Question

Determine the maximum number of electrons that can have each of the following designations: $2f,2d_{xy},3p,5d_{yz},$ and 4$p.$

Short answer

The maximum number of electrons for each designation is: - 2f: 14 electrons - 2d_{xy}: 2 electrons - 3p: 6 electrons - 5d_{yz}: 2 electrons - 4p: 6 electrons

Step-by-step solution

Determine the quantum numbers for each designation

For each designation, let's identify the principal quantum number (n), angular momentum quantum number (l), magnetic quantum number (m_l), and spin quantum number (m_s). 1. 2f: n = 2, l = 3 (since f orbitals correspond to l = 3) 2. 2d_{xy}: n = 2, l = 2 (since d orbitals correspond to l = 2), m_l = -2 (xy refers to the 2nd d orbital) 3. 3p: n = 3, l = 1 (since p orbitals correspond to l = 1) 4. 5d_{yz}: n = 5, l = 2 (since d orbitals correspond to l = 2), m_l = -1 (yz refers to the 3rd d orbital) 5. 4p: n = 4, l = 1 (since p orbitals correspond to l = 1)

Apply Pauli Exclusion Principle

According to the Pauli Exclusion Principle, no two electrons in an atom can have the same set of quantum numbers. Since the spin quantum number (m_s) can have only two values (+1/2 and -1/2), at most, two electrons can occupy the same orbital with the same values of n, l, and m_l. To find the maximum number of electrons for each designation, we need to calculate the number of possible m_l values.

Calculate the number of possible magnetic quantum numbers

For each designation, let's calculate the number of possible m_l values. Note that for each l value, the range of m_l values is -l, -(l-1), ..., 0, ...., (l-1), l. 1. 2f: For l = 3, m_l values range from -3 to 3. There are a total of 7 possible m_l values. 2. 2d_{xy}: For l = 2 and m_l = -2, there's only one possible m_l value. 3. 3p: For l = 1, m_l values range from -1 to 1. There are a total of 3 possible m_l values. 4. 5d_{yz}: For l = 2 and m_l = -1, there's only one possible m_l value. 5. 4p: For l = 1, m_l values range from -1 to 1. There are a total of 3 possible m_l values.

Apply Hund's Rule and calculate the maximum number of electrons

According to Hund's Rule, electrons will fill the orbitals in such a way that they maintain the maximum possible multiplicity (maximum number of unpaired electrons). So, the maximum number of electrons for each designation can be determined by multiplying the number of m_l values by 2 (for the two possible spin quantum numbers, +1/2 and -1/2). 1. 2f: Maximum electrons = 7 m_l values * 2 spins = 14 2. 2d_{xy}: Maximum electrons = 1 m_l value * 2 spins = 2 3. 3p: Maximum electrons = 3 m_l values * 2 spins = 6 4. 5d_{yz}: Maximum electrons = 1 m_l value * 2 spins = 2 5. 4p: Maximum electrons = 3 m_l values * 2 spins = 6 In conclusion, the maximum number of electrons for each designation is as follows: - 2f: 14 electrons - 2d_{xy}: 2 electrons - 3p: 6 electrons - 5d_{yz}: 2 electrons - 4p: 6 electrons

Key concepts

Principal Quantum Number

The principal quantum number, denoted as $n$, plays a crucial role in defining the energy level of an electron in an atom. It is an integer starting from 1, which determines the size and energy of the atomic orbitals. The larger the value of $n$, the higher the energy and the further the electron is from the nucleus. For example, in the exercise above, the designations like 2f, 2d, and 3p all have specific principal quantum numbers:

• For 2f and 2d, $n=2$.
• For 3p, $n=3$.
• For 5d, $n=5$.
• For 4p, $n=4$.

Understanding the principal quantum number is essential for predicting where an electron might be found and its relative energy.

Angular Momentum Quantum Number

The angular momentum quantum number, denoted as $l$, is associated with the shape of the electron's orbital. It is an integer that ranges from 0 to $n-1$ for a given principal quantum number $n$. Each $l$ value corresponds to a specific type of orbital:

• $l=0$ corresponds to s orbitals,
• $l=1$ corresponds to p orbitals,
• $l=2$ corresponds to d orbitals,
• $l=3$ corresponds to f orbitals.

For instance, in the case of 2f, $l$ must be 3, referring to an f orbital, while for 3p, $l=1$. This quantum number helps us understand how the electron cloud spreads around the nucleus.

Magnetic Quantum Number

The magnetic quantum number, $m_l$, indicates the orientation of an orbital within a specific subshell defined by $l$. It can have values ranging from $-l$ to $+l$, including zero. This means for each different $l$, there are $2l+1$ possible $m_l$ values:

• For $l=0$ (s orbital): single $m_l$ value (0).
• For $l=1$ (p orbitals): three $m_l$ values (-1, 0, +1).
• For $l=2$ (d orbitals): five $m_l$ values (-2, -1, 0, +1, +2).
• For $l=3$ (f orbitals): seven $m_l$ values (-3 to +3).

In the given exercise, designations like 2f or 5d can have multiple $m_l$ values, correlating to multiple orientations within their respective subshells.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics stating that no two electrons in an atom can have identical sets of quantum numbers ($n,l,m_l,m_s$). This essentially means that each electron in an atom must be unique in some way, usually through its spin. Electrons can have a spin quantum number $m_s$ of +1/2 or -1/2. Because of this principle:

• Each orbital can hold a maximum of two electrons, each with opposite spins.
• This sets limits on how electrons are arranged in an atom.
• In orbitals such as those described in the exercise (e.g., 3p or 4p), the total electron capacity is based upon the number of possible $m_l$ values multiplied by the two possible spins.

The Pauli Exclusion Principle is essential in determining the electron configurations of elements.

Hund's Rule

Hund's Rule is an important principle that deals with the filling of electrons into orbitals at the same energy level. It states that electrons will fill degenerate orbitals (orbitals with the same energy) one by one, with parallel spins, before any pairing occurs. This helps to minimize electron repulsion and consequently provides the atom with a more stable electron configuration. Consider how this applies to the example of the 3p orbitals:

• The three p orbitals (with $m_l=-1,0,+1$) will each get one electron before any orbital gets a second one.
• This can lead to a maximum unpaired electron configuration (like what is seen with nitrogen with three unpaired electrons in the p orbitals).
• The implementation of Hund's Rule ensures that the electron configuration is as stable as possible.

By applying Hund's Rule, atoms achieve the lowest possible energy configuration.