1 Introduction Read through this information before proceeding on with the lab. 1.1 Energy Levels 1.1.1 Hydrogen Atom A Hydrogen atom consists of a proton and an electron which are bound together the proton (positive charge) and electron (negative charge) stay together and continually interact with each other. If the electron escapes, the Hydrogen atom (now a single proton) is positively ionized. Similarly, the Hydrogen atom can sometimes bind another electron to it. Such a Hydrogen atom is negatively ionized. In astronomy, the former kind of ionization is much more common. In heavier atoms, the proton is replaced with a mixture of protons and neutrons collectively called the nucleus. The nucleus of the Hydrogen atom is just one proton. Helium, on the other hand, has two protons and two neutrons for a total of four nucleons (a nucleon is a general term for particles which are either a proton or neutron). A Hydrogen atom is an electron and proton bound by the electromagnetic force (an attractive force between oppositely charged particles). Because the electron is so much less massive than protons, early physicists visualized the electron as being like a tiny planet which orbited the proton which acted like a tiny sun. Though this view has the advantage of being easy to visualize, it is just an approximate physical representation. A more physical view of the Hydrogen atom is one where the electron is not seen as orbiting the proton like a planet around a sun, but exists as a diffuse cloud surrounding the nucleus. Only by measuring its location can one know where the electron is (or rather was, as once the electron s position is measured it moves to a different place). The region where the electron is probably located is called the electron cloud. In some cases, the single most probable electron-proton distance happens to correspond to the distance of the more planet-like model. (a) Early and incorrect view of the Hydrogen atom. (b) A slightly more accurate view of Hydrogen. (c) Probability density of the electron for the ground and two excited states of Hydrogen. Brighter regions are regions where the electron is more likely to be found. Figure 1: Models of the Hydrogen atom. 1.1.2 Orbitals The density of this electron cloud at any location measures the probability of finding the electron there. In the basic hydrogen atom, shown in Figure 1(c), the cloud is densest in the center and thins out with distance from the nucleus, which means the electron is most likely to be found near the nucleus, in a region about 1/20 nm in size. When additional energy is stored in the atom, the electron cloud takes on expanded patterns with low-density nodal surfaces corresponding to the dark rings on the right two panels of Figure 1(c). We call these electron cloud patterns orbitals (a term inherited from the early planet-like visualization) and each corresponds to a specific amount of energy stored in the atom. How much (kinetic) energy a planet has determines how far it is away from the star it orbits. Likewise, how much energy an electron has determines how far it is way from the proton (or more accurately, how far out the electron cloud extends). Planets can have essentially any energy and thus orbit at any distance. Electrons bound to the 1 / 9
nucleus, however, can not have just any value of energy. The electron can only occupy certain orbital states, each with a specific amount of stored energy. There is a lowest energy an electron can have and it corresponds to the state called the ground state. When the electron (or atom) has higher energy than this lowest energy, it is said to be in an excited state. An early model which mixed this discrete structure of electron energies states and the planet model is called the Bohr model. Though the Bohr model doesn t describe the electrons as clouds, it does a fairly good job of describing the discrete energy levels. The Hydrogen Atom Simulator presented in this module is strongly based on the Bohr Model. 1.1.3 Energy Levels Because the states an electron occur only at discrete energy levels, they are said to be quantized. The word quantum comes from a Latin word meaning how much. The branch of physics that provides the current model of the Hydrogen atom is called quantum mechanics. The electron in a Hydrogen atom can only have certain energies. These energies are called the Hydrogen s energy levels. The different energy levels of Hydrogen are denoted by the quantum number n where n varies from 1 for the ground state (the lowest energy level) to, corresponding to unbound electrons. In practice, electrons with high n (e.g. 100 or more) are so weakly bound that even weak disturbances will pull the electron away. Because it takes a minimum amount of energy, called the ionization energy to strip or ionize a bound electron from the Hydrogen atom, energy levels are usually referred to as being negative quantities. In both classical physics and quantum mechanics the absolute value of energy is irrelevant; only energy differences matter. It is convenient to say that when ionized