Wave Mechanical Model

At this point in your chemistry journey, you have probably heard about the different models trying to explain the behavior of electrons surrounding an atom's nucleus. Here, we will focus on the wave mechanical model of the atom! 

Wave Mechanical Model Wave Mechanical Model

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Contents
Table of contents
    • First, we will talk about the Bohr model of the atom.
    • Then, we will look at the history of the wave mechanical model and its definition.
    • After, we will talk about some features of the wave mechanical model.

    Bohr vs. Wave Mechanical Model

    Before diving into the wave mechanical model of the atom, let's talk about the Bohr model of the atom.

    Niels Bohr was a Danish physicist who was born in 1885, and died in 1962 at the age of 77 years. Bohr was a very influential physicist, known for his contribution to Atomic Structure and quantum mechanics. But, his greatest discovery was the Bohr model of the atom.

    In this model, Bohr proposed that electrons would go around the nucleus in fixed circular orbits, and each orbit would have a specific energy and distance from the nucleus. In other words, an electron found in a specific orbit would have that specific amount of energy.

    Bohr came up with an equation to describe the energy of each orbit, relating it to the energy level, \(n\). This equation says that the energy of a particular orbit (\(E_{n}\)) is equal to:

    $$ \frac{(-2.18 \times 10^{-18} J)}{n^{2}} $$

    where,

    • \(n\) is the energy level (1, 2, 3,4, ...)

    How much energy would an electron with an orbital \(n\)=2 possess?

    This is a very simple problem. We only need to plug in the number 2 instead of \(n\) in the equation above!

    $$ E_{n} = \frac{(-2.18 \times 10^{-18} J)}{n^{2}} = \frac{(-2.18 \times 10^{-18} J)}{2^{2}} = -5.45\times 10^{-19}J $$

    The Bohr model also states that the closer an electron is to an atom's nucleus, the closer its potential energy will be. Moreover, Bohr suggested that electrons could jump between orbits, absorbing or releasing a certain amount of energy in the process. To calculate the change in energy, \( \Delta E\), from going from one orbit to another, Bohr used the formula below:

    $$ \Delta E =(-2.18 \times 10^{-18} J)\times (\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}) $$

    Where:

    • \(n_{f} \) is the final orbit
    • \(n_{i}\) is the initial orbit

    However, this model was proved wrong because it could only be applied to hydrogen (H). Plus, electrons don't actually move around the nucleus in fixed circular orbits. This brings us to another model, a model that aims to describe the wave-like behavior of electrons in an atom.

    Wave Mechanical Model History

    The modern theory of the atom is given by the wave mechanical model, which was proposed by the work of three important physicists: Werner Heisenberg, Louis de Broglie, and Erwin Schrödinger.

    In 1924, Louis de Broglie proposed that an electron (previously considered to be a particle) showed the properties of a wave. He was able to make this discovery by seeing how electrons could be bent or diffracted when passing through a crystal.

    The de Broglie wavelength equation states that electrons (and all matter) have both particle and wave-like characteristics, and its wavelength is equal to Planck's constant divided by the mass of the particle times its velocity.

    $$ \lambda = \frac{\text{h}}{m\times v} $$

    Based on this information, in 1925, Werner Heisenberg noticed that it was impossible to know an electron's position and speed at the same time, due to the dual particle/wave nature of the electron. So, Heisenberg came up with Heisenberg's uncertainty principle.

    The Heisenberg Uncertainty Principle states that one cannot know the momentum (mass x velocity) and position of an electron simultaneously.

    For a more in-depth explanation on this, check out "Heisenberg Uncertainty Principle"!

    Then, in 1926, using both insights, Schrödinger came up with the wave mechanical model after noticing that an electron bound to the nucleus indeed seemed similar to a standing wave. This model consisted of a mathematical equation involving wave functions (\(\psi\)) as a way to describe the behavior of electrons as a wave. The simplest form of the Schrödinger's wave equation is shown below. This equation was used to predict the probable location of an electron around the nucleus.

    $$ Hψ = Eψ $$

    Where:

    • H is equal to numerous mathematical functions called "operators".
    • ψ is equal to a wave function.

    To be able to find this electron's possible location, the wave mechanical model suggested that each energy level/shell (given by the Bohr model) was subdivided into a specific number subshells.

    A subshell is a region where a group of electrons in an atom are located within the same shell.

    Atomic Orbital

    Now, electrons are distributed among atomic orbitals in each subshell. These orbitals are sometimes called charge clouds or electron clouds.

