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Chapter 2: Problem 76

In defining the sizes of orbitals, why must we use an arbitrary value, such as \(90 \%\) of the probability of finding an electron in that region?

Short Answer

Expert verified
In short, an arbitrary value like 90% probability is used to define the sizes of orbitals because atomic orbitals are derived from continuous probability density plots without clear boundaries. This arbitrary value provides a reasonable approximation for the orbital size and shape, which is crucial for understanding atomic structure and properties.

Step by step solution

01

Understanding Atomic Orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons with specific energy levels and angular momenta are most likely to be found. These orbitals arise from the solutions of the Schrödinger equation, which describes the wave-like behavior of electrons in atoms. Each orbital can be represented by a probability density plot, which is essentially a 3D map showing the likelihood of finding the electron at different points around the nucleus.
02

Probability Density Plots

A probability density plot represents the probability of finding an electron in a specific region of space. The electron density is highest at the center of the orbital and decreases as we move away from it. The plots are continuous, with no sharp boundaries between regions where the electron is more likely to be found and regions where it is less likely to be found. This leads to the problem of defining the size of an orbital, as it does not have a fixed shape or clear-cut borders.
03

The Need for an Arbitrary Value

Due to the continuous nature of probability density plots, there is no exact point at which we can say that the orbital ends and another region of space begins. To overcome this issue, scientists use an arbitrary value to define the size of an orbital. A common arbitrary value used is 90%, which means that the size of the orbital is defined as the region where there is a 90% probability of finding the electron. This value is not fixed and can be changed to any other reasonable percentage, but 90% is widely accepted since it provides a good balance between including the most probable regions and limiting the overall size of the orbital.
04

Conclusion

In summary, defining the sizes of orbitals requires the use of an arbitrary value, such as 90% of the probability of finding an electron in that region, because atomic orbitals are derived from probability density plots that have no clear boundaries. The choice of an arbitrary value allows scientists to provide a reasonable approximation for the size and shape of an orbital, which is essential for understanding atomic structure and properties.

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Most popular questions from this chapter

Octyl methoxycinnamate and oxybenzone are common ingredients in sunscreen applications. These compounds work by absorbing ultraviolet (UV) B light (wavelength \(280-320 \mathrm{nm}\) ), the UV light most associated with sunburn symptoms. What frequency range of light do these compounds absorb?

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 orbital designations are incorrect: \(1, s$$1 p, 7 d, 9 s, 3 f, 4 f, 2 d ?\)

Give the maximum number of electrons in an atom that can have these quantum numbers: a. \(n=4\) b. \(n=5, m_{\ell}=+1\) c. \(n=5, m_{s}=+\frac{1}{2}\) d. \(n=3, \ell=2\) e. \(n=2, \ell=1\)

A certain oxygen atom has the electron configuration \(1 s^{2} 2 s^{2} 2 p_{x}^{2} 2 p_{y}^{2} .\) How many unpaired electrons are present? Is this an excited state of oxygen? In going from this state to the ground state, would energy be released or absorbed?
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