Question

How many orbitals in total are associated with \(4^{\text {th }}\) energy level? (a) 4 (b) 9 (c) 16 (d) 7

Short answer

There are 16 orbitals associated with the 4th energy level.

Step-by-step solution

Understanding energy levels and orbitals

Each energy level 'n' has 'n^2' orbitals. The energy level given is the 4th energy level, so we use 'n=4'.

Calculating the number of orbitals

Calculate the number of orbitals using the formula 'n^2'. For n=4, it is '4^2'.

Finding the answer

Square the value of n: '4^2 = 16'. Thus, there are 16 orbitals associated with the 4th energy level.

Key concepts

Quantum Numbers

To comprehend the complex world of atomic structure, a set of four quantum numbers are introduced which describe the unique state of an electron in an atom. These are the principal quantum number (), the azimuthal or orbital angular momentum quantum number (), the magnetic quantum number (), and the spin quantum number ().

The principal quantum number () signifies the energy level and size of the orbital where the electron resides. As we saw in the exercise, by knowing , we can determine the total number of orbitals in an energy level simply by calculating ^2. The azimuthal quantum number (), which can range from 0 to -1, identifies the shape of the orbital (s, p, d, or f). The magnetic quantum number (), with values ranging from - to +, pinpoints the orientation of the orbital within a sublevel. Lastly, the spin quantum number () denotes the direction of the electron's spin, with possible values of +1/2 or -1/2.

In essence, these quantum numbers not only help us understand where electrons are likely to be found, but also how they interact within the atom. Understanding these numbers is key to grasping electron configurations and how atoms achieve a stable state.

Atomic Structure

At its most fundamental level, atomic structure consists of a nucleus containing protons and neutrons, which are surrounded by electrons in orbitals. These orbitals are not just simple rings around the nucleus; they're best visualized as clouds of probability where an electron might be found, defined by quantum numbers.

Electrons within an atom occupy orbitals in a well-organized manner known as the electron configuration. This order is not random: electrons fill the lowest energy orbitals first, a principle known as the Aufbau principle. For example, the 1s orbital, being the closest to the nucleus, is the lowest in energy and is filled before the 2s orbital. The pattern continues with increasing complexity and higher energy levels.

The fourth energy level that we see in the exercise is further divided into sublevels, corresponding to different types of orbitals: s, p, d, and f, with each capable of holding a different maximum number of electrons. Understanding this organized structure reveals much about the chemical properties of an element and dictates how it will interact with other atoms.

Electron Configuration

Electron configuration is the arrangement of electrons in an atom's orbitals. It is written as a notation that combines the energy level and type of orbital with the number of electrons in that orbital. For instance, helium's electron configuration is 1s2, indicating 2 electrons in the 1s orbital.

The distribution of electrons across different orbitals determines an element's chemical behavior. Hund's rule states that electrons will fill unoccupied orbitals of the same energy before pairing up. This minimizes electron repulsion and contributes to the stability of the atom.

In multi-electron atoms, electrons fill the principal energy levels and sublevels in ascending order of energy, as explained by the Aufbau principle. This order is s, p, d, f, with 's' being the lowest and 'f' being the highest in energy. The exercise highlights how these concepts come together: the 4th energy level's electron configuration could start as 4s, then fill 4p, 4d, and possibly 4f, with each sublevel containing a distinct number of orbitals, in the sequence of s (1), p (3), d (5), and f (7), totalling 16 orbitals for the 4th level.