    Orbitals are 3D regions of space within a subshell where an electron might be found 90 percent of the time.

    Each orbital contains two electrons, and are further classified based on their shape. You can learn more about this by reading "Electron shells, Sub-shells, and Orbitals"!

    The image below shows the difference between orbits (as seen in the Bohr model) and orbitals. Orbits are 2D circular paths that possess a fixed distance from the nucleus and contain 2n2 electrons per orbit. Orbitals, on the other hand, are 3D regions of space (no fixed path) with a variable distance from the nucleus, and 2 electrons per orbital.

    Wave Mechanical Model Definition

    The definition of the quantum (wave) mechanical model of the atom is written below.

    The quantum (wave) mechanical theory states an electron behaves as a standing wave. It also describes an electron's possible location in an orbital.

    Standing waves are waves that do not propagate through space and are fixed at both ends.

    To describe the theoretical behavior of electrons, following the wave mechanical model, Quantum Numbers are used.

    Quantum Numbers are specific values that describe the energy levels and, ultimately, the location of a specific electron.

    Quantum numbers basically give us the "coordinates" to find the theoretical location of an electron. There are four quantum numbers you need to be familiar with:

    • The principal quantum number (\(n\)) deals with the energy and size of atomic orbitals.

    • The angular momentum (azimuthal) quantum number (\(ℓ\)) deals with the shape of an orbital within a subshell.

    • The magnetic quantum number (\( \text{m}_{ℓ } \)) gives us the approximate location of electrons in a set of atomic orbitals.

    • The spin quantum number (\( \text{m}_{s} \)) tells us the spin of electron in an orbital.

    To describe an orbital using Schrödinger's equation, we would need three quantum numbers: the principal quantum number (\(n\)), the magnetic quantum number (\( \text{m}_{ℓ } \)) , and the azimuthal quantum number (\(ℓ\))!

    Wave Mechanical Model Features

    The wave mechanical model has the following features:

    • Electrons do not follow fixed/definite paths (as proposed by Bohr).

    • Electrons are found in a cloud of negative charge around the nucleus called the electron cloud.

    • There are areas around the nucleus that correspond to certain energy levels (as suggested by Bohr).

    • The 3D region where an electron can probably be found is called an orbital.

    Now, I hope that you were able to understand the wave mechanical model!

    Wave Mechanical Model - Key takeaways

    • Bohr proposed that electrons would go around the nucleus in fixed circular orbits. However, this was later proved wrong thanks to the wave mechanical model.
    • The de Broglie wavelength equation states that electrons (and all matter) have both particle and wave-like characteristics
    • The Heisenberg uncertainty principle states that one cannot know the momentum (mass x velocity) and position of an electron simultaneously.
    • The quantum (wave) mechanical theory states an electron behaves as a standing wave. It also describes an electron's possible location in an orbital.
    • To describe the theoretical behavior of electrons, following the wave mechanical model, quantum numbers are used.

    References

    1. Zumdahl, S. S., Zumdahl, S. A., & Decoste, D. J. (2019). Chemistry. Cengage Learning Asia Pte Ltd.
    2. Theodore Lawrence Brown, Eugene, H., Bursten, B. E., Murphy, C. J., Woodward, P. M., Stoltzfus, M. W., & Lufaso, M. W. (2018). Chemistry : the central science (14th ed.). Pearson.
    3. Randall Dewey Knight, Jones, B., & Field, S. (2019). College physics : a strategic approach. Pearson.
    4. Moore, J. T., & Langley, R. H. (2021c). 5 Steps to a 5: AP Chemistry 2022 Elite Student Edition. McGraw Hill Professional.
    5. Swanson, J. (2021). Everything you need to ace chemistry in one big fat notebook. Workman.
    Frequently Asked Questions about Wave Mechanical Model

    Who made the wave mechanical model?

    The wave mechanical model was proposed by Erwin Schrödinger.

    What is the difference between the wave mechanical model and the quantum mechanical model? 

    The quantum mechanical model is just another name for the wave mechanical model of the atom. 

    Why did scientists develop the wave mechanical model?

    Scientists developed the wave mechanical model to describe the behavior of electrons as a wave. 

    When was the wave mechanical model created?

    The wave mechanical model was created in 1926. 

    What is the wave mechanical model required to explain?

    The wave mechanical model aims to explain the wave nature of electrons. 

    Test your knowledge with multiple choice flashcards

    True or False: The principal quantum number (n) tells us how far an electron is from the nucleus

    Which \(l\) value corresponds to the d-orbital?

    What is the maximum value of \(l\)?

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