Question

According to the Pauli exclusion principle, the electrons within a given orbital must have _______ spins.

Short answer

According to the Pauli exclusion principle, the electrons within a given orbital must have opposite spins.

Step-by-step solution

Recall the Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of quantum numbers (n, ℓ, mℓ, ms), which describe the properties of the electron's state.

Understand the Spin Quantum Number

The spin quantum number (ms) refers to the intrinsic angular momentum of the electron. There are only two possible values for the spin quantum number: +1/2 and -1/2, corresponding to "spin up" and "spin down" states.

Apply the Principle to the Orbital

Because there can only be two distinct values for the spin quantum number (ms), it is necessary for electrons within a given orbital to have opposite spins to satisfy the Pauli Exclusion Principle.

Conclusion

According to the Pauli exclusion principle, the electrons within a given orbital must have opposite spins.

Key concepts

Spin Quantum Number

Electrons possess a property called spin, which can be thought of as an internal form of angular momentum. The spin quantum number, denoted as \(m_s\), helps to define this property. For electrons, the spin quantum number can only take one of two possible values: \(+1/2\) or \(-1/2\). These values indicate the "spin up" and "spin down" states, respectively.

When imagining electron spin, think of it as a tiny bar magnet with its north or south pole pointing up or down. However, it's important to remember that electron spin is not literally a spinning motion like a planet. Instead, it's a quantum mechanical property closely tied to magnetism and quantum behavior.

• Two electrons occupying the same orbital must have opposite spins
• The Pauli Exclusion Principle demands uniqueness in their set of quantum numbers

Quantum Numbers

Quantum numbers are essential in defining the state of an electron within an atom. These numbers provide a comprehensive "address" for the electrons, indicating their energy level, shape of orbitals, orientation, and spin direction. The four quantum numbers are crucial for understanding electron configurations and chemical behavior.

• Principal Quantum Number \( (n) \): Determines the electron's energy level and shell.
• Azimuthal Quantum Number \( (ℓ) \): Dictates the shape or type of orbital (e.g., s, p, d, f).
• Magnetic Quantum Number \( (m_ℓ) \): Specifies the orientation of the orbital in space.
• Spin Quantum Number \( (m_s) \): Represents the direction of the electron's spin, as mentioned earlier.

These quantum numbers combine to describe the unique position and behavior of each electron, thereby maintaining the balance within an atom. The Pauli Exclusion Principle relies on these numbers to ensure no two electrons share an identical set, promoting unique properties and stability.

Electron Configuration

Electron configurations are a clear representation of how electrons are distributed within an atom's orbitals. These configurations help in understanding the chemical properties and reactivity of elements. A typical electron configuration uses the principal quantum number and subshell labels to show energy levels. For example, \(1s^2\) indicates two electrons in the first energy level's 's' subshell.

The arrangement follows a specific sequence guided by the Aufbau principle, Hund's Rule, and the Pauli Exclusion Principle:

• Aufbau Principle: Electrons fill orbitals starting with the lowest energy level.
• Hund's Rule: Electrons occupy degenerate orbitals singly before pairing.
• Pauli Exclusion Principle: No two electrons in an atom can have identical quantum numbers—meaning paired electrons in an orbital must have opposite spins.

Understanding how to write and interpret electron configurations is fundamental in predicting element behavior and bonding characteristics. This notation not only communicates a wealth of information but also reveals the intricacies of atomic structure.