In defining the sizes of orbitals, why must we use an arbitrary value, such as \(90 \%\) of the probability of finding an electron in that region?

An orbital is a region in an atom where an electron with a certain energy level is most likely to be found. They are usually represented by shapes and symbols (such as s, p, d, f) and are determined by the wave function of electrons in quantum mechanics. However, it is important to note that electrons in orbitals do not follow fixed paths, and their exact position cannot be precisely determined.

Since it is impossible to determine the exact location of an electron due to the uncertainty principle in quantum mechanics, we rely on the probability distribution of electrons. The probability distribution represents where an electron is most likely to be found by considering the wave function. In other words, electron probability describes the likelihood of finding an electron in a particular region of space.

When we attempt to represent orbitals graphically or define their sizes, we need to set a boundary to display them. The problem lies in the nature of electron probability distributions, which diminish gradually and have no sharp borders. Therefore, an arbitrary value, such as 90% probability, is needed to represent orbitals' form or size while still having most of the electron's probability density covered.

The choice of 90% probability is arbitrary, but it is widely accepted because it provides a reasonable balance between coverage and simplicity. By considering 90% of the electron density probability, orbitals can be visualized and understood relatively easily while still representing most of the electron's probability distribution. Other values could be chosen, but they might make the orbital representation too small or overly complicated to visualize and understand. In summary, the use of an arbitrary value like 90% probability when defining the size of orbitals is necessary because it allows us to represent and visualize an orbital's shape while taking into account the probability distribution of electrons to a considerable extent. It strikes a balance between simplicity and accuracy.