Problem 84 - Chemistry: An Atoms-Focused Approach

Question

How many electrons could occupy orbitals with the following quantum numbers? a. \(n=3, \ell=2\) b. \(n=5, \ell=4\) c. \(n=3, \ell=0\) d. \(n=4, e=1, m_{\ell}=1\)

Step-by-Step Solution

Answer

a. \(n=3, \ell=2\) Answer: 10 electrons b. \(n=5, \ell=4\) Answer: 18 electrons c. \(n=3, \ell=0\) Answer: 2 electrons d. \(n=4, \ell=1, m_{\ell}=1\) Answer: 2 electrons

Step 1: List the possible values of \(m_{\ell}\) for \(\ell=2\)

Since \(\ell=2\), the possible magnetic quantum numbers are \(m_{\ell}=-2, -1, 0, 1, 2\).

Step 2: Count the possible combinations of \(m_{\ell}\) and \(m_s\)

For each value of \(m_{\ell}\), there can be two electrons with different spins: - \(m_{\ell}=-2\): 2 electrons (1 spin up, 1 spin down) - \(m_{\ell}=-1\): 2 electrons (1 spin up, 1 spin down) - \(m_{\ell}=0\): 2 electrons (1 spin up, 1 spin down) - \(m_{\ell}=1\): 2 electrons (1 spin up, 1 spin down) - \(m_{\ell}=2\): 2 electrons (1 spin up, 1 spin down) Thus, there can be a total of 5(2) = 10 electrons. #b. \(n=5, \ell=4\)#

Step 1: List the possible values of \(m_{\ell}\) for \(\ell=4\)

Since \(\ell=4\), the possible magnetic quantum numbers are \(m_{\ell}=-4, -3, -2, -1, 0, 1, 2, 3, 4\).

Step 2: Count the possible combinations of \(m_{\ell}\) and \(m_s\)

For each value of \(m_{\ell}\), there can be two electrons with different spins: There are 9 values of \(m_{\ell}\), so there are 9(2) = 18 electrons possible. #c. \(n=3, \ell=0\)#

Step 1: List the possible values of \(m_{\ell}\) for \(\ell=0\)

Since \(\ell=0\), the only possible magnetic quantum number is \(m_{\ell}=0\).

Step 2: Count the possible combinations of \(m_{\ell}\) and \(m_s\)

For the single value of \(m_{\ell}\), there can be two electrons with different spins: There is just 1 value of \(m_{\ell}\), so there are 1(2) = 2 electrons possible. #d. \(n=4, \ell=1, m_{\ell}=1\)#

Step 1: Determine the possible combinations of \(m_s\) for the given \(m_{\ell}\)

Since \(m_{\ell}\) is specified, there can be only two electrons with different spins: There is only 1 value of \(m_{\ell}\) and 1 given value for \(m_s\), so there are 1(2) = 2 electrons possible.

Key Concepts

Quantum Mechanics

Quantum mechanics is a fascinating branch of physics that explores the behavior of matter and energy at the smallest scales, such as atoms and subatomic particles. This field challenges our classical understanding of the universe by introducing probabilities and the concept that particles can exist in multiple states at once. This is famously illustrated by the wave-particle duality, where particles like electrons exhibit both wave-like and particle-like properties.

In quantum mechanics, properties of particles like position, momentum, and energy are described by something called wave functions. The solutions to wave functions give rise to quantum numbers, which are essential for understanding electron configurations and the arrangement of electrons in atoms. This field is crucial for the development of modern technologies, including semiconductors and quantum computing, and plays a vital role in understanding the nature of the universe at the most fundamental level.

Electron Configuration

Electron configuration describes the arrangement of electrons around the nucleus of an atom. The electrons occupy regions called orbitals, which are defined by specific energy levels and shapes. These configurations are dictated by a set of rules derived from the principles of quantum mechanics.

Electrons fill orbitals starting with the lowest energy level, according to the Aufbau principle. The Pauli exclusion principle states that no two electrons in an atom can have the same set of all four quantum numbers, ensuring a unique position for each electron. Hund's rule suggests that electrons will occupy separate orbitals in the same subshell with parallel spins before pairing in orbitals.

Electron configuration plays a key role in determining the chemical properties of elements, such as their reactivity and the types of chemical bonds they can form. To fully understand electron configuration, it's crucial to grasp the concepts of subshells and quantum numbers, which describe the energy level and shape of each orbital.

Magnetic Quantum Number

The magnetic quantum number, denoted by \(m_\ell\), is one of the four quantum numbers used in quantum mechanics to specify the position and energy of an electron within an atom. It indicates the orientation of an electron's orbital in three-dimensional space relative to an external magnetic field and is derived from the azimuthal quantum number \(\ell\).

The values of \(m_\ell\) range from \(-\ell\) to \(+\ell\), including zero. For instance, if \(\ell = 2\), \(m_\ell\) can be \(-2, -1, 0, 1,\) or \(2\). Therefore, the magnetic quantum number determines how many orbitals exist within a subshell. Each orientation corresponds to one of these possible values and can hold two electrons with opposite spins. Understanding \(m_\ell\) is crucial in predicting how electrons are distributed within an atom and how they interact with external magnetic fields.

Spin Quantum Number

The spin quantum number, represented as \(m_s\), is a unique quantum number that describes the intrinsic angular momentum, or "spin," of an electron within an atom. Unlike other quantum numbers, the spin quantum number can only take on two possible values: \(+\frac{1}{2}\) or \(-\frac{1}{2}\). These values indicate the two fundamental spin states of an electron: "spin up" and "spin down."

Electron spin is a fundamental property, much like an electron's charge or mass, and it plays a crucial role in the arrangement of electrons in atoms. This property is what allows each orbital, identified by the other quantum numbers, to hold a pair of electrons. Because of the Pauli exclusion principle, no two electrons in an atom can share the same four quantum numbers, which means that electrons within the same orbital must have opposite spins.

Understanding the spin quantum number is essential for explaining the behavior of electrons in magnetic fields and for understanding phenomena such as electron pairing and the magnetic properties of materials.