Lecture 4: The Bohr Model and Electronic Transitions

Lecture 4: The Bohr Model and Electronic Transitions Summary The Bohr model is a framework used to describe the quantized nature of atoms. By assuming that electrons orbited around atomic nuclei—like planets around the sun—Bohr showed that the electrons bound to atomic nuclei can only exist in discrete energy levels. By equating the attractive force of the Coulomb interaction between the negatively charged electron and the positive nucleus, FCoulomb = −Ze2 /r2 , and the repulsive force due to rotation, F = mv 2 /r, and substituting the quantized angular momentum, L = mvr = nh/2, we find formulas for the quantized radius and the quantized energy. The energy of an electron in the nth energy level of an atom which has Z protons is E = −13.6Z 2 /n2 [eV ] The integer n, which can have any positive value ≥ 1, is called a quantum number because it makes it so that the energy can only take on certain, discrete values as opposed to a continuum of possibilities. The state n=1 is the ground state, and higher energy levels are called excited states. When an electron goes from a state with a lower quantum number to one with a higher quantum number, the energy of the electron becomes less negative: it gains energy. We know that energy must be conserved, so the energy gained in such a transition must come from somewhere. In fact, a photon with exactly the ΔE between two energy states in the atom can be absorbed, exciting the electron. Similarly, if the transition happens in the opposite direction, the excess ΔE can be emitted in the form of a photon. The kinds of photons capable of being absorbed or emitted by a Bohr model atom or molecule can be equivalently characterized by their energy, frequency, or wavelength. Einstein taught us that light can also be quantized in his work on the photoelectric effect: the energy of the photon can take discrete values: E = hf = hc/λ Where f is frequency, λ is wavelength, h is Planck’s constant, c is the speed of light, and the second equality is true because c=fλ. The wavelength of light that can interact with a particular energy transition can be found by equating the two expressions above. With this in mind, we see that light can be used to probe what electronic transitions are happening in an atom or molecule, and therefore what energy levels are present in a material. The Bohr model fit the spectral lines of stars observed previously by astronomers; in particular, the discrete wavelengths emitted by hydrogen gas that had been observed by spectrometer finally had an explanation. There are a few limitations to the Bohr model. We now know that electrons are not really orbiting the nucleus (although their angular momentum is quantized as predicted). Further, the Bohr model can only be used to describe electronic transitions with a single electron, such as H, He+ , or Li2+ . However, the key takeaways hold: if the energy states in an atom or molecule are known, the wavelengths of light that can be emitted or absorbed by the material are also determined. Likewise, if the discrete wavelengths emitted or absorbed by an atom or molecule are observed, the energy levels available to an electron are also known. 20 Why this matters How does absorption of photons by electrons that are transitioning from one energy level to another matter? One application this principle is connected to is the refrigerator(!) That might not seem obvious at first, but let’s see why. Take a look at the average energy used by a fridge in the U.S. over a 55-year period. It’s interesting to note that the average size of a refrigerator grew steadily for 35 years and then plateaued in the 1980’s. This, it turns out, was not because people didn’t want more fridge, but rather because they couldn’t fit anything larger through the kitchen door. More to the point, note that the energy use per unit increased along with the size for 30 years, until suddenly it started decreasing rapidly in the 1970’s, even though the size continued to increase. This is because during that time there was also an oil shortage and people were starting to get concerned about energy use. Some of these people had important titles like President of the United States (Nixon, Ford, and Carter), Average energy use vs. time © source unknown. and a whole lot of legislation aimed at energy conservation happened because of them. For example, President Ford signed the first law ever on this topic in 1975, the Energy Policy and Conservation Act, which led to the establishment of fuel economy standards for cars and efficiency standards for appliances. After that, more laws were signed that created incentives to lower energy consumption (towards a goal of making the U.S. an energy independent nation by 1985), with Carter presenting a plan to congress stating that, “conservation