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Chapter 7: 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 the probability density of finding the electron at that point in space. A higher value of \(\psi^2\) indicates that the electron is more likely to be found in that region, while a lower value signifies a lower probability of finding the electron there. This information is crucial for understanding the behavior and properties of electrons in atoms and predicting the outcomes of chemical reactions and interactions between atoms.

Step by step solution

01

Understanding the wavefunction \(\psi\)

In quantum mechanics, the wavefunction \(\psi\) is a mathematical function that describes the state of a quantum system, such as an electron in an atomic orbital. The wavefunction contains all the information about the system and can be used to calculate various properties of the system, such as energy and angular momentum.
02

Interpreting the square of the wavefunction \(\psi^2\)

The square of the wavefunction, \(\psi^2\), has physical significance in terms of probability. For an electron in an atomic orbital, the value of \(\psi^2\) at a particular point represents the probability density of finding the electron at that point in space. In other words, if we were to take a small volume around that point, the probability of finding the electron within that volume would be proportional to \(\psi^2\) multiplied by the volume.
03

Understanding probability density in atomic orbitals

In atomic orbitals, the electron's probability density is distributed throughout the orbital. Some regions have a higher probability density than others, meaning that the electron is more likely to be found in these regions. The probability density is highest near the nucleus and decreases as we move away from it, generally following a radial distribution function that depends on the specific orbital shape and quantum numbers.
04

Physical significance of \(\psi^2\) in atomic orbitals

To summarize, the physical significance of the value of \(\psi^2\) at a particular point in an atomic orbital is the probability density of finding the electron at that point in space. A higher value of \(\psi^2\) at a particular position indicates that the electron is more likely to be found in that area, while a lower value signifies a lower probability of finding the electron in that region. This information is essential for understanding the behavior and properties of electrons in atoms and for predicting the outcomes of various chemical reactions and interactions between atoms.

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Most popular questions from this chapter

In the ground state of element 115, Uup, a. how many electrons have \(n=5\) as one of their quantum numbers? b. how many electrons have \(\ell=3\) as one of their quantum numbers? c. how many electrons have \(m_{\ell}=1\) as one of their quantum numbers? d. how many electrons have \(m_{s}=-\frac{1}{2}\) as one of their quantum numbers?

Which of the following statements is(are) true? a. The \(2 s\) orbital in the hydrogen atom is larger than the \(3 s\) orbital also in the hydrogen atom. b. The Bohr model of the hydrogen atom has been found to be incorrect. c. The hydrogen atom has quantized energy levels. d. An orbital is the same as a Bohr orbit. e. The third energy level has three sublevels, the \(s, p\), and \(d\) sublevels.

Calculate the wavelength of light emitted when each of the following transitions occur in the hydrogen atom. What type of electromagnetic radiation is emitted in each transition? a. \(n=3 \rightarrow n=2\) b. \(n=4 \rightarrow n=2\) c. \(n=2 \rightarrow n=1\)

It takes \(476 \mathrm{~kJ}\) to remove 1 mole of electrons from the atoms at the surface of a solid metal. How much energy (in kJ) does it take to remove a single electron from an atom at the surface of this solid metal?

Which of elements \(1-36\) have one unpaired electron in the ground state?
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