The modern quantum mechanical model of the atom is built on the work of many physicists. The idea of a dense central nucleus surrounded by orbiting electrons was first proposed by Rutherford in 1911. However, his model had several shortcomings, such as its inability to explain the observed emission spectra of elements. In 1913, Bohr extended Rutherford’s model by introducing the concept of angular momentum quantization, which successfully addressed many of these issues.
Bohr's Model of the Atom
Bohr’s atomic model is based on four postulates:
- Postulate 1: Electrons revolve about the nucleus in well-defined, allowed orbits (shells), where the Coulomb force between the electrons and the nucleus balances the centripetal force due to the electron's orbital motion.
- Postulate 2: An electron in orbit does not lose energy despite being constantly accelerated, contradicting classical physics predictions (which stated that accelerated charged particles would emit radiation and lose energy).
- Postulate 3: The angular momentum of an electron in an allowed orbit is quantized and takes values of \(n\hbar\), where \(n\) is an integer and \(\hbar = \frac{h}{2\pi}\), with \(h\) being Planck’s constant.
- Postulate 4: An atom emits radiation when an electron transitions from an initial orbit with quantum number \(n_i\) to a final orbit with quantum number \(n_f\), where \(n_i > n_f\).
Although Bohr's model was revolutionary, it could not explain more complex atomic systems. This led to further advancements in atomic theory.
Quantum Mechanical Model
In the early 20th century, the quantum mechanical model of the atom was developed by physicists such as Heisenberg, Schrödinger, Dirac, and Pauli. According to quantum mechanics, electrons do not follow precise orbits but instead occupy specific energy states defined by four quantum numbers:
- Principal quantum number \(n\): Specifies the main energy shell of the electron and can take integer values \(n = 1, 2, 3, \dots\).
- Azimuthal quantum number \(l\): Determines the shape of the orbital and can take integer values between 0 and \(n-1\).
- Magnetic quantum number \(m\): Specifies the orientation of the orbital and can take integer values between \(-l\) and \(+l\).
- Spin quantum number \(s\): Specifies the direction of the electron's spin, which can be either \(-\frac{1}{2}\) or \(+\frac{1}{2}\).
According to the Pauli exclusion principle, no two electrons can occupy the same quantum state. Therefore, the number of possible electron states in a given shell is \(2n^2\), where \(n\) is the principal quantum number.
Energy Levels and Quantum Numbers
The energy levels of electrons in an atom can be represented in energy level diagrams. These diagrams show the different energy shells, with each principal quantum number \(n = 1, 2, 3, \dots\) corresponding to a distinct shell or band. The energy associated with these shells increases with the atomic number \(Z\) and decreases with the quantum number \(n\). The outermost shell (valence shell) of an atom determines its chemical properties.
For example, the energy bands associated with \(n = 1, 2, 3, \dots\) are labeled K, L, M, etc. Each energy band is further divided into smaller levels due to the azimuthal quantum number \(l\) and the spin quantum number \(s\).
Quantum Mechanical Energy Level Diagram
In the energy level diagram for a hydrogen atom, the energy levels associated with quantum number \(n = 1\) correspond to the K shell, \(n = 2\) to the L shell, and so on. As the quantum number