is the quickest, cheapest, most practical source of energy,” requiring federal agencies to develop energy conservation plans, and creating the Department of Energy (DOE). In this Why This Matters, I want to emphasize the importance of government policy. It was the government stepping in that led to the technological innovation that led to greater efficiency and lower costs. In 1992, the Energy Star program was introduced which led to even greater reduction in the energy consumption of appliances and critically increased consumer awareness. Not only did refrigerators themselves become cheaper because of this innovation, but the energy they used continued to decrease. In 2019 in the U.S., we save about $20 billion annually on energy consumption alone for our refrigerators compared to the 1970’s. Sadly, since 2017, the Trump administration has been trying to eliminate the Energy Star program, which many consider to be one of the most successful voluntary energy efficiency programs in the world. It has saved Americans $430 billion in energy costs across hundreds of appliances, not to mention the lower CO2 emissions that have resulted. 21 Figure above: All rights reserved. This content is excluded from our Creative Commons license. For more information, see: https://ocw.mit.edu/fairuse. Now we still haven’t connected back to our chemistry lesson, and while I could do that for the energy savings alone, there’s another example involving a fridge-related policy decision that very clearly brings Bohr back to the table. That is the Montreal Protocol, which was signed in 1987 and was the first treaty to be ratified by all countries in the world. It represented universal agreement to protect the stratospheric ozone layer by phasing out both production and consumption of ozone-depleting chemicals, like chlorofluorocarbons (CFCs). Taking this action led to the recovery of the earth’s ozone layer that protects life from harmful UV radiation. Millions of lives have been saved as a result. But what is it about CFC’s that makes them deplete ozone? And why is ozone so important for blocking UV radiation? The answer, of course, comes from chemistry. Ozone is a molecule consisting of three oxygen atoms, O3 , and a common CFC molecule consists of a carbon atom, three chlorine atoms, and a fluorine atom, CCl3 F . Occasionally, the CFC molecule breaks down and gives up one of its Cl atoms, like this: CCl3 F (g) → CCl2 F (g) + Cl(g) Note the “(g)” subscripts are included to show that all this is happening in the gas phase. Once a chlorine atom is freed up, it can then react with ozone, like this: O3 (g) + Cl(g) → ClO(g) + O2 (g) And so each CFC molecule can deplete one ozone molecule. But it doesn’t stop there. A third reaction takes place, since there are also oxygen atoms in the stratosphere. When an oxygen atom interacts with the ClO molecule, it reacts with it to make O2 , like this: ClO(g) + O(g) → Cl(g) + O2 (g) And therein lies the real problem: the Cl atom is now free again and ready to attack another ozone molecule. Because of this catalytic cycle, one single CFC molecule can lead to the destruction of 100,000 ozone molecules! That brings us to the second question from above: why does losing ozone lead to more UV radiation? This is where electron transitions come in. Check out the plot in this figure of the solar spectrum. The x-axis is the wavelength of light coming from the sun, covering only the UV, visible and infrared (so it is just a sliver of the full EM spectrum we showed a few pages back). The y-axis is the intensity of light at a given wavelength, in Watts per area. A Watt is a measure of power, which is energy per time, or Joules/s. In plain words, this is a plot showing the power of sunlight as a function of its color. 22 Solar radiation spectrum by Robert A. Rohde as part of the Global Warming Art project. License: CC BY-SA. Source: Wikimedia Commons. The first thing to point out on the plot is that solid black curve, which corresponds to a theoretical prediction of the black body radiation emitted by an object at the temperature of the sun. This temperature varies a lot depending on where in the sun it’s measured, but near the surface, T=5250◦ C is a decent approximation. Note that it’s a nice continuous spectrum, and it has a shape typical of black body radiation (remember, that’s the kind of radiation we mentioned earlier – the kind that Planck introduced the concept of quantization to explain). Next, there’s the yellow shaded parts that correspond to the actual measured power from the sun before it interacts with our atmosphere. It follows the black body radiation quite well, which validates the prediction. And then, there’s the red shaded region. This is the power from the sun as measured on earth, so the only difference between the red and the yellow is that the