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Wave Mechanical Model

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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!

• 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.

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.

The wave mechanical model was proposed by Erwin Schrödinger.

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

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

The wave mechanical model was created in 1926.

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

Wave Mechanical Model Quiz - Teste dein Wissen

Question

What are quantum numbers?

Quantum numbers describe the size, shape, and orientation in space of orbitals. Each electron in an atom has a unique set of quantum numbers

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Question

What are orbitals?

Orbitals are a 3-D space that describe the area where and electron is likely to be

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Question

What are the four quantum numbers?

The four quantum numbers are:

• Principal quantum number (n)
• The angular quantum number ($$l$$)
• The magnetic quantum number ($$m_l$$)
• The electronic spin quantum number (ms)

Show question

Question

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

True

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Question

Which $$l$$ value corresponds to the d-orbital?

2

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Question

What is the maximum value of $$l$$?

n-1

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Question

If the $$l$$ value is 2, what are the possible $$m_l$$ values?

-2,-1,0,1,2

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Question

What does the magnetic quantum number ($$m_l$$) tell us?

The magnetic quantum number tells us the orientation of an orbital with a given n and $$l$$ value.

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Question

What is the Pauli exclusion principle?

The Pauli exclusion principle states that no two electrons can have the same four quantum numbers.

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Question

What are the possible spins for an electron?

+1/2

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Question

What is an electron configuration?

Electron configuration is the distribution of electrons in an atom or molecule

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Question

What is an orbital diagram?

An orbital diagram is an illustration of the distribution and spin of an atom's electrons within its orbitals.

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Question

Which of the following answers do these quantum numbers match?

n=4

$$l=3$$

4f

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Question

An electron's possible location around a nucleus can be determined by its  ______

energy level

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Question

Electrons in the outermost level are known as _______.

valence electrons

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Question

In the process of moving from a higher to a lower energy level (closer to the nucleus), an electron _____ energy, emitting a photon.

releases

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Question

True or false: according to Louis de Broglie, matter possesses both particle and wave properties.

True

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Question

The ______  states that it is impossible to know the momentum and position of an object at the same time.

Heisenberg uncertainty principle

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Question

In Heisenberg's uncertainty principle, The object's velocity is related to its ____ nature.

wave

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Question

In Heisenberg's uncertainty principle, the position of the object is related to its ____ nature.

particle

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Question

The Heisenberg uncertainty principle tells us that the more accurately we know an object's position, the  _____ accurately we can know its momentum (and vice versa).

less

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Question

According to the ______, the uncertainty in position times the uncertainty in momentum is greater than/equal to Planck's constant divided by four π.

Heisenberg uncertainty formula

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Question

What are the steps to find an electron's uncertainty in velocity, given the electron's uncertainty in position.

1. Solve for uncertainty in momentum using Heisenberg's uncertainty formula

2. Solve for uncertainty in velocity using the general formula relating uncertainty in momentum, mass and velocity.

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Question

Who proposed Heisenberg's uncertainty principle?

Werner Heinsenberg

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Question

The ____ model of the atom 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.

Bohr

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Question

The _______  states that electrons (and all matter) have both particle and wave-like characteristics.

de Broglie wavelength equation

Show question

Question

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

quantum numbers

Show question

Question

The _______ states an electron behaves as a standing wave. It also describes an electron's possible location in an orbital.

quantum (wave) mechanical theory

Show question

Question

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

Heisenberg uncertainty principle

Show question

Question

True or false: according to the wave mechanical model, electrons do not follow fixed/definite paths.

True

Show question

Question

According to the wave mechanical model, electrons are found in a cloud of negative charge around the nucleus called the _____.

electron cloud

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Question

According to the wave mechanical model, the 3D region where an electron can probably be found is called an _____.

orbital

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Question

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

location

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Question

What are the four quantum numbers?

The principal quantum number (n), the azimuthal quantum number (l), the magnetic quantum number (ml) and the spin quantum number (ms)

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Question

True or false: standing waves are waves that do not propagate through space and are fixed at both ends.

True

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Question

_____ are 2D circular paths that possess a fixed distance from the nucleus and contain 2n2 electrons per orbit.

Orbits

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Question

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

Orbitals

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Question

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

subshell

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Question

Schrödinger came up with the ______ after noticing that an electron bound to the nucleus indeed seemed similar to a standing wave.

wave mechanical model

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Question

What is the De Broglie hypothesis?

De Broglie hypothesized that matter acts as a wave

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Question

What is another name for the De Broglie hypothesis?

De Broglie's wavelength

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Question

What are matter waves?

Matter waves are the wave-like behavior exhibited by matter

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Question

True or False: Matter behaves as both a particle and a wave

True

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Question

True or False: The wavelengths of everyday objects are large/strong enough to have a visible effect

False

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Question

What do matter waves show us mathematically?

The probability of the particle being at "x" is spread out like a wave, as shown by a matter wave.  There is no definite position of the particle.

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Question

What is the De Broglie wave equation?

$$\lambda=\frac{h}{p}$$

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Question

True or False: The De Broglie wavelength equation applies to particles without mass

False

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Question

What is the relationship between particle size/mass?

The larger/heavier the particle, the smaller the wavelength

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Question

The De Broglie waves for electrons form what shape?

A closed loop

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Question

True or False: Electrons can only exist as standing waves at certain energy levels

True

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