# Principal Quantum number: n to infinity?

1. Jun 28, 2012

### CAF123

I am aware that n is the principal quantum number and determines the energy of a specific energy level of an atom. In my notes, I see that n goes from 1,2,3... which implies to me all the way to infinity. If this is the case, why doesn't this imply that there can be infinitely many shells in an atom and consequently make the atom infinitely big?

I have also read about there being a 'series limit' and know that the atom is about 10^-10m big, so there must be a 'cut off' number of shells somewhere?

Many thanks.

2. Jun 29, 2012

### Many_S_Theory

It's an interesting question, if the universe was made of but one hydrogen atom, could it have an infinite number of energy levels (at absolute zero)? Well, regardless, in a hydrogen-like atom the energy of a given level goes like $$E_n=\frac{-13.6 \mathrm{eV}}{n^2}.$$ The key thing being the negative sign. The electron, when in an orbital is in a BOUND STATE (a state whose energy is LESS THAN what it would be if the electron was infinitely far away, or ionized in this case). The amount of energy then needed to "ionize" or strip that electron away is the negative energy value. Thus you can see that as n gets bigger the amount of energy required to remove the electron gets smaller like $$\propto \frac{1}{n^2}.$$ So consider two things: one, an electron at a non-zero temperature has energy associated with this temperature, and two, no atom is in complete isolation and there are always other forces perturbing this perfect system. Thus, for all intents and purposes there is indeed an effective n_max and thus a maximum radius but I'd imagine what that radius is depends on the temperature and the environment. Hope that helps.

3. Jun 29, 2012

### xlines

Yes, actually there is no reason that I'm aware of for n to be limited above and, should that be true, expectation of the electron's distance from the nucleus rises without limits, with n2 I believe. Consequence is that with n = 1 000 000 you'd have hydrogen atom larger than Earth. Of course, this states are very unstable, good starting point on this topic is wikipedia.

Reason that atom is "10^-10m big" is that electrons dominantly populate ground state.

4. Jun 29, 2012

### CAF123

Ah, ok thanks.
Perhaps it is due to the energy levels coming closer and closer together to the first ionisation energy and then subsequently to the second ,third and so on to how many applicable ionisation energies there are for certain atoms. Once we reach the last ionisation energy, the atom exists as an ion, stripped of all electrons. What is the size of an ion?

I don't understand however, 'why the closely spaced levels converge to the first ionisation energy' as Wikipedia puts it. Since as n increases, E decreases since E is proportional to 1/n^2.

5. Jun 30, 2012

### Einstein Mcfly

The binding energy is an energy DIFFERENCE between the state it starts in and the final state (dissociated or limit of n--> infinity) so as n increases the energy difference between n and n+1 gets smaller and, by definition, as n gets larger you approach the ionization energy.