the electron will have zero binding energy to the proton. With this convention, the different energy levels of a Hydrogen atom are given by the equation: E = E 0 n 2 (1) where E 0 = 13.6 electron volts (ev) and n = 1, 2, 3... and so on so that the ground state has energy E 1 = 13.6 ev and the second energy level (the first excited state) has energy E 2 = 13.6/4 ev = 3.4 ev. The equivalence statements for electron volts, Joules (J), and calories (cal) are given by 1 ev = 1.602 10 19 J (2) 1 J = 0.2390 cal (3) Note that the number of calories given in the Nutrition Facts on food is actually kilocalories, e.g. a snack bag of Cool Ranch Doritos says it has 260 calories, which actually means it has 260 kilocalories, or 2.60 10 5 cal = 1.09 10 6 J = 6.79 10 24 ev. 1.1.4 Excitation A hydrogen atom with excess energy is said to be excited. The two primary ways to excite an atom are through absorbing light and through collisions. When two atoms collide energy is exchanged. Sometimes, some of that energy is used to excite an electron from a lower energy level to a higher energy level. How many collisions and how energetic the collisions are will depend on how tightly the hydrogen atoms are spaced and their average temperature. How absorbing light causes transitions is discussed more in Section 1.3. Another way to excite an atom is to absorb electromagnetic energy, or in the terminology of quantum mechanics, absorb a photon. 1.2 Light 1.2.1 Properties of Light We are all familiar with light. Through light we see the world. But what is it? Light can be thought of as particles of electromagnetic energy called photons. Photons exhibit many properties that we are familiar with as particles. 2 / 9
Figure 2: Components of a Wave. But light also exhibits wavelike properties (which property wave or particle it will manifest depends on how the light is being observed). It is frequently convenient to express the properties of the light in wave terminology. For light, the relationship between wavelength (λ) and frequency (f) is straightforward, their product is equal to the speed of the wave, which for light is ( c 3.00 10 8 m/s ). That is c = λ f (4) Another relationship that was instrumental in the formation of quantum mechanics was the relationship between the energy of a photon at a certain frequency. It is E = h f (5) where h = 6.626 10 34 J s is called Planck s constant. Since the energy goes up as the frequency increases, the energy is directly proportional to the frequency. Because frequency and wavelength are related by a constant (c) the energy can also be written in terms of wavelength: E = h c/λ. When the energy increases the wavelength decreases and vice versa. That is, energy in inversely proportional to wavelength. In short, a photon can be described by either its energy, frequency, or wavelength. All three methods are frequently used. 1.2.2 Electromagnetic Spectrum Light can have a large range of values of energy/wavelength/frequency covering what is called the electromagnetic spectrum. We often split the full spectrum into smaller regions and talk about the different kinds of light. Radio waves are the least energetic kind of photons while gamma rays are the most energetic kind of photons. Figure 3 shows the electromagnetic spectrum. Note how small of a region the visible light we see with the human eye covers. Figure 3: The full spectrum of light. 3 / 9
1.3 Transitions 1.3.1 Absorbing Photons According to the theory quantum mechanics, an electron bound to an atom can not have any value of energy, rather it can only occupy certain states which correspond to certain energy levels. The formula defining the energy levels of a Hydrogen atom are given by the Equation (1). The energy is expressed as a negative number because it takes that much energy to unbind (ionize) the electron from the nucleus. It is common convention to say an unbound electron has zero (binding) energy. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies exactly matched to the energy difference, or quantum leap, between two energy states. When an electron absorbs a photon it gains the energy of the photon. Because an electron bound to an atom can only have certain energies the electron can only absorb photons of certain energies. For example an electron in the ground state has an energy of 13.6 ev. The second energy level is 3.4 ev. Thus it would take E 2 E 1 = 3.4 ev ( 13.6 ev) = 10.2 ev to excite the electron from the ground state to the first excited state. If a photon has more energy than the binding energy of the electron then the photon will free the electron from the atom ionizing it. The ground state is the most bound state and therefore takes the most energy to ionize. 1.3.2 Emitting Photons Generally speaking, the excited state is not the most stable state of an atom. An electron has a certain probability to spontaneously drop from one excited state to a lower (i.e. more negative) energy level. When an electron drops from a higher level to a lower level it sheds the excess energy, a positive amount, by emitting a photon in a random direction. 