6.2: Development of Quantum Theory

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Skills to Develop

Extend the concept of wave–particle duality that was observed in electromagnetic radiation to matter as well
Understand the general idea of the quantum mechanical description of electrons in an atom, and that it uses the notion of three-dimensional wave functions, or orbitals, that define the distribution of probability to find an electron in a particular part of space
List and describe traits of the four quantum numbers that form the basis for completely specifying the state of an electron in an atom

Bohr’s model explained the experimental data for the hydrogen atom and was widely accepted, but it also raised many questions. Why did electrons orbit at only fixed distances defined by a single quantum number n = 1, 2, 3, and so on, but never in between? Why did the model work so well describing hydrogen and one-electron ions, but could not correctly predict the emission spectrum for helium or any larger atoms? To answer these questions, scientists needed to completely revise the way they thought about matter.

Behavior in the Microscopic World

We know how matter behaves in the macroscopic world—objects that are large enough to be seen by the naked eye follow the rules of classical physics. A billiard ball moving on a table will behave like a particle: It will continue in a straight line unless it collides with another ball or the table cushion, or is acted on by some other force (such as friction). The ball has a well-defined position and velocity (or a well-defined momentum, p = mv, defined by mass m and velocity v) at any given moment. In other words, the ball is moving in a classical trajectory. This is the typical behavior of a classical object.

When waves interact with each other, they show interference patterns that are not displayed by macroscopic particles such as the billiard ball. For example, interacting waves on the surface of water can produce interference patters similar to those shown on Figure \(\PageIndex{1}\). This is a case of wave behavior on the macroscopic scale, and it is clear that particles and waves are very different phenomena in the macroscopic realm.

A photograph is shown of ripples in water. The ripples display an interference pattern with each other. — *Figure \(\PageIndex{1}\): An interference pattern on the water surface is formed by interacting waves. The waves are caused by reflection of water from the rocks. (credit: modification of work by Sukanto Debnath)*

As technological improvements allowed scientists to probe the microscopic world in greater detail, it became increasingly clear by the 1920s that very small pieces of matter follow a different set of rules from those we observe for large objects. The unquestionable separation of waves and particles was no longer the case for the microscopic world.

This figure includes a circle formed from a dashed line. A sinusoidal wave pattern indicated with a solid red line is wrapped around the circle, centered about the edge of the circle. Line segments extend outward from the circle extending through 2 wave crests along the circle. A double ended arrow is drawn between these segments and is labeled, “wavelength, lambda.” A dashed double headed arrow is drawn from the center to the edge of the circle and is labeled, “radius r.” — *Figure \(\PageIndex{1}\): If an electron is viewed as a wave circling around the nucleus, an integer number of wavelengths must fit into the orbit for this standing wave behavior to be possible.*

Understanding Quantum Theory of Electrons in Atoms

The goal of this section is to understand the electron orbitals (location of electrons in atoms), their different energies, and other properties. The use of quantum theory provides the best understanding to these topics. This knowledge is a precursor to chemical bonding.

As was described previously, electrons in atoms can exist only on discrete energy levels but not between them. It is said that the energy of an electron in an atom is quantized, that is, it can be equal only to certain specific values and can jump from one energy level to another but not transition smoothly or stay between these levels.

The energy levels are labeled with an n value, where n = 1, 2, 3, …. Generally speaking, the energy of an electron in an atom is greater for greater values of n. This number, n, is referred to as the principal quantum number. The principal quantum number defines the location of the energy level. It is essentially the same concept as the n in the Bohr atom description. Another name for the principal quantum number is the shell number. The shells of an atom can be thought of concentric circles radiating out from the nucleus. The electrons that belong to a specific shell are most likely to be found within the corresponding circular area. The further we proceed from the nucleus, the higher the shell number, and so the higher the energy level (Figure \(\PageIndex{5}\)). The positively charged protons in the nucleus stabilize the electronic orbitals by electrostatic attraction between the positive charges of the protons and the negative charges of the electrons. So the further away the electron is from the nucleus, the greater the energy it has.

*Figure \(\PageIndex{5}\)*: *Different shells are numbered by principal quantum numbers.*

This quantum mechanical model for where electrons reside in an atom can be used to look at electronic transitions, the events when an electron moves from one energy level to another. If the transition is to a higher energy level, energy is absorbed, and the energy change has a positive value. To obtain the amount of energy necessary for the transition to a higher energy level, a photon is absorbed by the atom. A transition to a lower energy level involves a release of energy, and the energy change is negative. This process is accompanied by emission of a photon by the atom. The following equation summarizes these relationships and is based on the hydrogen atom:

\[ \begin{align*} ΔE &=E_\ce{final}−E_\ce{initial} \\[4pt] &=−2.18×10^{−18}\left(\dfrac{1}{n^2_\ce f}−\dfrac{1}{n^2_\ce i}\right)\:\ce J \end{align*} \]

The values n_f and n_i are the final and initial energy states of the electron.

