Question

How is the order of the energy of the atomic orbitals established?

Short answer

Question: Explain the order of the energy of atomic orbitals using the principles of quantum numbers, Aufbau Principle, and Pauli Exclusion Principle. Answer: The order of energy of atomic orbitals is determined by considering the values of the principal quantum number (n) and the azimuthal quantum number (l). Electrons fill atomic orbitals in order of increasing energy, based on the Aufbau Principle, starting with the lowest energy orbital and following the sequence 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, and so on. The Pauli Exclusion Principle states that each atomic orbital can hold a maximum of two electrons with different spin quantum numbers, ensuring that each electron has a unique set of quantum numbers.

Step-by-step solution

Understand atomic orbitals

Atomic orbitals are regions around the nucleus of an atom where electrons are most likely to be found. Each atomic orbital is specified by a set of quantum numbers: the principal quantum number (n), the azimuthal quantum number (l), and the magnetic quantum number (m_l). The energy of each orbital depends primarily on the principal quantum number (n), and to a lesser extent, the azimuthal quantum number (l).

Define the quantum numbers

In order to understand how atomic orbitals are ordered, we must first define the quantum numbers that determine their energy: 1. Principal quantum number (n): It determines the energy level of an electron. It can have integer values (n=1, 2, 3...). Higher values of n mean that the electron is further from the nucleus and has a higher energy. 2. Azimuthal quantum number (l): It determines the shape of the orbital and can have integer values from 0 to n-1. The value of l indicates the type of orbital (0 = s, 1 = p, 2 = d, 3 = f). 3. Magnetic quantum number (m_l): It determines the orientation of the orbital in space and can have integer values from -l to +l.

Aufbau Principle

The Aufbau Principle states that electrons fill atomic orbitals in order of increasing energy, starting with the lowest energy orbital. An electron will always occupy the lowest available energy state. The order of filling can be determined by following the rules of the Aufbau Principle, which can be summarized in a diagram, like the one below: 1s 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p 6s 4f 5d 6p 7s 5f 6d 7p... Electrons fill the orbitals from left to right, starting with the lowest energy level.

Pauli Exclusion Principle

The Pauli Exclusion Principle states that no two electrons in an atom can have the same set of four quantum numbers (n, l, m_l, and m_s, where m_s is the electron's spin quantum number). In other words, each electron must have a unique set of quantum numbers. This means that each atomic orbital can hold a maximum of two electrons, one with a spin quantum number of +1/2 and one with a spin quantum number of -1/2.

Establish the order of energy of the atomic orbitals

Based on the information about quantum numbers and the principles discussed above, we can establish the order of the energy of atomic orbitals in the following way: 1. Orbitals with the same principal quantum number (n) have similar energies, but the energy increases with the value of the azimuthal quantum number (l) within the same value of n. 2. As the principal quantum number (n) increases, the energy of the atomic orbitals also increases. 3. The order of the energy of the atomic orbitals can be determined by considering both the values of n and l, and following the Aufbau Principle: 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p, and so on.