Posted: Fri Oct 14, 2016 2:24 am
"How many orbitals are in subshells with l equal to (a) 0; (b) 2; (c) 1; (d) 3?"
I know that this is a fairly basic question, but could some please explain how they got to their answer in detail and the reasoning behind the answer? Thank you!
Re: Problem 2.17
Posted: Fri Oct 14, 2016 12:31 pm
In the lecture, I believe Professor Lavelle covered the topic on quantum number sets (n,l,ml,ms). So the "n" number describes the energy level of the shell when no external electric or magnetic field is around. "l" number is related to the subshell (s,p,d,f...) derived from the "n" number. The question here asks for the number of orbitals in the subshells for a given "l" number, which is the "ml" value.
To answer this question, understanding of the relationship between "l" and "ml" is a must. And a mathematical description of this is ml = +l, +l-1, +l-2, ... , 0, ... , -l +2, -l +1, -l. So how and where is the equation derived? I think that is beyond the scope of this class, but if you decide to pursue chemistry further to physical chemistry or quantum mechanics, you will find out why. For now this equation is enough to answer the question, and it makes physical sense in the following explanation.
Following the equation above, ml = +l, +l-1, +l-2, ... , 0, ... , -l +2, -l +1, -l, we see that for a given l value of 0 (s orbital), the only possible ml value is 0. What this 0 means is an orbital of ml value = 0. So in total, we have 1 orbital in the s subshell or 1 orbital in the l=0 subshell.
So okay, then let's try l = 1, we get ml = +1, 0, -1, so a total of 3 orbitals for the p subshell or l =1 subshell. Interesting that we have 1 orbital for s and 3 orbitals for p, just as the physical world's electron configuration for s orbital and p orbitals. From here, I encourage you to attempt l =2 and 3, does your answer make physical sense?
This is not a complete explanation of the l and ml quantum numbers, since there are more to the equations and derivations, but it is enough for this question.
Re: Problem 2.17
Posted: Fri Oct 14, 2016 4:57 pm
I think Chem_Mod explained it pretty thoroughly, but this diagram helps for visualizing it: