### Video Transcript

In this video, we will learn about the key principles of modern atomic theory and compare it to the Bohr model that came before it. Every model of the atom was developed to explain something. Dalton’s hard sphere model accounted for the way atoms packed together. J. J. Thomson’s plum pudding model accounted for the presence of negatively charged particles in the atom. These are electrons. Geiger, Marsden, and Rutherford demonstrated the existence of the nucleus, a dense positively charged part of the atom in the very center. It’s the nucleus that contains protons. And Chadwick later demonstrated that nuclei also contain uncharged particles. These were neutrons.

But even before Chadwick demonstrated the existence of neutrons, Niels Bohr and Ernest Rutherford produced a new model of the atom. In the Rutherford–Bohr model, often simply called the Bohr model, electrons occupy orbits around the nucleus like the planets around the sun. The nuclear sits in the middle, the electrons go around the nucleus in a flat orbit, and the distance from the nucleus of each orbit is determined by the number of electrons orbiting and the nuclear charge. And it’s forbidden for an electron to orbit the nucleus in between the fixed orbits.

The Bohr model accounted for the fact that when we excite atoms or ions, they emit repeatable patterns of light, individual lines, at specific wavelengths. The strict differences in energy between orbits accounted for this behavior. An electron can be promoted to a higher orbit and emit light with a specific energy when it drops down again. For each nucleus, there was a list of specific allowed transitions, and these transitions for at least one element matched up with the results. For atomic hydrogen, the Bohr model was very accurate; however, for atoms with more than one electron, the Bohr model was incapable of reproducing experimental results accurately.

Naturally, there’s a lot of mathematics involved in the Bohr model, but we won’t be looking into the specifics. Instead, we’d be looking at the specific limitations of the Bohr model and then moving on to modern atomic theory. As we’ve already mentioned, the Bohr model in terms of the mathematics did not accurately describe multielectron systems, which is most of chemistry. So the model was inappropriate even for a simple system like a helium atom with two electrons.

It’s also important that the theory of the atom allows us to move on to describing bonding in molecules and ionic substances. However, the Bohr model didn’t offer a clear way forward. One clear indication that Bohr model wasn’t complete was that certain features of line spectra were not accounted for, such as some lines being more intense than others, distinct lines that was so very close together that no transition in the Bohr model could explain them, and in a magnetic field, line spectra can split into two. This effect could not be explained by the Bohr model.

A final nail in the coffin was that the Bohr model violated Heisenberg’s uncertainty principle. Heisenberg’s uncertainty principle is a part of quantum theory that came in later, and it states the more accurately you determine a particle’s position in space, the less you can determine about the particle’s momentum. This is a very deep, fundamental feature of the universe, and we won’t be going into the details. But we can simplify it to mean that we can either know where a particle is or how fast it’s going. The more precisely we measure one, the more uncertain we are of the other. In the Bohr model, the position and momentum of each electron is precisely defined and known.

This is a very high-level piece of understanding, but we can boil it down to a very simple principle of matter. The Bohr model treated electrons purely as particles, which orbited in a predictable way. However, de Broglie demonstrated that electrons had wavelike properties as well as particle-like properties. Wave–particle duality is the name for the concept that we can describe things in particle-like ways and also in wavelike ways. The electron acts like a particle; for instance, it has mass, and electrons can bounce off one another. But electrons also behave like waves, creating diffraction patterns in the same way light does in a double-slit experiment. An electron exhibits wave–particle duality. It has particle-like properties and wavelike properties.

The understanding that particles also behave like waves is fundamental to modern atomic theory, which also relies on quantum theory. In the early 1900s, there were many theories and discoveries that revolutionized what we call classical physics. In classical physics, all types of energy are seen to be continuous. You can adopt any value. But quantum theory introduced the idea that certain types of energy came in discrete packets called quanta. The size of these quanta depends on the system we’re dealing with. The details of this are complicated. But the key thing to understand is that this understanding allowed for a better model of the atom.

Quantum theory had a significant effect in our understanding of light. Light comes in quanta called photons. One of these green photons has roughly the right amount of energy to break a chlorine-chlorine single bond. A photon of blue light has even more energy, while a photon of red light has less. Armed with this understanding, we can appreciate why some wavelengths of light have an effect, while others don’t. If the energy of the photon doesn’t match the energy gap for the transition, the transition will not occur.

