What are the electronic configurations of the first three elements in the periodic table, and how are they determined using quantum mechanics?

The electronic configurations of the first three elements in the periodic table—hydrogen (H), helium (He), and lithium (Li)—are fundamental to understanding the arrangement of electrons in atoms. These configurations are determined using quantum mechanics, specifically the quantum numbers and rules that govern the distribution of electrons within the electron cloud surrounding the nucleus. Let's explore the electronic configurations of these elements and the quantum mechanical principles behind their determination:

1. Hydrogen (H):
- Hydrogen has one proton and one electron. Its electronic configuration is written as 1s^1.
- This configuration is determined using quantum mechanics as follows:
- Principal Quantum Number (n): For hydrogen, n = 1, indicating the first energy level (shell).
- Azimuthal Quantum Number (l): Since n = 1, l can only be 0. Thus, the electron is in the 1s orbital.
- Magnetic Quantum Number (m_l): For l = 0, m_l can only be 0, so there is one 1s orbital.
- Spin Quantum Number (m_s): Electrons can have two possible spin values: +1/2 (spin-up) and -1/2 (spin-down). There is one electron in the 1s orbital, so the spin quantum number can be either +1/2 or -1/2.

2. Helium (He):
- Helium has two protons and two electrons. Its electronic configuration is written as 1s^2.
- The electronic configuration is determined using the same quantum mechanical principles as for hydrogen. Helium's second electron fills the same 1s orbital, following the Pauli Exclusion Principle, which states that no two electrons in an atom can have the same set of quantum numbers.

3. Lithium (Li):
- Lithium has three protons and three electrons. Its electronic configuration is written as 1s^2 2s^1.
- The electronic configuration is determined using quantum mechanics as follows:
- Principal Quantum Number (n): For lithium, the first two electrons occupy the 1s orbital, so n = 1 for these electrons. The third electron goes into the next available energy level, which is the second energy level (n = 2).
- Azimuthal Quantum Number (l): The first two electrons in the 1s orbital have l = 0. The third electron in the 2s orbital also has l = 0.
- Magnetic Quantum Number (m_l): For l = 0, m_l can only be 0, so there is one 1s orbital and one 2s orbital.
- Spin Quantum Number (m_s): There are two electrons in the 1s orbital with opposite spins (+1/2 and -1/2) due to the Pauli Exclusion Principle. The third electron in the 2s orbital can have either spin +1/2 or -1/2.

In summary, the electronic configurations of the first three elements in the periodic table are determined using quantum mechanics, particularly the principles of quantum numbers. These configurations provide insights into the arrangement of electrons in energy levels, orbitals, and subshells around the nucleus of each atom. The Pauli Exclusion Principle ensures that no two electrons within an atom have the same set of quantum numbers, leading to the filling of orbitals according to specific rules. Understanding electronic configurations is crucial for predicting chemical properties and reactivity in elements and compounds.