Quantum Mechanical Model Explained

The quantum mechanical model provides a detailed description of the behavior of electrons within atoms, including the simplest atom, hydrogen. This model uses a set of quantum numbers to precisely define the state and properties of an electron. Understanding these quantum numbers is fundamental to comprehending atomic structure.

Let's focus on the two main quantum numbers relevant to this question:

• Principal Quantum Number (n): This quantum number is denoted by 'n' and specifies the electron's main energy level or shell. It can take any positive integer value, starting from 1 (i.e., n = 1, 2, 3, ...). A higher value of 'n' indicates a higher energy level and, on average, a greater distance of the electron from the nucleus. In the given question, the principal quantum number is 4.
• Orbital Quantum Number (l): Also known as the azimuthal or angular momentum quantum number, 'l' describes the shape of an electron's orbital and defines the subshell within an energy level. The possible values of 'l' are directly dependent on the principal quantum number 'n'. For any given 'n', the orbital quantum number 'l' can take integer values from 0 up to \((n-1)\).

Determining Permitted Orbital Quantum Numbers

The question asks for the number of different permitted orbital quantum numbers when the principal quantum number for the hydrogen atom is 4.

We are given:

• The principal quantum number, \(n = 4\)

According to the rules of the quantum mechanical model, the allowed values for the orbital quantum number (l) are calculated using the formula:

\(l = 0, 1, 2, ..., (n-1)\)

Substituting the given value of \(n=4\) into this formula, we get:

\(l = 0, 1, 2, ..., (4-1)\)

\(l = 0, 1, 2, 3\)

These are the distinct values that the orbital quantum number can take when the principal quantum number is 4. Each of these 'l' values corresponds to a specific type of subshell:

• If \(l=0\), it corresponds to an 's' subshell, which has a spherical shape.
• If \(l=1\), it corresponds to a 'p' subshell, typically having a dumbbell shape.
• If \(l=2\), it corresponds to a 'd' subshell, which has more complex shapes.
• If \(l=3\), it corresponds to an 'f' subshell, with even more intricate shapes.

Summary of Quantum Numbers Relationship

The relationship between the principal quantum number (n) and the permitted orbital quantum numbers (l) can be summarized in the following table:

Conclusion on Permitted Orbital Quantum Numbers

Therefore, for the hydrogen atom, when the principal quantum number is 4, the number of different permitted orbital quantum numbers is the count of the values {0, 1, 2, 3}, which is four.