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Chapter 9: Problem 5

For the \(\mathrm{H}_{2}\) molecule, why must the spins the electron spins be antiparallel?

Short Answer

Expert verified
The electron spins in the H₂ molecule must be antiparallel due to the formation of the σ bonding orbital and the σ* antibonding orbital from their atomic orbitals and the Pauli Exclusion Principle. When the two electrons from the hydrogen atoms combine to form a hydrogen molecule, they occupy the σ bonding orbital, and their spins must be antiparallel (one ↑ and the other ↓) to satisfy the Pauli Exclusion Principle.

Step by step solution

01

Molecular Orbitals

In molecular orbital theory, atomic orbitals (AOs) from individual atoms combine to form molecular orbitals (MOs). MOs are formed from linear combinations of atomic orbitals, and the number of MOs formed is equal to the number of AOs. For a hydrogen atom, the 1s AO is used to describe the electron's behavior. When two hydrogen atoms interact to form a hydrogen molecule (H₂), their 1s orbitals combine to form two MOs. One MO is a lower energy (bonding) orbital, called the σ bonding orbital, and the other is a higher energy (antibonding) orbital, called the σ* antibonding orbital.
02

Electron Spin and Pauli Exclusion Principle

Electron spin is a fundamental property of electrons, much like their charge. An electron can have two possible values of its spin quantum number (ms), which are ±½. These are often referred to as "spin up" and "spin down" states, indicated as ↑ or ↓ arrows, respectively. According to the Pauli Exclusion Principle, no two electrons in the same atom or molecule can have the same values for all of their quantum numbers (n, ℓ, mℓ, and ms). In other words, if any two electrons are in the same orbital, sharing the same n, ℓ, and mℓ values, their ms values (spins) must be different, i.e., one must be "spin up" and the other must be "spin down."
03

Formation of H₂ Molecule

When two hydrogen atoms combine to form the H₂ molecule, their AOs overlap, and their 1s orbitals form the σ bonding and σ* antibonding molecular orbitals. The H₂ molecule has two electrons – one from each hydrogen atom. These two electrons will occupy the lowest energy MO, which is the σ bonding orbital. Since both electrons cannot have the same quantum numbers, their ms values must be different (one ↑ and the other ↓) due to the Pauli Exclusion Principle. This arrangement of electron spins is known as "antiparallel."
04

Conclusion

The electron spins in the H₂ molecule must be antiparallel because of the formation of the σ bonding orbital and the σ* antibonding orbital from their atomic orbitals and the Pauli Exclusion Principle. Thus, when the two electrons from the hydrogen atoms combine to form a hydrogen molecule, they will occupy the σ bonding orbital, and their spins must be antiparallel (one ↑ and the other ↓) to satisfy the Pauli Exclusion Principle.

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