2.6.1. Orbitals and Quantum Numbers

A large number of orbitals are possible in an atom. Qualitatively these orbitals can be distinguished by their size, shape and orientation. An orbital of smaller size means there is more chance of finding the electron near the nucleus. Similarly shape and orientation mean that there is more probability of finding the electron along certain directions than along others.

Orbitals are regions in an atom where there is a high probability of finding electrons. Each orbital has distinct characteristics, such as size and shape. A smaller orbital indicates a higher probability of the electron being found closer to the nucleus. Similarly, the orientation of the orbital matters because it shows in which directions the electron is most likely to be located. This basic understanding of orbitals is crucial as we explore how electrons behave in atoms.

Think of an orbital like a playground. A smaller playground allows kids (electrons) to gather closer to the center (nucleus) more consistently, indicating they like to play near the center. Larger playgrounds might allow them to spread out more, but those areas far from the center aren't as active.

Atomic orbitals are precisely distinguished by what are known as quantum numbers. Each orbital is designated by three quantum numbers labelled as n, l and ml.

Quantum numbers are values that describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number 'n' indicates the energy level and size of the orbital, while 'l' determines the shape of the orbital. The magnetic quantum number 'ml' describes the orientation of the orbital in space. For example, an 'n' value of 2 means the electron is in the second energy level, while an 'l' value allows us to understand whether the orbital is spherical (s), dumbbell-shaped (p), or more complex (d, f).

Imagine you’re organizing a party. The quantum number 'n' tells you which floor of a building (energy level) the party is on. The 'l' quantum number specifies the room shape (spherical, dumbbell, etc.) on that floor, and the 'ml' quantum number shows the specific arrangement of tables and chairs (spatial orientation) within that room.

The principal quantum number ‘n’ is a positive integer with value of n = 1,2,3....... The principal quantum number determines the size and to large extent the energy of the orbital. For hydrogen atom and hydrogen like species (He+, Li2+, .... etc.) energy and size of the orbital depends only on ‘n’.

The principal quantum number 'n' is crucial as it sets the stage for the energy and size of orbitals. The larger the number, the higher the energy and the larger the orbital. For example, when 'n' equals 1, the electron is closest to the nucleus and has the lowest energy. When 'n' equals 2, the electron is further away and has higher energy. This relationship is particularly straightforward in hydrogen and similar atoms, where energy strictly depends on 'n'.

Think of the principal quantum number as the floors of a building. Ground floor (n=1) is close to the elevator (nucleus) and less costly to rent (lower energy). The higher you go (n=2, n=3, etc.), the more space you get, but you also pay more, as it costs more to live higher up in the building.

Azimuthal quantum number. ‘l’ is also known as orbital angular momentum or subsidiary quantum number. It defines the three-dimensional shape of the orbital. For a given value of n, l can have n values ranging from 0 to n – 1, that is, for a given value of n, the possible values of l are: l = 0, 1, 2, .......... (n–1).

The azimuthal quantum number 'l' helps us understand the shape of the orbital. It can take on values ranging from 0 to (n-1). The value of 'l' dictates whether the orbital is spherical (s, if l=0), dumbbell-shaped (p, if l=1), or more complex (d, if l=2, etc.). This shape is essential for predicting how electrons will behave in atoms, including how they form bonds with other atoms.

Imagine you’re a sculptor. The principal quantum number 'n' tells you how big the sculpture will be, while the azimuthal quantum number 'l' defines the sculpting style. A spherical sculpture (s) is simpler and more traditional, whereas a more intricate shape (d) requires advanced skills and techniques.

Magnetic orbital quantum number. ‘ml’ gives information about the spatial orientation of the orbital with respect to standard set of coordinate axes. For any sub-shell (defined by ‘l’ value), 2l+1 values of ml are possible and these values are given by: ml = – l, – (l–1), – (l–2)... 0, 1... (l–2), (l–1), l.

The magnetic quantum number 'ml' provides insight into how orbitals are oriented in space. Each sub-shell defined by 'l' has multiple orientations based on the value of 'ml'. For example, with 'l=1' (p orbitals), there are three possible orientations corresponding to 'ml' of -1, 0, and +1, meaning each p orbital is aligned differently in space. This orientation plays a significant role in how orbitals overlap during chemical bonding.

Imagine you're setting up chairs for a seminar in a room. The shape of the chairs (p for ‘dumbbell’ shaped) shows you where to place them (the magnetic orientation). So, if you have three types of chairs (p orbitals), you need to decide how to place each type to maximize space and interaction—a plan that could be compared to how electrons orient themselves in their respective orbitals.

Electron spin ‘s’: The three quantum numbers labelling an atomic orbital can be used equally well to define its energy, shape and orientation. But all these quantum numbers are not enough to explain the line spectra observed in the case of multi-electron atoms. Thus, in 1925, George Uhlenbeck and Samuel Goudsmit proposed the presence of the fourth quantum number known as the electron spin quantum number (ms).

The electron spin quantum number 'ms' is introduced to explain the additional energy levels seen in multi-electron atoms due to electron spin. This quantum number can take on two values: +1/2 or -1/2, indicating that electrons possess a property called spin that makes them behave like tiny magnets. In any given orbital that can hold two electrons, they must have opposite spins, which adds another layer to understanding how electrons are arranged.

Think of the spins of electrons like magnetic tops. Each top can either spin clockwise (+1/2) or counterclockwise (-1/2). If you try to fit two tops in the same small box (an orbital), one must spin one way and the other the opposite to fit! This ensures they can coexist peacefully without colliding.

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