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Chapter 7: Problem 81

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 the probability density of finding an electron at that specific location in space. Higher values of \(\psi^{2}\) indicate a higher likelihood of finding the electron in that region, while lower values suggest a lower probability. It is crucial to remember that \(\psi^{2}\) represents the probability density, not the actual probability itself. To determine the probability of locating the electron within a specific volume, the probability density function (\(\psi^{2}\)) must be integrated over that volume.

Step by step solution

01

Identify the wave function (\(\psi\))

In quantum mechanics, the wave function (\(\psi\)) is a mathematical function that describes the wave-like behavior of a particle, such as an electron in an atomic orbital. The wave function is obtained by solving the Schrödinger equation for the specific system under consideration.
02

Calculate the probability density function (\(\psi^{2}\))

The probability density function (\(\psi^{2}\)) is obtained by squaring the wave function (\(\psi\)). The value of \(\psi^{2}\) at a particular point in an atomic orbital represents the likelihood of finding the electron at that location.
03

Interpret the meaning of \(\psi^{2}\) in the context of atomic orbitals

In atomic orbitals, the value of \(\psi^{2}\) at a particular point gives the probability density of finding an electron at that point in space. A higher value of \(\psi^{2}\) means that there is a higher probability of finding the electron in that region, whereas a lower value indicates a lower probability. It is important to note that \(\psi^{2}\) represents the probability density, and not the actual probability itself. To find the probability of finding the electron within a certain volume, you would need to integrate the probability density function (\(\psi^{2}\)) over that volume.
04

Visualize the wave function and probability density function

It is often helpful to visualize the wave function and probability density function to better understand the electron distribution in an atomic orbital. Different graphing techniques, such as isosurface plots or contour plots, can be used to represent the probability density and help understand where the electron is more likely to be found within the atomic orbital.

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Most popular questions from this chapter

Identify the following elements. a. An excited state of this element has the electron configu- ration 1$s^{2} 2 s^{2} 2 p^{5} 3 s^{1}$ b. The ground-state electron configuration is [Ne \(] 3 s^{2} 3 p^{4}\) . c. An excited state of this element has the electron configu- ration \([\mathrm{Kr}] 5 s^{2} 4 d^{6} 5 p^{2} 6 s^{1}\) d. The ground-state electron configuration contains three unpaired 6\(p\) electrons.

In the ground state of cadmium, Cd, a. how many electrons have \(\ell=2\) as one of their quantum numbers? b. how many electrons have \(n=4\) as one of their quantum numbers? c. how many electrons have \(m_{\ell}=-1\) as one of their quantum numbers? d. how many electrons hav \(m_{s}=-\frac{1}{2}\) as one of their quantum numbers?

Assume that a hydrogen atom's electron has been excited to the \(n=5\) level. How many different wavelengths of light can be emitted as this excited atom loses energy?

Rank the elements Be, B, C, N, and O in order of increasing first ionization energy. Explain your reasoning.

How many unpaired electrons are present in each of the following in the ground state: $\mathrm{O}, \mathrm{O}^{+}, \mathrm{O}^{-}, \mathrm{Os}, \mathrm{Zr}, \mathrm{S}, \mathrm{F}, \mathrm{Ar}$ ?
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