1.3.3 Lyman, Balmer, Paschen Lines Long before the Hydrogen atom was understood in terms of energy levels and transitions, astronomers had being observing the photons that are emitted by Hydrogen (because stars are mostly Hydrogen). Atomic physicist Balmer noted, empirically, a numerical relationship in the energies of photons emitted. This relationship was generalized and given context by the Rydberg Formula. But the various discrete photon energies/wavelengths that were observed by Balmer were named the Balmer series. It was later understood that the Balmer lines are created by energy transitions in the Hydrogen atom. Specifically, when a photon drops from an excited state to the second orbital, a Balmer line is observed. The Balmer series is important because the photons emitted by this transition are in the visible regime. The Balmer series is indicated by an H with a subscript α, β, γ, etc. with longest wavelength given by α. As there are other transitions possible, there are other series. All transitions which drop to the first orbital (i.e. the ground state) emit photons in the Lyman series. All transitions which drop to the 3rd orbital are known as the Paschen series. Figure 4 shows some of the Lyman and Balmer transitions graphically. 2 Pre-Lab Questions Complete the following questions before coming to class. 1. Complete Table 1 which relates the parameters of two different photons by circling the appropriate relationship. The first row has been completed for you: A red photon has a larger wavelength, smaller frequency, smaller energy, and the same velocity through space as a blue photon. 4 / 9
Figure 4: Energy Levels and Transition Lines. Photon A Wavelength Frequency Energy Velocity Photon B (in space) Red the same the same the same the same Blue Green the same the same the same the same Orange Infrared the same the same the same the same Visual Visual the same the same the same the same Microwave X-Rays the same the same the same the same Gamma Rays Table 1: Circle the appropriate relationships for each pair of photons. 5 / 9
2. Scientists often say A is proportional to B if B increases when A increases. They also say A is inversely proportional to C if C decreases when A increases. Inspect the table above for evidence of such relationships and use these terms to describe the relationships between wavelength, frequency, energy, and velocity below. Solution: Wavelength is inversely proportional to frequency and energy. Frequency and energy are directly proportional to each other. The velocity is constant regardless of any of the others. 3 Hydrogen Atom Simulator Open the NAAP Hydrogen Atom Simulator from this link: http://astro.unl.edu/naap/hydrogen/animations/hydrogen_atom.html The Hydrogen Atom Simulator, shown in Figure 5, allows one to view the interaction of an idealized Hydrogen atom with photons of various wavelengths. This atom is far from the influence of neighboring atoms and is not moving. The simulator consists of four panels. Below gives a brief overview of the basics of the simulator. Figure 5: NAAP Hydrogen Atom Simulator. The panel in the upper left shows the Bohr Model: the proton, electron, and the first six orbitals with the correct relative spacing. The electron can absorb photons and jump higher energy levels where it will remain for a short time before emitting a photon(s) and drop to lower energy level (with known probabilities fixed by quantum mechanics). The electron can also be ionized. The simulator will a short time later absorb an electron. For convenience you can drag the electron between levels. Once it is released it will behave physically once again as if it had gotten to that present level without being dragged. The upper right panel labeled energy level diagram shows the energy levels vertically with correct relative spacing. The Photon Selection panel (bottom left) allows one to shoot photons at the Hydrogen atom. The slider allows the user to pick a photon of a particular energy/wavelength/frequency. 6 / 9
Note how energy and frequency are directly proportional and energy and wavelength are inversely proportional. On the slider are some of the energies which correspond to levels in the Lyman, Balmer, and Paschen series. Clicking on the label will shoot a photon of that energy. If the photon is in visual band, its true color is shown. Photons of longer wavelengths are shown as red and shorter wavelengths as violet. The Event Log in the lower right lists all the photons that the atom has encountered as well as all the electron transitions. The log can be cleared by either using the button or manually dragging the electron to a particular energy level. For any particular level of the Hydrogen atom one can think of the photons that interact with it as being in three groups: Increasing Energy Range 1: Range 2: Range 3: None of the photons have enough Some of the photons have the All the photons have enough energy to affect the atom. right energy to make the electrons energy to ionize the atom. jump to a higher energy level (i.e. excite