red is a measurement of photons from the sun that have passed through the atmosphere. It follows the yellow pretty well in general, except that now there are gaps in the spectrum: discrete wavelengths where there is no light observed on earth at all! The plot shows the reason why: next to some of the larger gaps, there is a molecule. H2 O is listed in several spots. One gap is attributed to O2 and another to CO2 , and one of those gaps is due to our nice, friendly, no-longer-getting-depleted ozone molecule, O3 . It’s the gap all the way over on the left side of the plot, and it’s right where the UV light would have been shining down on us. We can see that it does indeed shine down on us outside of the atmosphere (the yellow), but not at the earth’s surface (the red). This tiny sliver of absence of UV light due to the O3 molecule is absolutely essential for life on earth as we know it, and it’s all thanks to electrons being pumped up in energy from one state to another. In this case it happens in a molecule, not an atom, but the same principle we learned in this lecture for electron transitions according to the Bohr model applies. There are discrete energy states where electrons are allowed to be in the O3 molecule, and transitions between these energy levels can happen when an electron either absorbs or emits a photon. Just like the Bohr model for the hydrogen atom, the discrete nature of electronic transitions in molecules means that absorption of light occurs only at distinct points in the spectrum. For ozone, there just happens to be an energy difference between two electron states that has the energy of a UV photon, which is why this particular molecule absorbs well in the UV. Why this employs I was thinking about mentioning jobs in spectral astronomy, since the Bohr model explained the observation of spectral lines from hydrogen and other light elements out in space. The same type of analysis, using basically just a (much) fancier version of the spectroscope you got in your Goodie Bag, is used today to understand the stuff that makes up distant stars, nebulae, galaxies, quasars, all sorts of other intergalactic matter, including whether exoplanets have earth-link atmospheric compositions. But instead of going down that outer space route, I’d like to pick up here on the point I made in Why This Matters, regarding how solving the ozone depletion problem is a beautiful example of how important science policy can be in making the world a better place. What employment 23 opportunities are there along such lines? “Science policy” is a broad term that can apply to a wide array of jobs and activities, from deciding how science is funded to how it can (or cannot) be translated into commercial products, to how science impacts human health to protecting the health of the environment. The jobs that result from this direction often involve a connection to politics and Washington (if in the U.S.) but they don’t necessarily have to. One way to get involved is through fellowships, where students can engage with policy-makers in DC for a summer or a semester. There are a number of options along these lines, like the AAAS Diplomacy fellow (check out https://www.aaas.org/programs/science-technology-policy-fellowships), or fellowships at the Science and Technology Policy Institute, or the White House Office of Science and Technology Policy (OSTP). And then beyond such general ones there are many specialized fellowships in science policy that go by the sub-field, like the American Geophysical Union Congressional Science Fellowship, or the various NOAA Sea Grants for ocean-related policy, or the John Bahcall Public Policy Fellowship for astronomy. And many more. Beyond fellowships there are tons of jobs out there, like at science policy centers, think tanks or other organizations, from the Union of Concerned Scientists, to the RAND Corporation, to UNESCO, to the Federation of American Scientists, to the Center for Science in the Public Interest, to University-based centers like Arizona State’s Consortium for science policy and outcomes. These are only a very few examples – there are so many more. You could get involved directly with the science funding agencies like the NSF, DOE, or NIH (if you become a program manager at any of those please give me a call). Science policy is an incredibly important part of what science is, what it can become, and how it can be used most effectively, wisely, and for the good of the world. Extra practice 1. What is the energy associated with the ground state in a lithium ion? 2. If a light wave has the same magnitude of energy as the lithium ground state, what are the frequency and wavelength of this light? 24 Solutions 1. En = −Ry 32 Z2 = −13.6 2 = −122.4eV 2 1 n 2. E = hν = 122.4[eV ] = hc λ (4.14x10−15 [eV s])(3x108 [m/s]) (λ[m]) λ = 9.9 × 10−7 [m] 25