# Why do shells begin to hold more electrons?

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1. Mar 19, 2016

### Metals

So I've been told about the 2n^2 rule, and how the number of electrons held in each shell goes 2, 8, 18, 32, 50, 72, etc... But I am not aware of why shells further out from the nucleus are able to hold more electrons. Does this have something to do with spdf or energy levels?

Thank you.

2. Mar 19, 2016

### Astronuc

Staff Emeritus
For neutral atoms, the number of electrons around the nuclear must provide the same charge as the number of protons (positive charges) in the nucleus. Note that the hydrogen atoms has a number of emission lines, so there are a number of 'orbitals' that an electron can occupy, but there is one ground state.

The 'orbits' are a reflection of the quantum states of electrons around a nucleus. They are described by a set of quantum numbers that address the potential energy, angular momentum and magnetic interaction of the electrons. The principal quantum number, n, determines principal energy level, and angular or azimuthal quantum number, l (s, p, d, f) is associated with the orbital angular momentum.

http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/quantum.html

http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydcol.html

3. Mar 20, 2016

### Metals

Why is it that the 'shells' further out from the nucleus have a higher energy level? And why can higher energy levels hold more electrons?

4. Mar 20, 2016

### Staff: Mentor

Because that's the way it is?

Seriously, "why" question is not the best one to ask here. In general it is often a bad question to ask in science.

I could answer "because when you calculate, you will find out, that the energy is proportional to -1/n2, so the higher the n, the higher the energy". Does it answer the question "why"? No, it doesn't, as you can ask "why" again. "Because the math says so". "Why?". "Get lost". Do you see the problem? ;)

5. Mar 23, 2016

### Metals

I see, fair enough. I'll try found out else where some time, thanks.

6. Mar 24, 2016

### DrDu

The energy of the electrons in shells further away from the nucleus increases, as the coulombic attraction by the nucleus is weaker (it falls off like 1/r with the distance r). That a higher shell can hold more electrons is a consequence of quantum mechanics. Namely, we know from Heisenberg's uncertainty relation that an electron will occupy a certain volume in phase space (which is spanned by the ordinary spatial coordinates + the momentum values). I will try to estimate very roughly how the total volume in phase space available to the electrons varies with energy. I shall use that potential and kinetic energy are of the same order of magnitude as the total energy E. Furthermore I drop all constants which do not depend on n or E:

Equating $E_\mathrm{pot}\sim E \sim 1/n^2 \sim 1/r$ we find that the volume of a sphere containing the electrons has a volume proportional to $n^6$. Equating Kinetic energy, we obtain $E_\mathrm{kin}\sim E \sim p^2 \sim 1/n^2$ or $p\sim 1/n$. Hence the total volume in phase space increases like $p^3 r^3\sim n^3$. As n is the total number of shells up to the given energy, the volume of the shells is proportional to $n^3/n=n^2$.