Question

Why are the \(s, p, d\), and \(f\) blocks in the periodic table \(2,6,10\), and 14 blocks wide, respectively?

Short answer

The widths of the \(s, p, d\), and \(f\) blocks in the periodic table correspond to the maximum number of electrons that can be accommodated in each type of orbital (\(s, p, d,\) and \(f\), respectively). The maximum number of electrons in each orbital is determined by the formula \(2(2l + 1)\), where \(l\) is the angular momentum quantum number. Applying this formula, we find that the widths of the \(s, p, d,\) and \(f\) blocks are \(2, 6, 10,\) and \(14\) elements wide, respectively.

Step-by-step solution

Understanding electron configuration

Electron configuration refers to the arrangement of electrons in the atomic orbitals of an atom. Orbitals are the spaces around the nucleus of an atom where electrons have the highest probability of being found. Each orbital has specific energy levels and shapes. The four types of orbitals we are concerned with in this question are the \(s\), \(p\), \(d\), and \(f\) orbitals.

Principal quantum number and angular momentum quantum number

Orbitals are characterized by two quantum numbers: the principal quantum number (\(n\)) and the angular momentum quantum number (\(l\)). The principal quantum number determines the energy level and size of an orbital, while the angular momentum quantum number determines the shape of the orbital. The \(s, p, d\), and \(f\) orbitals correspond to the angular momentum quantum numbers \(l = 0, 1, 2,\) and \(3\), respectively.

Maximum number of electrons in each type of orbital

Each type of orbital can accommodate a specific number of electrons. The maximum number of electrons that can occupy an orbital is determined by the formula \(2(2l + 1)\), where \(l\) is the angular momentum quantum number. For the \(s\) orbital (\(l = 0\)): The maximum number of electrons is \(2(2 \cdot 0 + 1) = 2\). For the \(p\) orbital (\(l = 1\)): The maximum number of electrons is \(2(2 \cdot 1 + 1) = 6\). For the \(d\) orbital (\(l = 2\)): The maximum number of electrons is \(2(2 \cdot 2 + 1) = 10\). For the \(f\) orbital (\(l = 3\)): The maximum number of electrons is \(2(2 \cdot 3 + 1) = 14\).

Relating the maximum number of electrons to the block widths

The width of each block in the periodic table corresponds to the maximum number of electrons that can be accommodated by the respective type of orbital. This is because, as we move across a period (horizontal row) in the periodic table, we fill up electrons in the orbitals of elements. The number of elements in each block is determined by the number of electrons that can be accommodated in that specific orbital. Thus, the width of the \(s, p, d,\) and \(f\) blocks in the periodic table are \(2, 6, 10,\) and \(14\) elements wide, respectively, because these are the maximum numbers of electrons that can occupy the respective orbitals.

Key concepts

Electron Configuration

When we talk about electron configuration, we're diving into the fascinating world of how electrons are arranged around an atom's nucleus. Imagine each electron orbiting in specified lanes around a track. These lanes represent various energy levels and shapes, known as atomic orbitals. There are four main types of orbitals: s, p, d, and f.

Each type has a distinct shape and can hold a specific number of electrons. For instance, the s orbital is like a sphere centered around the nucleus and can hold up to 2 electrons. The p orbitals are shaped like dumbbells with room for 6 electrons. The d and f orbitals are more complex, accommodating 10 and 14 electrons, respectively. When you see a full set of electron configurations, you're essentially looking at a map that tells you where each electron 'lives' in an atom.

Quantum Numbers

Delving into quantum numbers feels a bit like stepping into a system of GPS coordinates for electrons. These numbers aren't just random digits; they're crucial for describing the position and behavior of each electron in an atom.

The first number, known as the principal quantum number (n), tells us the energy level and, by extension, the orbital's size. The second number, the angular momentum quantum number (l), gives us the shape of the orbital. Higher numbers translate to more complex orbitals and higher energy levels. For example, when l is 0, we're talking about an s orbital, and when l is 3, we're in the realm of an f orbital. It's these numbers that guide us in mapping an electron's address within an atom.

Atomic Orbitals

Atomic orbitals are the 'homes' of electrons, spaces where they're most likely to hang out. But they're not just spots on a map; they have designated shapes and sizes, defined by quantum numbers, which influence the atomic and chemical properties of an element.

Think of orbitals as different rooms in a house. The s orbital is a relatively small, spherical room, while the p orbital is larger and has a unique, two-lobed shape. Orbitals like d and f are even more spacious and intricately shaped, which is why they can accommodate more electrons. These rooms fill up in a specific order as the atom increases in size, following the rules of electron configuration.

Principal Quantum Number

The principal quantum number, denoted as n, is like the floor level in a high-rise building, indicating how far an electron's orbital is from the nucleus. A higher n value means the electron is in a higher energy level and further away from the center.

This number directly impacts an atom's energy and the size of electron orbitals; it starts at 1 and moves up as you climb to higher energy levels. Just like the floors in a building, each level can house a certain number of rooms—or in our case, orbitals. Thus, as n increases, so does the capacity to hold more electrons, which explains the varying widths of the different blocks on the periodic table.