Why Do 4s Orbitals Fill Before 3d as Atomic Number Increases?

[Habeebe]

Homework Statement

Explain why, as the atomic number increases, the 4s electronic states fill before the 3d states. The fact that they fill first means they are lower energy. You must explain why they are lower energy.

Homework Equations

The Attempt at a Solution

First, I'm aware that 4s is not always lower energy than 3d and they are only asking about the case where previous orbitals have been filled.

Everything I've dug up so far has been kinda vague, mentioning things like penetration and shielding. That doesn't sound like a good answer. More specifically, I've seen penetration described as the electron spending more time closer to the nucleus, however, the electron is really a standing wave, so that doesn't make sense to me.

We've only really discussed the Schrödinger Equation for the hydrogen atom, so that doesn't seem applicable here, but any reasonably simple explanation involving it would be very welcome. Thanks.

 

[Simon Bridge]

First, I'm aware that 4s is not always lower energy than 3d and they are only asking about the case where previous orbitals have been filled.

What's special about having the lower states filled?

Everything I've dug up so far has been kinda vague, mentioning things like penetration and shielding. That doesn't sound like a good answer.

Why not?

More specifically, I've seen penetration described as the electron spending more time closer to the nucleus, however, the electron is really a standing wave, so that doesn't make sense to me.

Where did you get the idea that the electron is "really" a standing wave?
It is the wavefunction that forms a stationary state - not the electron. "Spending more time close to the nucleus" means more likely to be detected close to the nucleus than far away from it ...

Compare the radial distribution of the 4s and 3d states.
What is the average distance of the electron from the nucleus in these states?

Have you had a look at what shielding may be doing?

What happens to the spacing of atomic energy levels as you increase n?
What does angular momentum do to each energy level?

We've only really discussed the Schrödinger Equation for the hydrogen atom, so that doesn't seem applicable here, but any reasonably simple explanation involving it would be very welcome.

It is, sort-of, applicable.
The pattern of energy levels for atoms in general is similar to that for hydrogen.

 

[DrClaude]

Have a look at this thread: https://www.physicsforums.com/showthread.php?t=712360

 

[Habeebe]

I discussed it with my professor, and in conjunction with your replies it makes a lot more sense now. Thanks.