Now we have learnt that electrons are not orbiting around the nucleus like our solar system does. They are indeed unpredictable and could only be described using probability of appearing at a certain position. And the area in space where electrons could appear is called orbital, and it is described by four quantum numbers (\( n \), \( l \), \( m \), \( m_s \)). We would like to learn more about the orbitals in this post.

## Energy levels

This is the same as the concept of energy level that we learnt in Bohr's model. We have mentioned in electron transit that, electrons would move between different energy levels, accompanied by energy absorbing or releasing.

We didn't explain why we called the energy levels as "level 1", "level 2", etc. in Bohr's model. But now we would know that energy levels are related to **principal quantum number** in the quantum mechanical model.

Scientists have found that in order for the wave functions to have solutions, i.e. certain positions of electrons could yield a high probability, the principal quantum number has to be integers starting from 1, whereas energy level \( n = 1 \) is the nearest to the nucleus and energy level with a greater \( n \) is farther away from the nucleus.

## Sublevels

One of the main differences between the energy levels in Bohr's model and those in quantum mechanical model is that, energy levels in quantum mechanical model are further divided into sublevels.

Sublevels are defined by angular quantum number \( l \), which could be any integer from \( 0 \) to \( n-1 \).

For example, for energy level 1, (\( n=1 \)), \( l \) must be \( 0 \), while for \( n=2 \), \( l \) could be \( 0 \) or \( 1 \).

We've also assigned symbols to sublevels so that we could easily identify sublevels. You could refer to the table below.

\( l \) | symbol |
---|---|

0 | s |

1 | p |

2 | d |

3 | f |

So we would use *number for energy levels, and symbol for sublevels*. For example, we could write *4p* which means the *p* sublevel in energy level *4*. Do take note that **the number before the symbol refers to the energy level** that the sublevel is in, but not telling you how many sublevels there are. In fact, *4p* is only one sublevel located in energy level 4.

## Orbitals

The third quantum number, magnetic quantum number \( m \), would further break down the sublevels into orbitals, which are the space where there are high probability of finding an electron.

You may imagine energy levels, sublevels, and orbitals as city, street, and house respectively. You must have all the three parameters to locate the electrons.

The magnetic quantum number could be any integer from \( -l \) to \( +l \). For example, for \( l = 2 \), \( m \) can be \( -2, -1, 0, +1, +2 \), therefore, 5 orbitals could be found in the sublevel \( l=2 \), or the *d* sublevel.

We would summarize the energy levels, sublevels, and orbitals in the table below.

energy level | sublevel | number of orbitals |
---|---|---|

1 | 1s | 1 |

2 | 2s | 1 |

2p | 3 | |

3 | 3s | 1 |

3p | 3 | |

3d | 5 | |

4 | 4s | 1 |

4p | 3 | |

4d | 5 | |

4f | 7 |

However, unlike sublevels, we don't have a general naming for the orbitals. Naming for orbitals is a little complicated and we'll talk about that in the next post. Nevertheless, we don't need to identify individual orbitals as much as sublevels. You'll know why is so in a moment.