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Chapter 2: Problem 9

Explain electron from a quantum mechanical perspective, including a discussion of atomic radii, probabilities, and orbitals.

Short Answer

Expert verified
In quantum mechanics, electrons in atoms reside in areas called orbitals, which are mathematical functions describing their wave-like behavior and probability distribution around the nucleus. The behavior of electrons is governed by four quantum numbers (principal, angular momentum, magnetic, and spin), which determine their energy level, orbital shape, and orientation. Atomic radii, or the distance from the nucleus to the outermost electron shell, increase with the principal quantum number, as electrons occupy higher energy orbitals further from the nucleus. Probability distributions depend on the shape of the orbital, which is determined by the angular momentum quantum number. Overall, atomic radii, probabilities, and orbitals are interconnected aspects of a quantum mechanical perspective on electrons, allowing us to understand atomic interactions and molecular formations.

Step by step solution

01

Defining atomic radii, probabilities, and orbitals

To understand electrons from a quantum mechanical perspective, we need to have a clear understanding of atomic radii, probabilities, and orbitals: 1. Atomic radii: The atomic radius is the distance from the nucleus of an atom to the outermost shell of electrons. It is usually measured in picometers (pm) or angstroms (Å). 2. Probabilities: In quantum mechanics, probabilities are used to describe the likelihood of finding an electron in a certain region around the nucleus. This is generally represented by a probability cloud or distribution. 3. Orbitals: An orbital is a mathematical function that describes the wave-like behavior of an electron within an atom. It is used to determine the probability of finding an electron in a specific region of space around the nucleus. Now that we have defined these concepts, let's explore how they apply to electrons in atoms.
02

Electron behavior in atoms

In quantum mechanics, electrons are treated as wave-particles that reside in areas around the nucleus called orbitals. Unlike the classical description of electrons as particles moving in defined circular paths (like planets around the sun), quantum mechanics describes electrons as wave-like entities with a range of possible locations within an atom.
03

Governed by quantum numbers

The behavior of electrons in atoms is governed by four quantum numbers that describe their energy, angular momentum, magnetic moment, and spin: 1. Principal quantum number (n): This number determines the energy level of an electron and its distance from the nucleus. As n increases, the electron occupies an orbital with more energy and is found further from the nucleus. 2. Angular momentum quantum number (l): This number defines the shape of the electron's orbital. It ranges from 0 to (n-1) and can have values such as s, p, d, and f for l = 0, 1, 2, and 3, respectively. 3. Magnetic quantum number (m_l): This number determines the orientation of the orbital in space. Values can be integers ranging from -l to +l. 4. Spin quantum number (m_s): Electrons have an intrinsic angular momentum called spin, and m_s can either be +1/2 (spin up) or -1/2 (spin down).
04

Atomic radii and electron probabilities

The atomic radius of an atom is influenced by the quantum numbers, particularly the principal quantum number (n). As n increases, the electron can occupy orbitals with higher energy and be found further from the nucleus, which leads to an increase in atomic radius. The probability distribution of an electron can be determined by the shape of the orbital, which is governed by the angular momentum quantum number (l). For example, an electron in an s orbital (l=0) is most likely to be found in a spherical region around the nucleus. The probability distribution of an electron can be visualized using a 3D plot or contour plot representing the areas with the highest likelihood of finding an electron.
05

Conclusion

In summary, electrons in atoms can be described from a quantum mechanical perspective by their behavior within atomic orbitals, which are determined by the quantum numbers n, l, m_l, and m_s. The atomic radii, probabilities, and orbitals are all interconnected aspects of this quantum mechanical description, with atomic radii and probabilities being influenced by the electron's presence in different orbitals. Understanding these aspects helps to build a deeper knowledge of how electrons interact within atoms and, ultimately, how atoms combine to form molecules and materials.

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Most popular questions from this chapter

The first ionization energies of As and Se are 0.947 and \(0.941 \mathrm{MJ} / \mathrm{mol},\) respectively. Rationalize these values in terms of electron configurations.

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?

Order the atoms in each of the following sets from the least negative electron affinity to the most. a. \(\mathrm{S}, \mathrm{Se}\) b. \(\mathrm{F}, \mathrm{Cl}, \mathrm{Br}, \mathrm{I}\)

Neutron diffraction is used in determining the structures of molecules. a. Calculate the de Broglie wavelength of a neutron moving at \(1.00 \%\) of the speed of light. b. Calculate the velocity of a neutron with a wavelength of \(75 \mathrm{pm}\left(1 \mathrm{pm}=10^{-12} \mathrm{m}\right)\)

An electron is excited from the \(n=1\) ground state to the \(n=\) 3 state in a hydrogen atom. Which of the following statements is/are true? Correct the false statements to make them true. a. It takes more energy to ionize (completely remove) the electron from \(n=3\) than from the ground state. b. The electron is farther from the nucleus on average in the \(n=3\) state than in the \(n=1\) state. c. The wavelength of light emitted if the electron drops from \(n=3\) to \(n=2\) will be shorter than the wavelength of light emitted if the electron falls from \(n=3\) to \(n=1\) d. The wavelcngth of light cmittcd when the clectron returns to the ground state from \(n=3\) will be the same as the wavelength of light absorbed to go from \(n=1\) to \(n=3\) e. For \(n=3,\) the electron is in the first excited state.
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