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Chapter 7: Problem 46

How is the concept of electron density used to describe the position of an electron in the quantum mechanical treatment of an atom?

Short Answer

Expert verified
In quantum mechanics, the electron density or probability density is used to depict the position of an electron in an atom. It represents the likelihood of finding an electron within a particular space. Instead of precise orbits like in classical physics, quantum mechanics describes the electron's location in the atom in terms of a probability cloud, or atomic orbitals. These atomic orbitals, defined by the distribution of electron density, are distinctive regions around the nucleus where the electron is most likely to be observed.

Step by step solution

01

Understanding the Particle in a Box Model

Begin with remembering the basics of quantum mechanics. In quantum mechanics, unlike in classical physics, an electron's precise location cannot be determined due to Heisenberg's Uncertainty Principle. Therefore, a model called 'the particle in a box' model is typically engaged to illustrate the principles of quantum mechanics, such as wave-particle duality and the probabilistic nature of quantum entities. This model sets the theoretical foundation of understanding the behavior of an electron in a potential energy well.
02

Conceptualize Electron Density

Electron density (also known as probability density) refers to the probability of an electron being located in a given volume of space, such as within an atomic or molecular orbital. Essentially, it illustrates the regions where one is likely to find an electron. The electron density is linked to the wave function of an electron (\( \psi \)), a mathematical function obtained by solving the Schr枚dinger equation. The electron density at a particular point in space is proportional to the square of the modulus of the wave function, \( |\psi|^2 \).
03

Apply Electron Density to Quantum Mechanical Treatment of an Atom

For an atom, the electron density or probability density distribution is used to describe, probabilistically, where the electron is likely to be observed. Instead of specifying an exact path for the electron, like an orbit, quantum mechanics describes the electron's position in terms of a probability cloud, also known as an atomic orbital. The distribution pattern of electron density within these orbitals gives the distinct shapes of s, p, d, and f orbitals. As a result, the concept of electron density serves an integral role in the quantum mechanical explanation of electron placements in an atom.

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Most popular questions from this chapter

The sun is surrounded by a white circle of gaseous material called the corona, which becomes visible during a total eclipse of the sun. The temperature of the corona is in the millions of degrees Celsius, which is high enough to break up molecules and remove some or all of the electrons from atoms. One way astronomers have been able to estimate the temperature of the corona is by studying the emission lines of ions of certain elements. For example, the emission spectrum of \(\mathrm{Fe}^{14+}\) ions has been recorded and analyzed. Knowing that it takes \(3.5 \times 10^{4} \mathrm{~kJ} / \mathrm{mol}\) to convert \(\mathrm{Fe}^{13+}\) to \(\mathrm{Fe}^{14+},\) estimate the temperature of the sun's corona.

Explain the statement, Matter and radiation have a "dual nature".

An electron in an excited state in a hydrogen atom can return to the ground state in two different ways: (a) via a direct transition in which a photon of wavelength \(\lambda_{1}\) is emitted and (b) via an intermediate excited state reached by the emission of a photon of wavelength \(\lambda_{2}\). This intermediate excited state then decays to the ground state by emitting another photon of wavelength \(\lambda_{3}\). Derive an equation that relates \(\lambda_{1}\) to \(\lambda_{2}\) and \(\lambda_{3}\)

List all the possible subshells and orbitals associated with the principal quantum number \(n,\) if \(n=6\).

When an electron makes a transition between energy levels of a hydrogen atom, there are no restrictions on the initial and final values of the principal quantum number \(n\). However, there is a quantum mechanical rule that restricts the initial and final values of the orbital angular momentum \(\ell\). This is the selection rule, which states that \(\Delta \ell=\pm 1,\) that is, in a transition, the value of \(\ell\) can only increase or decrease by one. According to this rule, which of the following transitions are allowed: (a) \(1 s \longrightarrow 2 s\), (b) \(2 p \longrightarrow 1 s\) (c) \(1 s \longrightarrow 3 d\) (d) \(3 d \longrightarrow 4 f\), (e) \(4 d \longrightarrow 3 s ?\)
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