An understanding of wave–particle duality and quantum theory led in part to a new model of the atom that we still use today. Lots of people contributed to the next model of the atom, and while it has undergone revisions since the 1900s, the simple version of the model has stayed the same. The first tweak to the Bohr model is that electrons can behave in wavelike ways. Electrons aren’t seen to have a clear fixed position or momentum but can be spread out in a way that’s not easy to understand. These electron waves act like standing waves on a string. We can compare it to the way a string on a guitar can vibrate, except electrons are doing this in three dimensions, which makes it quite hard to visualize. Don’t worry if you can’t imagine this; a lot of people struggle.

Just like with a guitar string, only specific electron waves are allowed, and one of the most crucial features to come out of the mathematics and the descriptions is that we don’t think so much about the position of an electron, but where it might be. We then do averages to understand the behavior of the electron. So while in the old description the distance of an electron from the nucleus remained constant, the wave mechanical model described electrons as having a range of possible positions. At any point in time, they may be closer to the nucleus or they may be further away.

What’s more important from a mathematical point of view is the distribution and its average. When we have positions with different probabilities, we use a distribution. We can think of the electron being a set distance from the nucleus for a certain proportion of the time. Sometimes the electron is close to the nucleus. Sometimes it’s further away. But most of the time the electron is in between. Just bear in mind that talking about position of a wavelike electron gets very complicated, so this is just a simplification.

When we consider these behaviors, we start to talk about probabilities, and the probability distribution for the position of an electron is known as its wave function. Some electron waves have interesting shapes, so wave functions have to include the angle as well as the distance from the nucleus. However, in this simple example, we’ll stick with distance from the nucleus.

Erwin Schrodinger came up with a formula that connected this probability distribution with the energy of the electron, which matched the energies of the Bohr model of the hydrogen atom. The details of this equation are far beyond the reach of this video, so don’t worry about remembering the equation. What is important is that this approach to probability and electron waves allowed for prediction of how systems behave with multiple electrons. It also allowed us to understand how electrons behave when they come together in chemical bonds.

We can think of the wave mechanical model like a building. The floors of the building are at fixed heights, just like the energy levels in an atom. Somebody working on the first floor probably spends most of their time at their desks, but sometimes they’ll be getting a drink from the water cooler, and sometimes they’re relaxed by the window. What’s important is that, on average, they’re at their desks doing their job. If they put enough energy in, they might get a promotion to a higher floor with a nicer desk. And if the reverse happens, there’ll be a release of energy. Now that’s an overview of the fundamentals. Let’s do a practice question.

The graph shows the probability of finding the electron at a distance from the nucleus in the 1s orbital of an atom of hydrogen. At what approximate distance is the electron most likely to be found from the nucleus?

Here is the graph. The probability of finding the electron is on the 𝑦-axis, and its distance from the nucleus is on the 𝑥-axis. Without units for the probability axis, we can assume we’re dealing with relative probability. So points that are higher up the 𝑦-axis have a higher probability than points that are lower. The question also tells us we’re dealing with an electron in the 1s orbital of an atom of hydrogen. An atom of hydrogen has the electron configuration 1s1. s-type orbitals are spherical. This means the probability of finding an electron doesn’t vary depending on which angle you’re pointing out. It’s only the distance from the nucleus that matters.

We need to find the approximate distance from the nucleus of this electron that’s most likely. The most likely event is the one with the highest relative probability. So we need to find the highest point of this curve and then draw down to the 𝑥-axis to find the distance. The highest point on this curve is here. This corresponds with a distance from the nucleus the 50 picometers. The question is only asking for the approximate distance. But to my eye, that peak is bang on the line of 50 picometers, so we can be comfortable that 50 picometers is the correct answer. What this means is if you were to check where the electron was in a 1s orbital of a hydrogen atom, more times than any other it would be in about 50 picometers.

Let’s finish up with the key points. The modern model of the atom is a step forward from Bohr’s model, which says that electrons are in circular orbits around the nucleus at fixed distances. The Bohr model was only able to account for the energies in atoms like hydrogen with one electron. The Bohr model also violated Heisenberg’s uncertainty principle, which is a fundamental feature of the universe. Heisenberg’s uncertainty principle tells us that the more accurately we know the position of a particle, the less accurately we can know its momentum and vice versa.

The modern theory of the atom, the wave mechanical model, tells us that electrons are like standing waves with probability distributions for their positions. These probability distributions are known as wave functions. The wave nature of an electron is part of an understanding called wave–particle duality. The key thing we need to understand about wave–particle duality is that particles like electrons have wavelike properties as well.