In defining the sizes of orbitals, why must we use an arbitrary value, such as 90 percent of the probability of finding an electron in that region?
The choice of using an arbitrary value, such as 90 percent, to define the sizes of orbitals is based on convention and practicality. Let's understand the reasoning behind it.
In quantum mechanics, orbitals are described by wave functions that represent the probability density of finding an electron at any given point in space. However, these wave functions do not have a definite boundary where the electron is either present or absent. Instead, the probability of finding the electron decreases gradually as you move away from the nucleus.
To define the size of an orbital, we need to draw a boundary that encloses most of the electron's probability density. The choice of the boundary value is somewhat arbitrary, but it is commonly set at 90 percent for practical purposes.
Using the 90 percent boundary means that we are defining the size of an orbital to the region where there is a 90 percent chance of finding the electron. This choice ensures that we capture the majority of the electron's presence while still allowing for some flexibility to account for the uncertainty associated with quantum mechanics.
It's worth noting that the exact choice of the boundary value can vary in different contexts or calculations. For example, in some discussions, a 95 percent or 99 percent boundary may be used. The choice depends on the specific needs of the analysis and the level of precision required.