In this explainer, we will learn how to describe the concepts of modern atomic theory.

Each successive model of the atom was developed to produce a theoretical explanation for a practical result.

- John Dalton’s hard-sphere model accounted for the way atoms pack together.
- J. J. Thomson’s plum pudding model accounted for the presence of negatively charged particles in the atom, which we call “electrons.”
- Geiger, Marsden, and Ernest Rutherford demonstrated the existence of the nucleus, a dense, positively charged part of the atom in the very center of it. It is the nucleus that contains protons.
- Niels Bohr and Ernest Rutherford proposed the Rutherford–Bohr model, often simply called the Bohr model, where electrons occupy orbits around the nucleus like the planets around the sun. This explained the features of the emission spectrum of hydrogen.
- James Chadwick later demonstrated that nuclei also contain uncharged particles. These are neutrons.

The Bohr model suggests the following about atoms:

- A dense positively charged nucleus sits in the middle of the atom.
- Electrons go around the nucleus in flat, circular orbits.
- The radius of each orbit is determined by the nuclear charge.
- It is impossible 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, when it drops down again, emit light with a specific energy.

For each nucleus, the Bohr model produced a list of allowed transitions and their energies. For atomic hydrogen, the Bohr model was very accurate. However, for atoms with more than one electron, the Bohr model did not reproduce experimental results.

These are the emission spectra for a few of these elements.

These are the main issues with the Bohr model that led to it being revised:

- It does not describe the energies of electron transitions in systems with more than one electron.
- It does not account for the fact that some emission lines in atomic emission spectra are more intense than others.
- It does not account for the fact that some emission lines are too close together to have come from distinct transitions.
- It does not provide a model of bonding.
- It violates Heisenberg’s uncertainty principle.

The Bohr model treated electrons as particles; we now know that electrons also have wave-like properties. Also, in the Bohr model, a hydrogen atom would be essentially two dimensional, with the electron moving in a single, flat plane.

Heisenberg’s uncertainty principle is a principle of quantum mechanics, which came later. It is a fundamental truth of the universe.

### Definition: Heisenberg’s Uncertainty Principle

The more accurately you determine a particle’s position in space, the more uncertain your measurement of its momentum becomes (and vice versa).

In simple terms, we can either know where a particle is or how fast it is going (and in what direction). This principle is only significant for subatomic particles, or larger objects moving at very high speeds.

The Bohr model treats electrons purely as particles that orbit in a predictable way. This model breaks Heisenberg’s uncertainty principle, so the Bohr model must be incorrect in some way.

Werner Heisenberg’s work on the uncertainty principle was part of a larger body of work that expanded how we describe the universe. Louis de Broglie provided one of the most crucial insights that redefined how we imagine how electrons behave. De Broglie demonstrated that electrons had wave-like properties as well as particle-like properties.

It turns out that everything we call a particle can be described in wave-like ways, and everything we call a wave can be described in particle-like ways. However, in general, we only observe wave-like behavior from particles moving at speeds close to the speed of light.

The name we give to this concept is wave–particle duality.

### Definition: Wave–Particle Duality

Physical objects can exhibit wave-like properties and particle-like properties. The degree to which they exhibit this behavior depends on what they are and how fast they are moving.

An electron can act like a particle since electrons have mass, and electrons can bounce off one another. However, an electron can also behave like a wave, 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 wave-like 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. Energies can be any value.

However, quantum theory introduced the idea that certain types of energy come in discrete packets called quanta. The size of these quanta depends on the system. The details of this are complex, but the key thing to understand is that this knowledge allowed for a better model of the atom.

Quantum theory had a significant effect on our understanding of light. Light comes in quanta called photons. A green photon 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 do not. If the energy of the photon is smaller than the energy gap for the transition, the transition cannot occur.

An understanding of wave–particle duality and quantum theory led to the modern model of the atom. 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 wave-like ways. Electrons are not seen to have a clear fixed position or momentum but can be spread out in a way that is not easy to describe or understand.

### Example 1: Identifying Which Statement Is Not a Limitation of the Bohr Model of the Atom

Which of the following is **not** a limitation of the Bohr model of the atom?

- Electrons move around the nucleus in circular, planar orbits.
- Electrons are only considered as particles and not as waves.
- It is possible to precisely determine the position and momentum of an electron simultaneously.
- Electrons within atoms can only occupy quantized energy levels.
- It explains the line emission spectrum of the hydrogen atom only.

### Answer

The Bohr model of the atom was at one point the best description of an atom. However, this model was not without its limitations. Discoveries in physics and the formulation of quantum mechanics highlighted inadequacies in the Bohr model of the atom. In this question, we are given five statements about the Bohr model and need to identify which of them is not a limitation.

In the Bohr model of the atom, electrons are described as having circular, planar orbitals around a central nucleus. By treating electrons as particles, the Bohr model implied that a hydrogen atom would be essentially two dimensional, with the electron moving in a single, flat plane. However, de Broglie demonstrated that electrons have wave–particle duality, while Schrödinger’s wave-mechanical model described electrons as having a range of possible positions. At any point in time, an electron may be closer to the nucleus or further away. Based on these two points, we can clearly label statements A and B as limitations to the Bohr model.

By describing particles that orbit in a predictable way, the Bohr model suggested that the position and momentum of an atom could be precisely determined simultaneously. However, Heisenberg’s Uncertainty principle states that the more accurately you determine a particle’s position in space, the more uncertain your measurement of its momentum becomes. In other words, we cannot accurately determine the position and momentum of a particle simultaneously. This is another limitation of the Bohr model, and so statement C is true.

