Question

Why can only two electrons occupy a particular orbital? What is this idea called?

Short answer

Only two electrons can occupy a particular orbital due to the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers. This limitation arises because electrons can have only two possible values for their spin quantum number, +\(\frac{1}{2}\) and -\(\frac{1}{2}\), representing opposite spin orientations.

Step-by-step solution

Understanding electron arrangement

In an atom, electrons occupy specific energy levels and orbitals. At any given energy level, electrons are found in various sub-shells and orbitals. Each orbital can accommodate a maximum of two electrons.

The Pauli Exclusion Principle

The idea that only two electrons can occupy a particular orbital is based on the Pauli Exclusion Principle. This principle was formulated by the Austrian physicist Wolfgang Pauli and states that no two electrons in an atom can have the same set of quantum numbers. In an atom, each electron is described by four quantum numbers: 1. Principal quantum number (n): Represents the electron's energy level. 2. Angular momentum quantum number (l): Represents the electron's sub-shell. 3. Magnetic quantum number (m_l): Represents the electron's orbital. 4. Spin quantum number (m_s): Represents the electron's spin. For two electrons to exist in the same orbital, they must have identical values for the first three quantum numbers (n, l, and m_l). However, the spin quantum number can have two possible values, +1/2 or -1/2, which correspond to the two opposite spin orientations. So, if one electron has a spin of +1/2, the other electron in the same orbital must have a spin of -1/2, and vice versa. Since only two values of m_s are possible, a maximum of two electrons can occupy a particular orbital, and they must have opposite spins.

Conclusion

Only two electrons can occupy a particular orbital due to the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers. This limitation arises because electrons can have only two possible values for their spin quantum number, +1/2 and -1/2, representing opposite spin orientations.

Key concepts

Electron Arrangement in Atoms

Understanding the electron arrangement within an atom is essential for grasping the foundational principles of chemistry and physics. Electrons, the negatively charged particles that orbit an atom's nucleus, are arranged in specific energy levels or shells. Each shell can be further divided into subshells, which are denoted by s, p, d, and f, depending on their shape. These subshells contain orbitals, the regions of space where there is a high probability of finding an electron.

For instance, the first energy level has one s orbital, the second has one s and three p orbitals, and so forth. The arrangement of electrons in these orbitals follows a set of rules including the Aufbau principle, which describes the progressive filling of orbitals from lower to higher energy levels; Hund's rule, emphasizing that electrons will fill empty orbitals before pairing up; and the Pauli Exclusion Principle, informing the specific limitation of two electrons per orbital with opposite spins. This structured arrangement ensures that atoms have the most stable, energy-efficient configuration possible.

Quantum Numbers

To describe the position and energy of an electron within an atom, we rely on four quantum numbers, precise values that result from the solution of quantum mechanical equations. The principal quantum number ()) indicates the electron's energy level and thus its relative distance from the nucleus. The higher the value of n, the higher the energy level and the larger the electron's orbit.

The angular momentum quantum number ()) represents the shape of the subshell where the electron is likely to be found and is determinative of the number of orbital shapes available at a given energy level. The magnetic quantum number ()) specifies the orientation in space of the orbital within a subshell. And lastly, the spin quantum number ()) designates the electron's spin direction, a fundamental property with only two allowed values: +1/2 and -1/2, denoting 'up' or 'down' spin, respectively.

These quantum numbers are vital because they ensure that each electron in an atom has a unique address, analogous to a person's home address, which prevents any two electrons from being in the same quantum state simultaneously.

Orbital Occupancy

Orbital occupancy refers to how electrons occupy the orbitals within an atom. Orbitals, based on their shapes and energy levels, fill according to the previously mentioned rules. The most crucial point is that each orbital can hold only two electrons. These two must have opposite spins, a phenomenon directly tied to the Pauli Exclusion Principle. This principle is a quantum mechanical rule that puts a constraint on the occupancy of electrons.

With the aid of a tool known as an electron configuration chart, chemists and physicists visualize how electrons populate the atom's orbitals. This tool showcases the orderly pattern in which orbitals fill and helps predict the chemical behavior of the atom, since the configuration of the outermost electrons, often termed valence electrons, determines an atom's ability to bond and interact with other atoms.

Thus, understanding orbital occupancy is not only foundational for atomic theory but also has significant implications for how elements combine and react to form molecules, the core of chemical transformation processes.