Problem 93 - Chemistry The Central Science In Si Units

Question

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) \(n=2, l=1, m_{l}=1, m_{\mathrm{s}}=-\frac{1}{2}\)

Step-by-Step Solution

Answer

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

Step 1: 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 \(-\frac{1}{2}\).

Step 2: 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.

Step 3: 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.

Step 4: 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.

Step 5: 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.

Step 6: 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.

Key Concepts

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.