The principal quantum number is one of three quantum numbers used to characterize an orbital. An atomic orbital, which is distinct from an orbit, is a general region in an atom within which an electron is most probable to reside. In addition, the principal quantum number defines the energy of an electron in a hydrogen or hydrogen-like atom or an ion (an atom or an ion with only one electron) and the general region in which discrete energy levels of electrons in a multi-electron atoms and ions are located.

The s subshell electron density distribution is spherical and the p subshell has a dumbbell shape. The d and f orbitals are more complex. These shapes represent the three-dimensional regions within which the electron is likely to be found.

*Figure \(\PageIndex{7}\)*: *Shapes of s, p, d, and f orbitals.*

*Figure \(\PageIndex{8}\)*: *The chart shows the energies of electron orbitals in a multi-electron atom.*

Figure \(\PageIndex{8}\) illustrates the energy levels for various orbitals. The number before the orbital name (such as 2s, 3p, and so forth) stands for the principal quantum number, n. The letter in the orbital name defines the subshell. In the case of a hydrogen atom or a one-electron ion (such as He⁺, Li²⁺, and so on), energies of all the orbitals with the same n are the same. This is called a degeneracy, and the energy levels for the same principal quantum number, n, are called degenerate energy levels. However, in atoms with more than one electron, this degeneracy is eliminated by the electron–electron interactions, and orbitals that belong to different subshells have different energies. Orbitals within the same subshell (for example ns, np, nd, nf, such as 2p, 3s) are still degenerate and have the same energy.

The Pauli Exclusion Principle

An Austrian physicist Wolfgang Pauli formulated a general principle that gives the last piece of information that we need to understand the general behavior of electrons in atoms. The Pauli exclusion principle can be formulated as follows: No two electrons in the same atom can have exactly the same quantum number and spin the same direction. What this means is that electrons can share the same orbital, but only if their spin have different values. Spin quantum numbers can only have two values \(\left(±\dfrac{1}{2}\right)\), no more than two electrons can occupy the same orbital (and if two electrons are located in the same orbital, they must have opposite spins). Therefore, any atomic orbital can be populated by only zero, one, or two electrons.

Summary

An atomic orbital is characterized by the principal quantum number, n, can be any positive integer. The general region for value of energy of the orbital and the average distance of an electron from the nucleus are related to n. Orbitals having the same value of n are said to be in the same shell. In addition, each electron has a spin number that can be equal to \(±\dfrac{1}{2}\). No two electrons in the same atom can have the same set of values for both quantum number and spin.

Glossary

atomic orbital: mathematical function that describes the behavior of an electron in an atom (also called the wavefunction), it can be used to find the probability of locating an electron in a specific region around the nucleus, as well as other dynamical variables

d orbital: region of space with high electron density that is either four lobed or contains a dumbbell and torus shape; describes orbitals with l = 2. An electron in this orbital is called a d electron

f orbital: multilobed region of space with high electron density, describes orbitals with l = 3. An electron in this orbital is called an f electron

p orbital: dumbbell-shaped region of space with high electron density, describes orbitals with l = 1. An electron in this orbital is called a p electron

Pauli exclusion principle: specifies that no two electrons in an atom can have the same value for the principle quantum number and spin

principal quantum number (n): quantum number specifying the shell an electron occupies in an atom

quantum mechanics: field of study that includes quantization of energy, wave-particle duality, and the Heisenberg uncertainty principle to describe matter

s orbital: spherical region of space with high electron density, describes orbitals with l = 0. An electron in this orbital is called an s electron

shell: set of orbitals with the same principal quantum number, n

spin quantum number (m_s): number specifying the electron spin direction, either \(+\dfrac{1}{2}\) or \(−\dfrac{1}{2}\)

subshell: set of orbitals in an atom with the same values of n but opposite spin

Contributors

Paul Flowers (University of North Carolina - Pembroke), Klaus Theopold (University of Delaware) and Richard Langley (Stephen F. Austin State University) with contributing authors. Textbook content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/85abf193-2bd...a7ac8df6@9.110).