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Chapter 7: Problem 82

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
The sizes of electron orbitals are defined using arbitrary probability values, such as 90%, because the exact position of an electron in an atom cannot be determined precisely due to the probabilistic nature of quantum mechanics. Instead, a specific probability threshold is chosen to "contain" a certain percentage of the total electron density within a region, allowing for visualization and comparison of electron orbitals while still acknowledging the inherent uncertainties associated with quantum mechanics. Using a 90% probability value provides a reasonably high level of certainty while taking into account these uncertainties.

Step by step solution

01

Understanding Electron Orbitals

In an atom, electrons are found in regions called orbitals. These orbitals describe the spatial distribution of electrons around the nucleus. Each orbital can accommodate up to two electrons and is characterized by a unique energy level, shape, and orientation.
02

Quantum Mechanics and Orbitals

According to quantum mechanics, the exact position of an electron in an atom cannot be determined precisely. Instead, we can only predict the probability distribution of the electron in a given region around the nucleus. This probability distribution is given by the square of the wavefunction, known as the electron density.
03

Defining Orbital Size

Since we cannot determine an exact boundary for electron orbitals due to the probabilistic nature of quantum mechanics, orbital size is generally defined using an arbitrary probability value. This means that we must set a specific probability threshold to "contain" a certain percentage of the total electron density within a region.
04

Arbitrary Probability Values

Choosing an arbitrary value, such as 90% probability, means that within the defined region, there is a 90% chance of finding an electron. A 90% probability is commonly used because it provides a reasonably high level of certainty while also taking into account the inherent uncertainties associated with quantum mechanics.
05

Significance of Arbitrary Values

The use of arbitrary probability values in defining orbital sizes is essential for simplifying complex quantum mechanical concepts and making them more understandable. By setting a threshold, we can better visualize and compare electron orbitals while still acknowledging the inherently probabilistic nature of electron distributions in atoms. However, it is important to keep in mind that these values are arbitrary and could be changed if needed for specific scenarios.

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Most popular questions from this chapter

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?

One of the emission spectral lines for \(\mathrm{Be}^{3+}\) has a wavelength of 253.4 \(\mathrm{nm}\) for an electronic transition that begins in the state with \(n=5 .\) What is the principal quantum number of the lower-energy state corresponding to this emission? (Hint: The Bohr model can be applied to one- electron ions. Don't forget the \(Z\) factor: \(Z=\) nuclear charge \(=\) atomic number.)

In the ground state of mercury, Hg, a. how many electrons occupy atomic orbitals with \(n=3 ?\) b. how many electrons occupy \(d\) atomic orbitals? c. how many electrons occupy \(p_{z}\) atomic orbitals? d. how many electrons have spin "up" " \(\left(m_{s}=+\frac{1}{2}\right) ?\)

Does a photon of visible light $(\lambda \approx 400 \text { to } 700 \mathrm{nm})$ have sufficient energy to excite an electron in a hydrogen atom from the \(n=1\) to the \(n=5\) energy state? from the \(n=2\) to the \(n=6\) energy state?

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