Question

The total number of orbital for \((n+l)=4\) is (a) 4 (b) 16 (c) 32 (d) 9

Short answer

The total number of orbitals for \(n+l)=4\) is 16.

Step-by-step solution

Understanding the Quantum Numbers

The principal quantum number (n) determines the energy level of an electron in an atom and the azimuthal quantum number (l) determines its subshell or orbital angular momentum. For a given value of the sum (n+l), multiple combinations of n and l can exist.

Finding Possible Combinations

Find all combinations of n and l that add up to 4. Possible combinations are (n=1, l=3), (n=2, l=2), (n=3, l=1), and (n=4, l=0).

Calculating the Number of Orbitals

Each value of l corresponds to a particular subshell and has a specific number of orbitals: s (l=0) has 1 orbital, p (l=1) has 3 orbitals, d (l=2) has 5 orbitals, and f (l=3) has 7 orbitals. Calculate the total number of orbitals by adding up orbitals corresponding to each (n, l) pair.

Adding up the Orbitals

Sum the orbitals for each (n, l) pair: (n=1, l=3) has 7 orbitals, (n=2, l=2) has 5 orbitals, (n=3, l=1) has 3 orbitals, and (n=4, l=0) has 1 orbital. Add them up to find the total: 7 + 5 + 3 + 1 = 16.

Key concepts

Principal Quantum Number

The principal quantum number, symbolized as \(n\), is a fundamental concept in understanding the structure of an atom. It indicates the energy level, or shell, in which an electron resides and plays a critical role in determining the size and energy of an atomic orbital.

An increase in the principal quantum number signifies a higher energy level and typically a larger orbital, meaning the electron is, on average, further away from the nucleus. Within the confines of quantum mechanics, \(n\) can have positive integer values starting from 1, corresponding to the closest shell to the nucleus. As \(n\) increases, the potential energy of electrons in those orbitals also increases, and they become less tightly bound to the nucleus.

It's crucial for students to remember that higher energy levels can accommodate more electrons, leading to more complex electron configurations. For instance, the first energy level (\(n=1\)) can only hold a maximum of 2 electrons, while the second (\(n=2\)) can hold up to 8.

Azimuthal Quantum Number

The azimuthal quantum number, labeled as \(l\), is associated with the shape of atomic orbitals and is sometimes referred to as the angular momentum quantum number. It is a direct consequence of the orbital angular momentum of an electron.

For any given principal quantum number \(n\), \(l\) can take on any integer value from 0 up to \(n-1\). Each value of \(l\) correlates to a specific subshell (s, p, d, f, and so on). For instance, \(l=0\) represents 's' orbitals which have a spherical shape, while \(l=1\) pertains to 'p' orbitals characterized by their dumbbell-shaped structure.

Different subshells can hold a different number of electrons: each 's' orbital can hold 2, each 'p' orbital can hold 6, 'd' can hold 10, and 'f' can hold 14 electrons. Students may find it useful to associate these subshells with their common shapes to better visualize electron configurations and orbital filling.

Orbital Angular Momentum

Orbital angular momentum relates to the movement of electrons around the nucleus, and with the azimuthal quantum number \(l\), its role becomes more defined. The value of \(l\) gives us the magnitude of the angular momentum for an electron within a particular orbital.

Electrons in orbitals with higher \(l\) values have greater angular momentum, which means they have more 'twisting' motion. To visualize this, imagine a figure skater spinning; as they pull their arms in (akin to lower \(l\) values), they spin more tightly and with less angular momentum. As they extend their arms out (comparable to higher \(l\) values), they exhibit greater angular momentum.

The quantized nature of angular momentum in quantum mechanics is central to why electrons exist in discrete orbitals and not between them. Understanding orbital angular momentum aids in comprehending the restrictions for electron movement within an atom, which in turn explains the structure of the periodic table and the chemical behavior of elements.

Electron Configuration

Electron configuration is a term used to describe how electrons are distributed among the orbitals of an atom. The distribution is governed by a set of principles, namely the Aufbau principle, Pauli exclusion principle, and Hund's rule, and it defines the chemical properties of an atom.

The Aufbau principle suggests that electrons fill orbitals starting from the lowest energy level, moving upwards. The Pauli exclusion principle states that no two electrons can have the same set of all four quantum numbers, ensuring each electron has a unique state. Hund's rule says that electrons will fill degenerate (equally energy) orbitals singly first, with parallel spins, before pairing up.

Electron configurations employ notation that includes the principal quantum number, the letter designating the subshell (s, p, d, or f), and a superscript showing the number of electrons in those subshells (e.g., \(1s^2 2s^2 2p^6\)). This notation helps in predicting and explaining the chemical bonding properties, reactivity, and even color of substances. Mastering electron configurations allows students to predict how atoms will interact in chemical reactions.