In this chapter, we will learn about Pauli Exclusion Principle. You have studied in the quantum mechanical model of atom, the state of each electron in an atom is described by four quantum numbers:
- Principal Quantum Number (n): This quantum number tells about the number of orbit/shell or energy level.
- Azimuthal Quantum Number (l): This quantum number defines the shape of the orbital (s, p, d, f).
- Magnetic Quantum Number (m): It tells about the orientation of the orbital in space.
- Spin Quantum Number (s): This tells about rotation or spin of the electron, which can be either +1/2 or -1/2.
An Austrian theoretical physicist Wolfgang Ernst Pauli in 1925 put forward an ingenious principle known as Pauli Exclusion Principle. This fundamental principle controls the assignment of values of four quantum numbers of an electron in an orbital. It applies certain restrictions on the values of quantum numbers of electrons in an atom and hence the name is “exclusion principle”. Let’s understand the concept of this principle in more detail.
Pauli’s Exclusion Principle
The Pauli Exclusion Principle is one of the most fundamental concepts in the quantum mechanical model of atom. This principle applies to many-electron atoms. It states that no two electrons in an atom may have the same set of quantum numbers simultaneously. In other words, no two electrons in an atom can have the same set of all four quantum numbers.
An orbital can accommodate a maximum of two electrons with opposite spins. These two electrons may have the same values for the principal, azimuthal, and magnetic quantum numbers, but they must have different spin quantum numbers. Thus, every electron in an atom differs from every other electron in at least one quantum number.
Examples of Pauli Exclusion Principle
Example 1: In order to understand this principle, let us consider helium atom which contains two electrons in 1s orbital. The atomic number of helium is two. According to the concept of quantum numbers and Pauli’s rule, the four quantum numbers for two electrons in 1s-orbital are as follows:
| Electron | n | l | m | s | Set of values of quantum numbers (n, l, m, s) |
|---|---|---|---|---|---|
| Electron 1 | 1 | 0 | 0 | +1/2 | (1, 0, 0, +1/2) |
| Electron 2 | 1 | 0 | 0 | -1/2 | (1, 0, 0, -1/2) |
The + and – sign before 1/2 represents the clockwise and anticlockwise spins of the electron. Thus, you can see that the set of values of quantum numbers for the first electron is (1, 0, 0, +1/2), while the set of values of quantum numbers for the second electron is (1, 0, 0, -1/2). Both electrons have the same n, l, and m, but the value of spin quantum number is different, i.e. +1/2 and -1/2.
Their spins or rotations are in the opposite directions. Both electrons cannot have the same value +1/2 because there will be more repulsion. The Pauli Exclusion Principle leads to a significant observation that each orbital can accommodate a maximum of two electrons, which must have opposite spins.
Example 2: Let us consider nitrogen atom which has seven electrons. Its electronic configuration and four quantum numbers are as follows:
N7 = 1s2, 2s2, 2p3
| Quantum Number | 1s2 | 2s2 | 2px1 | 2py1 | 2pz1 |
|---|---|---|---|---|---|
| Principal quantum number (n) | 1 | 2 | 2 | 2 | 2 |
| Azimuthal quantum number (l) | 0 | 0 | 1 | 1 | 1 |
| Magnetic quantum number (m) | 0 | 0 | +1 | -1 | 0 |
| Spin quantum number (s) | +1/2 -1/2 | +1/2 -1/2 | +1/2 | +1/2 | +1/2 |
As you can see in the above table out of seven electrons no two have same values of all four quantum numbers.
Application of Pauli’s Exclusion Principle
With the help of this principle, we can calculate the maximUm number of electrons which can be accommodated in an orbital, subshell, and main energy shell.
| Principal Q. No. (n) | Azimuthal Q. No. (l) | Magnetic Q. No. (m) | Spin Q. No. (s) | No. of electrons on a subshell | No. of electrons on a main shell |
|---|---|---|---|---|---|
| 1 | 0(s) | 0 | +1/2, -1/2 | 2 | 2 |
| 2 | 0(s) 1(p) | 0 -1 0 +1 | +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 | 2 6 | 8 |
| 3 | 0(s) 1(p) 2(d) | 0 -1 0 +1 -2 -1 0 +1 +2 | +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 | 2 6 10 | 18 |
| 4 | 0(s) 1(p) 2(d) 3(f) | 0 -1 0 +1 -2 -1 0 +1 +2 -3 -2 -1 0 +1 +2 +3 | +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 +1/2, -1/2 | 2 6 10 14 | 32 |
Conclusions:
The Pauli Exclusion Principle leads to the following conclusions:
- The maximum capacity of a main energy shell is equal to 2n2 electrons, where n is the principal quantum number.
- The maximum capacity of a subshell is equal to 2(2l + 1) electrons where l is azimuthal quantum number.
- Number of subshells in a main energy shell is equal to the value of principal quantum number n.
- Number of orbitals in a main energy shell is equal to n2.
- One orbital cannot have more than two electrons. If two electrons are present, their spins must be in opposite directions.
