Question

The greatest probability of finding the electron in a small-volume element of the 1 s orbital of the hydrogen atom is at the nucleus. Yet the most probable distance of the electron from the nucleus is \(53 \mathrm{pm}\). How can you reconcile these two statements?

Short answer

The biggest probability of finding the electron in a small volume is at the nucleus because this is where the electron’s probability density is the greatest due to the shape of the s orbital. However, the most probable distance is 53pm because, considering all the possible positions for the electron within the 1s orbital, it has a higher probability to be found somewhere in that volume around the nucleus than exactly at the nucleus.

Step-by-step solution

Understanding atomic orbitals

Atomic orbitals (like the 1s orbital in the hydrogen atom) can be thought of as regions in an atom where an electron is most likely to be found. The 1s orbital is the closest to the nucleus and, being spherically symmetrical, has the highest electron density near the nucleus

Define Electron Probability Density

Electron probability density refers to the likelihood of finding an electron in a particular region of an orbital. For the 1s orbital of a hydrogen atom, the electron probability density is maximum at the nucleus. This means that any small-volume element you choose at the nucleus will have the highest chance of containing the electron.

Explain the most probable distance

The most probable distance is where the electron can be found most of the time, considering all the possible positions in the 1s orbital. Because the electron has the whole volume of the orbital at its disposal, and due to the spherical nature of the 1s orbital, finding the electron exactly at the nucleus (a point) is actually less probable than finding it somewhere within the sphere of the orbital. This is why the most probable distance of the electron from the nucleus is not zero but \(53 \mathrm{pm}\) for a hydrogen atom.

Key concepts

Electron Probability Density

The concept of electron probability density is crucial for understanding where an electron is likely to be found in an atomic orbital. In essence, it describes how much of the electron's presence or charge is found in particular regions around the nucleus. Imagine an electron as a cloud surrounding the nucleus, where some parts of the cloud are denser. These denser regions indicate a higher likelihood of finding the electron there.
For a hydrogen atom's 1s orbital, this density is at its highest directly at the nucleus. That means, in terms of small-volume elements, the area right at the nucleus is where the electron is most densely located. However, keep in mind that this doesn't necessarily mean the electron is most probable to be found at this spot in terms of the total space it can occupy.

1s Orbital

The 1s orbital is a fundamental concept when discussing atomic structure, especially for the hydrogen atom, which only has one electron. An orbital is simply a term that means "electron cloud," and in the case of the 1s orbital, it's the closest orbital to the nucleus.
The defining feature of this orbital is its spherical shape. No matter where you look from the nucleus, the 1s orbital maintains a symmetrical shape. This symmetry is important because it creates a consistent environment where the electron probability density varies predictably.

• The electron is most dense at the nucleus.
• As you move away from the nucleus, the density decreases.
• This spherical distribution allows the electron to spread out, defining its more probable locations away from the nucleus.

Most Probable Distance

You might find it surprising that while the electron probability density is highest at the nucleus, the most probable distance for finding an electron isn't there. Instead, for a hydrogen atom's 1s orbital, this distance is about 53 pm from the nucleus.
Why is this? It's because the probability density accounts for the overall space available to the electron. While a single point at the nucleus has a high density, it's only one point. The entire volume of the spherical 1s orbital provides more locations for the electron.
Think of it like this: though the electron is densest at any single small-volume point, it spends more time on average within the larger volume bracket surrounding the nucleus, peaking at an average distance of 53 pm. This measurement considers both the probability and the physical area available to the electron, resulting in what we call the most probable distance.