Determining the Maximum Number of Electrons in an Atom Given Quantum Numbers n4, l3, and m_l2

When quantum numbers are specified for an atomic orbital, one can determine the maximum number of electrons that can occupy that orbital within an atom. In this article, we will explore the application of quantum numbers n4, l3, and ml2 to find the maximum electron capacity.

Understanding Quantum Numbers

Quantum numbers are crucial for defining the energy levels, subshells, and spatial orientations of atomic orbitals. There are four primary quantum numbers:

1. Principal Quantum Number (n)

The principal quantum number indicates the main energy level or shell, denoted by n. It takes positive integer values, starting from 1.

Example: n4 indicates the fourth energy level or shell.

2. Azimuthal Quantum Number (l)

The azimuthal quantum number, also known as the angular momentum quantum number, determines the shape of the subshell. It can take integer values from 0 to n-1.

Example: For n4, l can be 0, 1, 2, or 3.

3. Magnetic Quantum Number (ml)

The magnetic quantum number dictates the orientation of the orbital in space. It takes integer values from -l to l.

Example: For l3, ml can be -3, -2, -1, 0, 1, 2, or 3.

Calculation of Maximum Electrons in the Given State

The given quantum numbers are:

n4 l3 (f subshell) ml2

To determine the maximum number of electrons an atom can have in a given state, follow these steps:

Step 1: Determine Maximum Electrons per Orbital

According to the Pauli Exclusion Principle, each orbital can accommodate a maximum of 2 electrons, each with opposite spins.

Step 2: Determine the Number of Orbitals in the Given Subshell

The number of orbitals in a given subshell is determined by the range of ml values, which is from -l to l.

For l3, the possible ml values are -3, -2, -1, 0, 1, 2, and 3. This gives us 7 possible orbitals.

Step 3: Calculate the Maximum Number of Electrons

The maximum number of electrons that an atom can have in this state is calculated by multiplying the number of orbitals by 2 (since each orbital can hold 2 electrons). For l3, this is:

Maximum Electrons Number of Orbitals × 2 7 × 2 14

Therefore, for the quantum numbers n4, l3, and ml2, the maximum number of electrons that can be in this state is 14.

Conclusion

The quantum numbers provided (n4, l3, and ml2) define a specific orbital within the fourth shell, capable of accommodating up to 14 electrons, 2 in each of the 7 orbitals.

Additional Insights

1. Understanding Quantum Numbers: Quantum numbers are essential for describing the behavior of electrons in atoms. They help in predicting the energy levels, subshells, and spatial orientations of electrons.

2. Electron Configuration: Electron configuration can be determined using the Aufbau principle, which states that electrons fill orbitals starting from the lowest energy level to the highest.

3. Orbital Capacity: Each orbital holds a maximum of 2 electrons with opposite spins, following the Pauli Exclusion Principle.

References

Dirac, P.A.M. (1931). Quantum mechanics of many electron systems. Proceedings of the Royal Society A, 133(821), 69-77.