Question

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) \(n=3, m_{l}=-2 ;(\mathbf{b}) n=4, l=3 ;(\mathbf{c}) n=5, l=3, m_{l}=2\) (d) \(n=4, l=1, m_{l}=0\)

Short answer

\(a) 2 \ electrons; \ b) 14 \ electrons; \ c) 2 \ electrons; \ d) 2 \ electrons\)

Step-by-step solution

Determine the l value

Knowing the relation between m_l and l (that m_l varies between -l and l), we could find the possible values of l for m_l = -2. Since m_l = -2, l can be 2 (as m_l varies from -2 to +2).

Determine number of electrons

Considering the given principal quantum number n=3 and l=2, there is only one set of quantum numbers that satisfy the condition. There are 2 possible spin orientations (m_s) for this m_l value, so a maximum of 2 electrons can have these quantum numbers. #a) Answer: 2 electrons# #b) n=4, l=3#

Determine m_l values

Knowing the relationship between l and m_l (-l to +l), we could find the possible values of m_l for l = 3. The m_l values are -3, -2, -1, 0, 1, 2, and 3.

Determine number of electrons

Considering the given principal quantum number n=4 and l=3, there are 7 possible m_l values. For each m_l value, there are 2 possible spin orientations (m_s), resulting in a maximum of 14 electrons that can have these quantum numbers. #b) Answer: 14 electrons# #c) n=5, l=3, m_l=2#

Determine number of electrons

In this case, all quantum numbers are given, except for the spin quantum number (m_s). Since there are 2 possible spin orientations, a maximum of 2 electrons can have these quantum numbers. #c) Answer: 2 electrons# #d) n=4, l=1, m_l=0#

Determine number of electrons

All quantum numbers are given, except for the spin quantum number (m_s). Since there are 2 possible spin orientations, a maximum of 2 electrons can have these quantum numbers. #d) Answer: 2 electrons#

Key concepts

Electron Configuration

Electron configuration is a fundamental concept in chemistry and physics. It describes how electrons are arranged in an atom. Knowing the correct electron configuration helps in understanding chemical properties and behaviors. Each electron in an atom has a unique set of quantum numbers that define its state.
These quantum numbers include the principal quantum number ( ), azimuthal quantum number ( ), magnetic quantum number ( ), and the spin quantum number ( ). The electron configuration notation combines these to summarize the distribution of electrons in the atom's orbitals.

• Electrons fill orbitals starting with the lowest energy level.
• The Aufbau principle guides this filling order.
• Electron configuration is written as a series of numbers and letters, indicating the energy level and type of orbital.

Understanding electron configuration is crucial for predicting the reactivity and bonding of atoms.

Principal Quantum Number

The principal quantum number ( ) is a key player in defining the electronic structure of atoms. It indicates the main energy level of an electron within an atom, and is always a positive integer (1, 2, 3...).
Higher values represent energy levels further from the nucleus. Thus, electrons in higher energy levels have greater energy.

• Relates to the size of the electron cloud.
• Affects the energy level of the electron.
• Supports understanding of atomic radius and periodic properties.

In summary, controls the major shell occupied by an electron, having a direct impact on the atom’s energy state.

Magnetic Quantum Number

The magnetic quantum number ( ) adds another layer of detail to our understanding of an electron's position. It determines the orientation of an orbital within a given sub-shell.
Values of range from to , where is the azimuthal quantum number. For example, if equals 2, then can take values from -2 to +2.

• Defines the specific orbital within a sub-shell an electron occupies.
• Helps determine the shape and size of electron clouds.
• Influences how electrons fill available orbitals and their spatial distribution.

Understanding is essential for visualizing an electron's probable distribution path and its effects on magnetic properties.

Spin Quantum Number

The spin quantum number is nature's twist on quantum mechanics, introducing the idea that electrons have an intrinsic spin. This spin quantum number ( ) can either be +1/2 or -1/2, indicating two possible spin states.
This quantum number is central in the Pauli Exclusion Principle, which states no two electrons can share the same set of quantum numbers within the same atom.

• Accounts for the electron's magnetic moment due to its spin.
• Necessary to understand bonding and molecular structures.
• Explains why electrons pair up in atomic orbitals.

Through spin, we complete the detailed description of electron states within an atom, allowing predictions of atom interactions and stability.