Question

Only two electrons can occupy a given orbital in an atom, and to be in the same orbital, they must have opposite ________.

Short answer

Only two electrons can occupy a given orbital in an atom, and to be in the same orbital, they must have opposite \(\textit{spins}\).

Step-by-step solution

Define atomic orbitals and electron configuration

An atomic orbital is a region in an atom where there is a high probability of finding an electron. Electrons in atoms are organized into energy levels or shells, and within these levels, they occupy orbitals. The electron configuration describes the distribution of electrons in the atomic orbitals.

Recall the Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental principle in quantum mechanics, stating that no two electrons in a given atom can have the same set of quantum numbers. Electrons in an atom are described by four quantum numbers (n, l, ml, and ms), representing the electron's energy level, shape of the orbital, orientation of the orbital, and its spin, respectively.

Understand electron spin

The electron spin (ms) is a fundamental property of electrons. It can have two possible values: +1/2 (called "spin-up") and -1/2 (called "spin-down"). In order to share the same orbital, electrons must have opposite electron spins.

Complete the exercise

Given this information, we can conclude that to be in the same orbital, electrons must have opposite spins. So, the correct term to fill in the blank space is "spins". Answer: Only two electrons can occupy a given orbital in an atom, and to be in the same orbital, they must have opposite spins.

Key concepts

Atomic Orbitals

Imagine an atom as a tiny, bustling city with electrons as its citizens, hovering in various neighborhoods. These neighborhoods are what we call atomic orbitals. They're not fixed paths like the orbits of planets, but cloudy regions buzzing with electron activity, where you're most likely to find an electron chilling out.

An orbital can be visualized as a three-dimensional shape around the nucleus. The most common shapes are spherically symmetrical (s orbitals) or dumbbell-shaped (p orbitals). Each orbital has a unique size, energy, and orientation in space, characterized within the atom's electron ‘address system’, or quantum numbers. The first three quantum numbers, n (principal), l (azimuthal), and m_l (magnetic), tell us the energy level, shape, and orientation of the orbital respectively.

This 3D zip-code of atomic orbitals ensures electrons have a pretty organized residential system, maximizing space and energy efficiency within the bustling atomic metropolis. Simple as spots on a dice, atomic orbitals hold the secrets to an atom’s reactivity, bonding, and even its magnetic properties.

Electron Configuration

Filling out the atomic neighborhoods with their electron residents is more than just a game of quantum musical chairs. It's a structured process, following the rules of quantum mechanics. Electron configuration provides a roadmap for how electrons settle into atomic orbitals.

The most relaxed electrons, like tourists seeking the most comfortable hotel, naturally fill the lowest energy orbitals first. This follows a competing occupation order known as the Aufbau Principle. As if following a grocery list, the electrons fill the '1s' orbital, then '2s', followed by '2p', and so on. The 's', 'p', 'd', and 'f' labels indicate the shape of the orbitals.

For example, an oxygen atom, with 8 electrons, fills its first 2 electrons in the 1s orbital (1s²), the next two in the 2s orbital (2s²), and the remaining 4 fall into two 2p orbitals (2p⁴). Each new orbital fills up like seats in a theater, ensuring everyone gets a clear view of the screen before someone plops down in front obstructing the view. This organized distribution outlines the chemical character and the stability of atoms, much like a city planner meticulously allocating space for urban development.

Electron Spin

Electrons are not only defined by where they live but also how they twirl. This twirling is the electron spin, a quantum feature that's as integral to electrons as toppings are to a pizza — it just wouldn't be complete without it. The spin of an electron can take one of two values: '+1/2', commonly termed 'spin-up', or '-1/2', called 'spin-down'.

Engaging in a quantum dance, two electrons living on the same orbital must have opposite spins — a graceful act of balance and symmetry. It's like having two figure skaters spinning in opposite directions on the same rink; their opposing spins prevent them from crashing into one another.

Why is this important? The rule that two electrons in the same orbital must have opposite spins is part of the Pauli Exclusion Principle, a fundamental law that keeps electrons from stepping on each other's quantum toes. Thanks to this principle, every electron in an atom has its own unique set of quantum numbers (n, l, m_l, and m_s), contributing to a diversity of elements and the complexity of chemistry.