mariush said:
Chemistry for dummies says: "The principal quantum number n describes the average distance of the orbital form the nucleus (...) Chemists sometimes call the orbital electron shells". My high school chemistry book states: "The areas around the nucleus that are likely to hold electrons, are called shells. (...) Electrons in the same shell with different energies, are called different orbitals".
How can that be correct? Isn't the energy the same for an electron in e.g. 3s and 3d the same since the principal quantum number is the same?
It's not really correct. Or rather, it's a bit oversimplified. The reality here is this:
The state of an electron in an atom is an orbital, each orbital represents a unique pattern of motion of the electron (you could say 'orbits', but that's a misleading term from the QM point of view). Each orbital has a corresponding energy, although different orbitals can have the same energy. Each orbital can have two electrons (with opposite 'spin') in them.
An orbital is specified not by one but by
three 'quantum numbers': n, l, m. Which are named 'principal', 'angular momentum', 'magnetic' and 'spin', respectively. They follow the rule that n is 1,2,3..., and for each n there's an l which is an integer between 0 and n-1, and for each l there are m values which go from -l to +l.
n=1, l=0, m=0 --> 1s
n=2, l=0, m=0 --> 2s
n=3, l=1, m=-1,0,1 --> 2p
x, 2p
y, 2p
z
The principal quantum number represents the total linear momentum of the electron, in other words, its kinetic energy. The angular momentum quantum number describes its angular momentum, and the "magnetic" quantum number describes the spatial direction of the angular momentum. (hence the x,y,z) So these three numbers are required to know the energy of the electron.
The 'shell' is determined by the principal quantum number, n=1,2,3.. are the K,L,M shells and so on (note the capital letters - don't confuse the shell designations with the quantum numbers. In practice, the 'shell' concept isn't really used much, though). The orbital 'type' is determined by l. l=0,1,2,3.. are s,p,d,f orbitals. Now, you might not have seen a designation like '2p
x' before, that includes the magnetic quantum number (the subscripts). The reason is that for a single atom, it's spherically symmetric, all directions are equal, so all orbitals with the same n and l will have the same energy. So there's no need to specify it.
So, the energy is not only determined by 'n' but by 'l' too. Different orbitals in the same shell will have different energies if they have different angular momenta, which you can also see for hydrogen http://upload.wikimedia.org/wikipedia/commons/0/08/Energy_levels_of_hydrogen_atom.png" )
Now, Rydberg's formula and the Bohr model give the energies as proportional to 1/n^2. What's up with that? The answer is that they're wrong. Or rather, they only describe the energies of the 's'-states, where l=0. As for the description of the geometry, you have to be careful. Unlike the early 'planetary' models of the atom, orbitals (as opposed to 'orbits' or the visual idea of 'shells') http://chemlinks.beloit.edu/Stars/images/orbitals.jpg" , except for the s-orbitals, and electrons don't stay in a specific location, or specific distance from the nucleus.
It's true that the
average distance to the nucleus increases as n increases, but two electrons with the same n and different l values
don't share the same average distance, so the previous statement is only true when you're comparing orbitals with the same 'l'. So a shell is 'n', a 'subshell' is the set of orbitals with a given l value, and an orbital has all of n,l,m (although for reasons stated, the orbitals in a particular subshell are not always distinguished, so if you say 'excitation from the 1s to 2p level', you mean
any 2p level).
This is how 'orbitals' are usually taught in chemistry, and it's all based on analogy to hydrogen. However, hydrogen is a single-electron atom, which easily leads to some confusions. One is the difference between an orbital and an energy state of the atom. For hydrogen, these are the same. For atoms with more than one electron, they're not. The energetic state depends on the energy of
all the electrons. Since every electron interacts with every other electron, the difference between two energy levels in a many-electron atom can't be described as simply as being the difference between the two states the excited electron transitioned between. You can describe it that way as an approximation, but it's not strictly true.
Strictly, the orbital picture is an approximation for many-electron systems (or even more strictly, the picture where each electron occupies one orbital is an approximation). But from the chemistry point of view, it's a good enough approximation to qualitatively describe most chemistry. Much as viewing the solar system as a set of planets independently orbiting the sun is a good enough picture to understand the general concept of planetary motion, even though you can't treat them independently if you want an exact description of what's going on.