them). Note that the ranges are different for each energy level. Below is an example of the ranges for an electron in the ground state of a Hydrogen atom. Ground State Electron of H Range 1: Range 2: Range 3: 0 ev to 10.1 ev 10.2 ev to 13.6 ev > 13.6 ev (10.2 ev needed to excite electron (some will excite, some won t) (anything greater than this will to 1st orbital) ionize the electron) When the simulator first loads, the electron is in the ground state and the slider is at 271 nm. Fire a 271 nm photon. This photon is in range 1. Gradually increase the slider to find a photon which is between range 1 and range 2 (for a ground state electron). This should be the Lyman-α line (which is the energy difference between the ground state and the second orbital). Increase the energy a bit more from the Lyman-α line and click fire photon. Note that nothing happens. This is a range 2 photon but it doesn t have the right energy. Increase the energy more until photons of range 3 are reached. In the simulator this will be just above the L ɛ line. Technically there are photons which would excite to the 7th, 8th, 9th, etc. energy levels, but these are very close together and those lines not shown on the simulator. The L ɛ line has an energy of 13.22 ev and is in range two. The ionization energy for an electron in the ground state is 13.6 ev and so that is the correct range 3 boundary. 4 Analysis 1. Which photon energies will excite the Hydrogen atom when its electron is in the ground state? (Hint: there are 5 named on the simulator, though there are more.) Solution: L α = 10.2 ev, L β = 12.09 ev, L γ = 12.75 ev, L δ = 13.06 ev, L ɛ = 13.22 ev 7 / 9
2. Starting from the ground state, press the L α button twice in succession (that is, press it a second time before the electron decays). What happens to the electron? Solution: The first photon causes the electron to jump from the ground state (1st level) to the first excited state (2nd level). The second photon knocks the electron away; it ionizes the atom. 3. Complete the energy range values for the 1st excited state (i.e. the second orbital) of Hydrogen. Use the simulator to fill out ranges 2 and range 3. The electron can be placed in the 1st orbital by manually dragging the electron or firing an L α photon once when the electron is in the ground state. Note also that the electron will de-excite with time and so it may need to be placed in the 1st orbital repeatedly. 1st Excited State (2nd Level) Electron of H Range 1: Range 2: Range 3: 0 ev to 1.8 ev 1.9 ev 3.4 ev 3.5 ev (anything less than 1.9 ev will fail to excite the atom) 4. What is the necessary condition for Balmer Line photons (H α, H β, etc.) to be absorbed by the Hydrogen atom? Solution: Per Figure 4, the electron must be at the 1st excited state (2nd level) for it to absorb a Balmer Line photon. 5. Complete the energy range values for the 3rd orbital (2nd excited state) of Hydrogen. The electron can be placed in the 3rd orbital by manually dragging the electron or firing an L β photon once when the electron is in the ground state. Note also that the electron will de-excite with time and so it may need to be placed in the 3rd orbital repeatedly. 2nd Excited State (3rd Level) Electron of H Range 1: Range 2: Range 3: < 0.65 ev 0.66 ev 1.5 ev > 1.5 ev (anything more than this will ionize the atom) 6. Starting from the ground state, press two and only two buttons to achieve the 6th orbital in two different ways. One of the ways has been given in Figure??. Illustrate your transitions with arrows on the energy level diagram in Figure?? and label the arrow with the button pressed. Solution: L β and P γ photons will jump the electron from the ground (1st level) state to the 5th excited state (6th level). 7. Press three buttons to bring the electron from the ground state to the 4th orbital. Illustrate the transitions as arrows on the energy level diagram in Figure?? and label the arrow with the button pressed. Solution: L α, H α, and P α photons will jump the electron from the ground (1st level) state to the 3rd excited state (4th level). 8. How does the energy of a photon emitted when the electron moves from the 3rd orbital to the 2nd orbital compare to the energy of a photon absorbed when the electron moves from the 2nd orbital to the 3rd orbital? Solution: It s exactly the same. 9. Find the amount of energy needed for the following 3 transitions. L α : Level 1 to Level 2 10.20 ev H α : Level 2 to Level 3 1.89 ev P α : Level 3 to Level 4 0.66 ev 8 / 9
10. How much energy does a L γ photon carry? What transition does the L γ photon cause? How does the total energy needed to undergo all three transitions in question 9 compare to the energy in a single L γ photon? Solution: L γ = 12.75 ev, and it causes the electron to jump to the 3rd excited state (4th level). The sum of the energies from all three transitions in question 9 equals the energy of one L γ photon. So one L γ photon can cause the jump from 1st to 4th, or all three photons in this order L α, H α, P α will cause the same transition from 1st to 4th. 9 / 9