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Can electrons be found anywhere within the space described by a 3D orbital "90% of the time" (as stated in my textbook)? But that would mean they can be found right next to the nucleus or in the space of a lower energy level "90% of their time" (since the spheres and other shapes overlap starting right next to the nucleus).

We also know that electrons cannot jump from one energy level to another without absorbing or releasing energy and that they maintain a specific average distance from the nuceleus as denoted by a 2D depiction of their electron shells in the form of concentric circles (also written in the same textbook).

Even if they only skimmed the surface of the 3D spheres or dumbell shaped orbitals, the latter, e.g. a 2p orbital, still starts close to the nucleus while it should be farther away from the nucleus than a 1s orbital.

PS Thank you for your answers. But please elaborate on them since I did not understand at all. I just finished high school but am extremely curious and confused by the new information we are studying, and I want to understand...

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    $\begingroup$ Note that even the Bohr model, when modified to adapt electron orbital angular momentum quantization, stopped using the original constant radius of circular electron orbits. It was replaced by elliptical orbits. The the whole Bohr model is already for long time obsolete after creation of the quantum atomic model, based on Schrödinger wave equation. It is used only for education purposes in context of history of science. $\endgroup$
    – Poutnik
    Jan 13 at 14:50

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Yes. The electron has a small but not zero possibility to stay quite near the nucleus, and for example at a smaller distance than the traditional radius of the first 1s orbital. The probability of finding an electron is a number that looks a bit like the deviation of a vibrating rope fixed on the nucleus and going to an infinite distance, if all its points are attracted by the nucleus. Hmmm ! Beg the pardon from all real scientists !! It is not like this. It just looks like.

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Classical analogies about simple orbits don't describe electron behaviour well

Thinking about electrons as having "orbits" is rarely helpful as their behaviour only makes sense in a quantum mechanical (QM) approach. In QM it is meaningless to talk about position and velocity separately and you can't know both precisely at the same time.

Orbitals describe the probability of finding electrons in a particular region of space. And, indeed, in some types of orbitals (eg 1s) there is finite probability of the electron overlapping the nucleus (2p orbital have a node there so this isn't true for them): this matters for some other quantum effects involving nuclei and electrons. But the probability described by the shapes of orbitals is not uniform, some areas "inside" the area shown in most pictures have a higher probability density than others.

So saying that "they maintain a specific average distance from the nucleus" is not really correct: That is a classical analogy that is a poor description of what electrons do. The average distance is a vague way to describe the average of the probability cloud of electron location, but the idea that a distance is "maintained" contradicts the quantum picture.

And the picture involving orbits where being further from the nucleus is "higher energy" is far too simple. The energy is a function of the whole probability distributions, not the specific average distance. The electron location clouds of different orbitals do overlap.

Classical analogies about electron behaviour are not usually helpful. Unfortunately, QM pictures are often not simple until you have reached a certain ability to the complex mathematics behind QM. Until then "fuzzy clouds of probability" are the best you can do.

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  • $\begingroup$ Classical analogies are not intended to describe electron behavior. They just point out some shared principles, while omitting the difference. Otherwise, they would not be analogies, but identities. Both work with potential and kinetic energies. $\endgroup$
    – Poutnik
    Jan 13 at 14:33
  • $\begingroup$ Thank you matt_black. $\endgroup$
    – Falak
    Jan 13 at 16:11
  • $\begingroup$ @Poutnik That might not be the intent of classical analogies, but that is how many untutored people read them, causing confusion. $\endgroup$
    – matt_black
    Jan 13 at 16:49
  • $\begingroup$ The essential thing is to know the general purpose of any analogy. $\endgroup$
    – Poutnik
    Jan 13 at 17:51
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Can electrons be found anywhere within the space described by a 3D orbital "90% of the time" (as stated in my textbook)?

Yes, they can.

But that would mean they can be found right next to the nucleus or in the space of a lower energy level "90% of their time" (since the spheres and other shapes overlap starting right next to the nucleus).

Yes, it means that. There is intense overlapping of orbitals.

Particular points of 3D space around a central force do not belong to any particular energy level, being it around an atom nucleus or a star, because the sum of potential and kinetic energy must be considered.

By other words, the electron energy level does not determine the electron distance to nucleus. It only determines the probability of being at that distance.

The direct analogy is in planetary orbits around the Sun. The Earth is about $\pu{150000000 km}$ away from the Sun on near circular orbit with orbiting speed near $\pu{30 km s-1}$. But Earth could have at the same distance higher speed up to $\pu{42 km s-1}$, being at its perihelion (being the closest to the Sun on an elliptic orbit). Or a much lower speed, being at its aphelion (The most distant point of an elliptic orbit).

Consider classical gravitational analogy of the probe orbit. The mechanical energy alone(+) of the probe determines the farther distance from the planet the probe can reach. The closest distance(+) is not limited ( but by the radius of the planet ).

Similarly for orbitals. Electrons from all s orbitals can all occur near nucleus, as their zero orbital angular momentum is no limitation. 6s electrons of gold move at relativistic speed there, leading to orbital energy shift and color of gold.

