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Chapter 6: Problem 93

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: (a) \(n=3, l=3, m_{l}=2, m_{\mathrm{s}}=+\frac{1}{2}\) (b) \(n=4, l=3, m_{l}=-3, m_{s}=+\frac{1}{2}\) (c) \(n=3, l=1, m_{l}=2, m_{s}=+\frac{1}{2}\) (d) \(n=5, l=0, m_{l}=0, m_{s}=0\) (e) \(n=2, l=1, m_{l}=1, m_{\mathrm{s}}=-\frac{1}{2}\)

Short Answer

Expert verified
Sets (a), (c), and (d) have invalid quantum numbers. Set (b) and (e) are valid.

Step by step solution

01

Understand Quantum Numbers

Quantum numbers describe the state of a quantum system. They include: the principal quantum number \(n\), which must be a positive integer; the azimuthal quantum number \(l\), which can range from 0 to \(n-1\); the magnetic quantum number \(m_l\), which ranges from \(-l\) to \(l\); and the spin quantum number \(m_s\), which can be \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
02

Analyze Set (a)

For set (a): \(n=3, l=3, m_l=2, m_s=+\frac{1}{2}\). Since \(l\) must be in the range 0 to \(n-1\), for \(n=3\), valid \(l\) values are 0, 1, or 2. Thus, \(l=3\) is invalid.
03

Analyze Set (b)

For set (b): \(n=4, l=3, m_l=-3, m_s=+\frac{1}{2}\). All quantum numbers are valid: \(l=3\) is within 0 to \(n-1\) (0 to 3), \(m_l=-3\) is between \(-l\) and \(l\) (-3 to 3), and \(m_s=+\frac{1}{2}\) is acceptable.
04

Analyze Set (c)

For set (c): \(n=3, l=1, m_l=2, m_s=+\frac{1}{2}\). Here, \(m_l\) can only be in the range \(-l\) to \(l\) (-1, 0, 1) for \(l=1\). Thus, \(m_l=2\) is not valid.
05

Analyze Set (d)

For set (d): \(n=5, l=0, m_l=0, m_s=0\). Since \(m_s\) must be \(+\frac{1}{2}\) or \(-\frac{1}{2}\), \(m_s=0\) is invalid. All other numbers are valid.
06

Analyze Set (e)