The Bohr model of the atom introduced the idea of electrons occupying orbits of fixed energies. As a result, it is impossible for an electron to orbit the nucleus in between the fixed orbits. This gives rise to a list of allowed transitions between orbits with a specific energetic value. Based on these transitions and characteristic energies, the line emission spectrum of a hydrogen atom could be very accurately explained. However, it was less successful for atoms with more than one electron. This makes statement E a limitation of the Bohr model of the atom.

However, the notion of electrons occupying quantized energy levels is still part of the modern atomic theory and so statement D is not a limitation of the Bohr model.

The correct answer is therefore statement D.

One of the most crucial features to come out of the calculations and the descriptions is that we do not think so much about the position of an electron, but where it might be. We then calculate 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, an electron may be closer to the nucleus or further away.

What is 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, and sometimes it is further away. However, most of the time the electron is in between, although the position of a wave-like electron gets very complicated, so this is just a simplification.

### Example 2: Determining the Distance from the Nucleus at Which an Electron in a 1s Orbital is Most Likely to be Found

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?

### Answer

Modern atomic theory, or more specifically quantum theory, describes an electron as not occupying a circular orbit at a fixed distance from the nucleus, unlike in the Bohr model. Instead, the electron can occupy a range of different positions, although with differing probabilities. We can think of the electron being a set distance from the nucleus for a certain proportion of the time.

The graph given in the question shows the probability of finding an electron at a distance from the nucleus in the 1s orbital of an atom of hydrogen. The -axis gives the distance from the nucleus in units of picometres (pm), and the -axis gives us the probability of finding the electron at that distance. The higher the value on the -axis, the greater the probability of finding the electron.

Here, the electron is in a 1s orbital. s-type orbitals are spherical and so we only need to consider the distance from the nucleus and not the angle.

From the graph, we can see that the highest point on the curve, and hence the highest value on the -axis, occurs at a distance of around 50 pm from the nucleus. As the question only asks for the approximate distance, we do not need to give our answer to greater precision.

The correct answer is 50 pm.

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. The probability distribution of electrons can have complex shapes; however, in this simple example, we will stick with distance from the nucleus.

Erwin Schrödinger 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 equation looks deceptively simple, but the details behind it are incredibly complicated. What is important is that this approach to probability and electron waves allowed for the prediction of how systems behave with multiple electrons. It also allowed us to understand how electrons behave when they come together in chemical bonds. The modern model that Schrödinger ushered in is known as the wave-mechanical model of the atom. By solving the Schrödinger equation, it is possible to determine the energy levels and the most probable location, from the nucleus, of an electron in each energy level. Schrödinger’s work led to the concept of the electron cloud, the region of space around the nucleus where an electron might be found. This gave rise to atomic orbitals, which were much more complicated shapes than the circular, planar orbits of the Bohr model. However, the Schrödinger equation can only be solved exactly for the hydrogen atom.

### Definition: Electron Cloud

The electron cloud describes the region of space around the nucleus of an atom where there is a probability of finding an electron.

### Definition: Atomic Orbitals

Atomic orbitals are three-dimensional mathematical expressions that describe part of the electron cloud and are the most likely location of an electron in an atom.

### Example 3: Identifying the Density Map That Best Describes the Possible Location of the Electron of a Hydrogen Atom according to Quantum Theory

Which of the density maps most accurately represents the possible location of the electron in an atom of hydrogen according to the quantum theory of the atom?

### Answer

Each diagram consists of a large black dot (the nucleus) and some small blue dots (the electron density). The closer the blue dots are to each other, the greater the likelihood of finding the electron in that position. All this uncertainty comes from the wave-like nature of electrons when they are around a nucleus.

A hydrogen atom has 1 electron in its 1s orbital. This electron will be strongly attracted to the hydrogen nucleus and is likely to be found close to the nucleus. As we get further from the nucleus, the chance of finding the electron goes down.

Answer A is incorrect because there is a chance that the electron could be found at any distance from the nucleus, not just at fixed distances. This diagram is closer to the Bohr model of the atom.

Answer B is quite good. The closer to the nucleus we get, the denser the blue dots get, indicating that the chance of finding the electron increases as we approach the nucleus.

In answer C, the dots have an unusual pattern, with areas that are dotty and areas that are not. This might occur for an excited-state electron, but in the ground state of a hydrogen atom, the electron density map will be spherically symmetrical—this is the lowest-energy configuration. Therefore, C is not plausible.

Answer D has strange local patches of increased electron density; there is no reason for these to form as there is nothing to attract the electron besides the nucleus.

The most accurate density map is the one showing increasing electron density as we approach the nucleus. The answer is B.

### 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 is only able to account for the energies in atoms like hydrogen, with one electron.
- The Bohr model assumed electrons have only particle-like properties, but electrons have wave-like properties as well.
- In the Bohr model, the electron of a hydrogen atom would orbit the nucleus in a 2D plane, making the hydrogen atom essentially planar.
- The Bohr model violates 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 wave-like nature of an electron is part of a concept called wave–particle duality: particles, like electrons, have wave-like properties as well.
- 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.
- By solving the Schrödinger equation, it is possible to determine the allowed energy levels of a system and define the regions of space around the nucleus where an electron might be found, known as the electron cloud.