(+) - It can be further limited by the conservation of the probe orbital angular momentum.

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  • $\begingroup$ If I got it right, from the example of Earth at its perihelion and aphelion, the Earth increases its speed close to the sun and decreases it away from the sun so that the sum of potential and kinetic energies remains the same... Similarly electrons increase their speed when they come close to the nucleus and decrease it away from it. Is that right? Also considering 1s and 2s orbitals... do they overlap or does 2s begin where 1s ends? Because in the former case an electron would positively be jumping to a lower energy level for most of its time which is not possible... $\endgroup$
    – Falak
    Jan 13 at 13:37
  • $\begingroup$ But if the lower distance is not limited, why would electrons need to release energy to jump to a lower energy level... $\endgroup$
    – Falak
    Jan 13 at 13:40
  • $\begingroup$ What I wanted to point out it could have higher or lower energy than it has now and still be at some moment and with some speed at the same distance as is Earth now. All ( or near all? ) orbital overlap. $\endgroup$
    – Poutnik
    Jan 13 at 13:42
  • $\begingroup$ Jumping between energy levels is not jumping between 2 different places. It means the original and new orbital overlap at that point, both with significant probability density at the3D region where the electron is. The energy change is via kinetic energy change ( together with angular momentum change) $\endgroup$
    – Poutnik
    Jan 13 at 13:42
  • $\begingroup$ Ok so when an electron enters a lower energy level it might increase its kinetic energy or as we know, it might release energy to stay in that shell..? Both things are possible? $\endgroup$
    – Falak
    Jan 13 at 16:10
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[OP ...] that would mean they can be found right next to the nucleus

They are close to the nucleus at times. The nucleus is tiny compared to the dimensions of a bond (or an atomic radius), so the electron can come pretty close. When it comes too close, it can get captured (electron capture, for certain radioactive nuclei), or gets scattered (by forces that only act at very short range and are usually ignored in chemistry because these are very rare events).

[OP ...] or in the space of a lower energy level "90% of their time" (since the spheres and other shapes overlap starting right next to the nucleus).

The most powerful way to explain this is by treating electrons as a wave (you might have heard orbitals mentioned together with wave function). Just as we can hear multiple musical notes at the same time in the same place, electrons can occupy the same space. The technical term is superposition of waves. Below are two standing one-dimensional waves, one with a single node and the other with four nodes, superimposed (amplitudes are added).

enter image description here

Even though the two waves are using the same space, they are independent of each other, and you can separate them mathematically (because they have different wavelength and frequency). Our ears do the same thing with music, taking a single input stream and separating out the different pitches of musical notes and overtones.

The waves describing electrons are more complicated (they are three-dimensional and have an exponential decay away from the nucleus rather than ending at a defined position), but the idea of superposition still holds. Here is a depiction of the 3p wave function (a snapshot, not showing the time-dependency like I did above), showing various nodes (zero probability) and that there is no defined boundary to where electrons might be found (exponential decay means it gets very unlikely for large distances from the nucleus).

enter image description here

Source: https://mathematica.stackexchange.com/questions/32378/is-there-something-like-densityplot3d-to-visualize-atomic-orbitals

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  • $\begingroup$ Thank you so much! This was very helpful... However, I didn't quite understand what you meant by "three dimensional waves," what they are or how to visualize them.. Could you please elaborate on that? Does that mean an electron vibrating in .. a certain way? Or.. the 3D shape of the electron itself? Does the three dimensional nature of these waves have anything to do with the three dimensional space occupied by the electrons? $\endgroup$
    – Falak
    Jan 13 at 21:05
  • $\begingroup$ You lost me with "exponential decay away from the nucleus...." too. Didn't understand the last paragraph actually.. $\endgroup$
    – Falak
    Jan 13 at 21:09
  • $\begingroup$ @Falak Just to be sure, have you read Wikipedia: Atomic orbital ? Or similar searched for on Libretexts.org or hyperphysics.phy-astr.gsu.edu $\endgroup$
    – Poutnik
    Jan 13 at 21:24
  • $\begingroup$ @Falak 1 dimensional waves are like the vibrations of a violin string. Most people are very familiar with those. But 2D surfaces also have vibrational patterns, but more complex ones (search for Chaldni patterns to see examples). 3D waves are even more complex: and the type represented bay electron orbitals are known as spherical harmonics which is also worth searching for. Different orbitals are, basically, spherical harmonics. $\endgroup$
    – matt_black
    Jan 13 at 21:47
  • $\begingroup$ 3D waves are hard to visualize - we run out of dimensions. Sound is a good example, though, and there are some cool effects like echo chambers or the Doppler effect you can appreciate without extra math. The deeper questions you are asking, though, require quite a bit more math for solid answers. BTW: The square of the wave function is proportional to finding the electron in a given spot, so there is a direct relationship. $\endgroup$
    – Karsten ♦
    Jan 13 at 22:23

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