Final answer:
The quantum mechanical model of an atom requires four quantum numbers to describe an electron's state, as opposed to the single quantum number used in the Bohr model. These four quantum numbers are the principal quantum number (n), the orbital angular quantum number, the orbital angular projection quantum number, and the spin quantum number.
Step-by-step explanation:
The quantum mechanical model of the atom describes the state of an electron using four quantum numbers. While the Bohr model of the atom was groundbreaking for its time, it only required one quantum number to describe the energy state of an electron. However, our current understanding of atomic structure, based on quantum mechanics, requires these four quantum numbers: the principal quantum number (n), the orbital angular quantum number (l), the orbital angular projection quantum number (m), and the spin quantum number (s).
The principal quantum number (n) plays a significant role in determining an electron's energy level and defines the electron shell that the electron occupies. For example, if n=1, the electron is in the first energy level or shell and is more tightly bound to the nucleus, resulting in lower energy. As n increases, the electron moves to higher energy levels or shells and is less tightly bound to the nucleus.
In contrast, the Bohr model suggested that atoms have fixed, unchangeable energy levels, and electrons reside in precise orbits. While there are similarities in the concept of quantized energy levels between the Bohr model and the quantum mechanical model, Bohr's model could not explain why only certain orbits were allowed nor could it predict the angular and spin properties of the electron, which are successfully explained by quantum mechanics.