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Chapter 2: Problem 75

What is the physical significance of the value of \(\psi^{2}\) at a particular point in an atomic orbital?

Short Answer

Expert verified
The physical significance of the value of \(\psi^2\) at a particular point in an atomic orbital is that it represents the probability density, which indicates the likelihood of finding an electron at that specific location within the orbital. A higher value of \(\psi^2\) implies a high probability of finding an electron at that point, while a lower value indicates a lower probability. This information is crucial for understanding the behavior and distribution of electrons within atoms.

Step by step solution

01

Define the Wave Function

A wave function, represented by the symbol \(\psi\), is a mathematical description of the physical state of a quantum system, such as an electron in an atom. It contains all the information about the behavior and properties of the system and can be used to predict the outcomes of physical measurements on the system.
02

Quantum Mechanics and Atomic Orbitals

Quantum mechanics, the branch of physics that deals with the behavior of atoms and subatomic particles, uses wave functions to describe and study atomic orbitals, which are regions within atoms where electrons are most likely to be found. Atomic orbitals can come in different shapes and sizes, depending on the energy level and the type of atom.
03

Square of the Wave Function: Probability Density

The square of the wave function, \(\psi^2\), is known as the probability density. Probability density represents the likelihood of finding an electron at a specific point within an atomic orbital. Mathematically, \(\psi^2\) can be described as the probability density function. A higher value of \(\psi^2\) at a particular point implies a high probability of finding an electron at that point, while a lower value indicates a lower probability.
04

Physical Significance of \(\psi^2\) in Atomic Orbitals

The physical significance of the value of \(\psi^2\) at a particular point in an atomic orbital is that it indicates the probability of finding an electron at that location within the orbital. In other words, the higher the value of \(\psi^2\), the more likely it is to find an electron at that point. This information helps us visualize and understand the behavior and distribution of electrons within atoms.

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Most popular questions from this chapter

Photogray lenses incorporate small amounts of silver chloride in the glass of the lens. When light hits the AgCl particles, the following reaction occurs: $$ \operatorname{AgCl} \stackrel{h v}{\longrightarrow} \mathrm{Ag}+\mathrm{Cl} $$ The silver metal that is formed causes the lenses to darken. The energy change for this reaction is \(3.10 \times 10^{2} \mathrm{kJ} / \mathrm{mol} .\) Assuming all this energy must be supplied by light, what is the maximum wavelength of light that can cause this reaction?

Consider the following ionization energies for aluminum: $$ \begin{aligned} \mathrm{Al}(g) \longrightarrow \mathrm{Al}^{+}(g)+\mathrm{e}^{-} & I_{1}=580 \mathrm{kJ} / \mathrm{mol} \\ \mathrm{Al}^{+}(g) \longrightarrow \mathrm{Al}^{2+}(g)+\mathrm{e}^{-} & I_{2}=1815 \mathrm{kJ} / \mathrm{mol} \\ \mathrm{Al}^{2+}(g) \longrightarrow \mathrm{Al}^{3+}(g)+\mathrm{e}^{-} & I_{3}=2740 \mathrm{kJ} / \mathrm{mol} \\ \mathrm{Al}^{3+}(g) \longrightarrow \mathrm{Al}^{4+}(g)+\mathrm{e}^{-} & I_{4}=11,600 \mathrm{kJ} / \mathrm{mol} \end{aligned} $$ a. Account for the trend in the values of the ionization energies. b. Explain the large increase between \(I_{3}\) and \(I_{4}\).

While Mendeleev predicted the existence of several undiscovered elements, he did not predict the existence of the noble gases, the lanthanides, or the actinides. Propose reasons why Mendeleev was not able to predict the existence of the noble gases.

How many valence electrons do each of the following elements have, and what are the specific valence electrons for each element? a. Ca b. O c. element 117 d. In e. Ar f. Bi

How many orbitals in an atom can have the designation \(5 p,\) \(3 d_{z}, 4 d, n=5, n=4 ?\)
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