Question

Dealing with Quantum Theory and Atomic Structure. Don’t understand how to do.

HW: Complete Table Shell # Subshell Orbital

Answer 1

each electron in an atom is described by four different quantum numbers. The first three (n, l, m_l) specify the particular orbital of interest, and the fourth (m_s) specifies how many electrons can occupy that orbital.

1 Principal Quantum Number (n): n = 1, 2, 3, …, ∞
Specifies the energy of an electron and the size of the orbital (the distance from the nucleus of the peak in a radial probability distribution plot). All orbitals that have the same value of n are said to be in the same shell (level). For a hydrogen atom with n=1, the electron is in its ground state; if the electron is in the n=2 orbital, it is in an excited state. The total number of orbitals for a given n value is n².

1. Angular Momentum (Secondary, Azimunthal) Quantum Number (l): l = 0, ..., n-1.
   Specifies the shape of an orbital with a particular principal quantum number. The secondary quantum number divides the shells into smaller groups of orbitals called subshells (sublevels). Usually, a letter code is used to identify l to avoid confusion with n:

l	0	1	2	3	4	5	...
Letter	s	p	d	f	g	h	...

The subshell with n=2 and l=1 is the 2p subshell; if n=3 and l=0, it is the 3s subshell, and so on. The value of l also has a slight effect on the energy of the subshell; the energy of the subshell increases with l (s < p < d < f).

1. Magnetic Quantum Number (m_l): m_l = -l, ..., 0, ..., +l.
   Specifies the orientation in space of an orbital of a given energy (n) and shape (l). This number divides the subshell into individual orbitals which hold the electrons; there are 2l+1 orbitals in each subshell. Thus the s subshell has only one orbital, the p subshell has three orbitals, and so on.

1. Spin Quantum Number (m_s): m_s = +½ or -½.
   Specifies the orientation of the spin axis of an electron. An electron can spin in only one of two directions (sometimes called up and down).
   
   The Pauli exclusion principle (Wolfgang Pauli, Nobel Prize 1945) states that no two electrons in the same atom can have identical values for all four of their quantum numbers. What this means is that no more than two electrons can occupy the same orbital, and that two electrons in the same orbital must have opposite spins.
   
   Because an electron spins, it creates a magnetic field, which can be oriented in one of two directions. For two electrons in the same orbital, the spins must be opposite to each other; the spins are said to be paired. These substances are not attracted to magnets and are said to be diamagnetic. Atoms with more electrons that spin in one direction than another contain unpaired electrons. These substances are weakly attracted to magnets and are said to be paramagnetic.

   n { shell}	l {subshell}	$m_{l}$	no.of orbitals	no.of electrons	orbital name
   4 (N shell)	0	0	1	2	4s
   	1	-1,0,+1	3	6	4p
   	2	-2,-1,0,+1,+2	5	10	4d
   	3	-3,-2,-1,0,+1,+2,+3	7	14	4f