For set (e): \(n=2, l=1, m_l=1, m_s=-\frac{1}{2}\). All quantum numbers are valid. \(l\) is 0 or 1 for \(n=2\), \(m_l=1\) is between \(-l\) and \(l\), and \(m_s=-\frac{1}{2}\) is valid.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hydrogen Atom
The hydrogen atom plays a pivotal role in understanding quantum numbers because it is one of the simplest atoms, consisting of only one proton and one electron. This simplicity makes it an excellent system for studying quantum mechanics. In a hydrogen atom, these quantum numbers are used to describe the behavior and properties of the electron. When we discuss the hydrogen atom's electron, we often refer to a model in which the electron orbits the nucleus in various quantum states. These states, characterized by different energy levels, are introduced through quantum numbers, which help predict the location and energy of the electron.
Principal Quantum Number
The principal quantum number, denoted as \(n\), is a fundamental part of the quantum description of an atom. It represents the main energy level or shell in which an electron resides. Similar to how floors are numbered in a building, the larger the \(n\), the higher the energy level and the further the electron is from the nucleus.
  • \(n\) can only take positive integer values: 1, 2, 3, and so on.
  • The value of \(n\) determines the size and energy of the orbitals.
  • Higher \(n\) values correspond to orbitals with higher energy and greater radii.
By understanding \(n\), you gain insight into the energy and spatial properties of the electron within an atom.
Azimuthal Quantum Number
The azimuthal quantum number, denoted as \(l\), defines the shape of the electron's orbital. Think of \(l\) as describing the 'sublevel' within a principal energy level. Not only does it help specify the shape, but it also helps determine angular momentum.
  • \(l\) can range from 0 to \(n-1\). For each value of \(n\), there are \(n\) possible \(l\) values.
  • If \(n = 3\), \(l\) could be 0, 1, or 2, corresponding to s, p, and d orbitals, respectively.
  • Each \(l\) value is associated with a specific shape: spherically symmetric (s), dumbbell-shaped (p), and more complex shapes for \(d\) and \(f\).
Understanding \(l\) provides insight into the orbital arrangements of electrons and their associated molecular interactions.
Magnetic Quantum Number
The magnetic quantum number, represented as \(m_l\), offers more details about an electron's orientation within an atom. Specifically, it describes the orientation of the orbital in space relative to an external magnetic field.
  • \(m_l\) can take values ranging from \(-l\) to \(+l\), including 0.
  • For example, if \(l=1\), \(m_l\) can be \(-1\), 0, or 1, reflecting the three p-orbitals in the same energy level.
  • This quantum number is crucial in understanding phenomena like the Zeeman effect, where energy levels split under a magnetic field.
By studying \(m_l\), we can predict how the electron's orbital will respond to external magnetic influences, enhancing our understanding of electron configurations.
Spin Quantum Number
The spin quantum number, denoted as \(m_s\), unlike the other quantum numbers, describes an intrinsic property of electrons. Electrons can be thought of as spinning on an axis, similar to a tiny top. This spin creates a magnetic field, allowing them to act like tiny magnets.
  • \(m_s\) can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\).
  • This duality reflects two possible orientations of the electron's spin: up or down.
  • Understanding \(m_s\) is vital for concepts such as the Pauli Exclusion Principle, which states no two electrons in an atom can have the same set of all four quantum numbers.
Knowing the spin quantum number is essential for determining electron pairing and magnetism in materials.
Quantum Mechanics
Quantum mechanics is the fundamental theory in physics that describes the behavior of matter and energy at the atomic and subatomic levels. This field of science revolutionized our understanding of how atoms and particles interact.
  • At its core, quantum mechanics explains phenomena that cannot be explained by classical mechanics, such as the dual nature (wave-particle duality) of electrons and photons.
  • Quantum mechanics relies heavily on mathematics to predict probabilistic outcomes rather than definite states.
  • Key principles include superposition, entanglement, and uncertainty, all of which paint a complex yet fascinating picture of the microscopic world.
Through quantum mechanics, we learn about the probability distributions of electrons and their interactions, which are crucial for developing technologies like semiconductors and lasers.

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Most popular questions from this chapter

One type of sunburn occurs on exposure to UV light of wavelength in the vicinity of \(325 \mathrm{nm}\). (a) What is the energy of a photon of this wavelength? (b) What is the energy of a mole of these photons? (c) How many photons are in a \(1.00 \mathrm{~mJ}\) burst of this radiation? (d) These UV photons can break chemical bonds in your skin to cause sunburn-a form of radiation damage. If the \(325-\mathrm{nm}\) radiation provides exactly the energy to break an average chemical bond in the skin, estimate the average energy of these bonds in \(\mathrm{kJ} / \mathrm{mol}\).

Einstein's 1905 paper on the photoelectric effect was the first important application of Planck's quantum hypothesis. Describe Planck's original hypothesis, and explain how Einstein made use of it in his theory of the photoelectric effect.

The energy from radiation can be used to rupture chemical bonds. A minimum energy of \(192 \mathrm{~kJ} / \mathrm{mol}\) is required to break the bromine- bromine bond in \(\mathrm{Br}_{2}\). What is the longest wavelength of radiation that possesses the necessary energy to break the bond? What type of electromagnetic radiation is this?

Titanium metal requires light with a maximum wavelength of \(286 \mathrm{nm}\) to emit electrons. (a) What is the minimum energy of the photons necessary to emit electrons from titanium via the photoelectric effect? (b) What is the frequency of this radiation? (c) Is it possible to eject electrons from titanium metal using infrared light? (d) If titanium is irradiated with light of wavelength \(276 \mathrm{nm}\), what is the maximum possible kinetic energy of the emitted electrons?

What is the maximum number of electrons that can occupy each of the following subshells? (a) \(3 s,(\mathbf{b}) 2 p\), (c) \(4 d\) (d) \(5 \mathrm{~s}\).
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