levels of electrons in orbit must be quantized as the Bohr Model predicted.

This experiment earned Franck and Hertz the Nobel Prize in physics in 1925, an honor Gustav's arguably more famous uncle never received.

See also

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Further reading

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References

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