Quantum Numbers & Orbitals

Chem 1211, chapter 6

What are the four quantum numbers and their symbols?

• principal quantum number (n)

• angular momentum quantum number (l)

• magnetic quantum number (m_l)

• electron quantum number (m_s)

Describe the principal quantum number (n)

• n relates to the energy and the probable distance of the electron from the nucleus

• the larger the n, the further the electron is from the nucleus

• analogous to n from the Bohr model

• describes the shell where the electron is located

• n can have a positive, non-zero, whole number value

• Ex. 1, 2, 3, 4, etc.

True or false: all electrons with the same value of n are in the same principal electron shell or level

True

Describe angular quantum number (l)

• l describes the shape of the orbital where the electron is located; what does the orbital look like?

• describes in which subshell the electron is located

• l can have a non-negative, whole number value (including zero), but cannot be greater than n-1

• ex. l = 0,1,2 …, n-1

True of false: all electrons with the same value of n and l are in the same subshell or sublevel

True

Describe sub shells

number of sub shells in a principle electronic shell is equal to the number of possible l values this is equal to the value of the principal quantum number (n) itself

we assign certain letters to the sub shells with the first four values of l:

• l = 0 → s

• l = 1 → p

• l = 2 → d

• l = 3 → f

Describe orbital designation

• To designate/name a certain subshell within a principal electronic shell, we use the n value and the letter assigned to the subshell in question

• Ex. n = 4, l = 2 → 4d

Describe the magnetic quantum number (m_l)

m_l, describes the orientation of the orbital where the electron is located; what direction is the orbital pointing in? how is it angled?

m_l cab have a negative, positive, or zero whole-number value, ranging from -l to +l

• m_l = -l, …, -2, -1, 0, 1, 2, …, +l

Describe the spin quantum number (m_s)

• m_s describes the orientation of the electron “spin”; the electrons aren’t actually spinning, this is just what it’s called

• m_s can only have two values: -1/2 and +1/2

• electron spin is taken advantage of in spectroscopic techniques, such as NMR (nuclear magnetic resonance) and its sister technology MRI (magnetic resonance imaging; learn more in org

What does the Pauli Exclusion Principle state?

• no two electrons can have the same set of four quantum numbers (n, l, m_l, m_s)

• m_s can either be +1/2 (“spin up”) or -1/2 (“spin down”)

What can we determine from the Pauli Exclusion Principle?

We can know two things:

• since there are only two possible spin states, there can be a maximum of two electrons in each orbital

• also, since no two electrons can have the exact same set of quantum numbers, two electrons in the same orbital must have opposite spins; one will be spin up, the other will be spin down

What are orbitals?

• the areas around the nucleus where you are likely to find an electron

• these are the result of the solutions to the Schrödinger Equation

• the number of orbitals within a certain subshell is equal to the number of allowed m_l values for that subshell (equal to 2l + 1)

• Ex. in the 10f subshell, since f → l = 3, there are 2(3) + 1 = 7 orbitals

Describe Electron Probability

• probability of finding the electrons within a certain space

• because of the Heisenberg Uncertainty Principle the electron is “smeared” everywhere all at once, but not uniformly

• orbitals do not have sharp boundaries. the probability of finding the electron at a large distance from the nucleus is tiny, but never zero

Describe nodes

an area where the electron density is 0 inside of orbitals

as n increases, the number of nodes increases

• total number of nodes = n - 1

What are the two types of nodes?

• radial nodes are spherical; number of radial nodes = n - l - 1

• angular nodes are planes; number of angular nodes = l

• s orbitals have no angular nodes

• p orbitals have 1 angular node

• d orbitals have 2 angular nodes, etc.

Describe p orbitals

• l = 1 → p orbital

• p orbitals have two lobes of electron density on both sides of the nucleus, like a dumbell

• a node is located on the nucleus, as a plane

Describe d orbitals

• l = 2 → d orbital

• d orbitals have multiple nodes and multiple lobes

• 4 of them look like a cloverleaf with angular nodes between the lobes

• the last looks like a dumbbell with a ring, with non Cartesian planes (or cones) as angular nodes around the vertical lobes

describe f, g, h … orbitals

• past f orbitals (l = 3), the orbitals are named in alphabetical order: g, h, … etc. for l = 4, 5, … respectively

Describe Energy diagrams in a single-electron atom

• for a single-electron atom only: all sub shells with the same n are degenerate, or are equal in energy (e.g. E_3s = E_3p = E_3d for hydrogen)

• the only forces acting on the electron are kinetic energy nd the electrostatic attraction between the negative electron and the positive nucleus, so we can use the Bohr model

Describe energy diagrams in a multi-electron atom

• in a multi-electron atom, sub levels within the same principal energy level are no longer degenerate. electrons interact with each other, changing the energy levels

• E_ns < E_np < E_nd < E_nf

Describe electron shielding in multi-electron atoms

• shielding - electrons closer to the nucleus “shield” further electrons from the nuclear attraction; this causes a lower effective nuclear charge (Z_eff) on those further electrons

• because of less attraction from the nucleus, this causes higher energy

Describe electron penetration

• electrons closer to the nucleus can have a greater attraction to the nucleus

• because of the greater attraction, the energy of these electrons is lower

Describe Orbital Energies

• the closer to the nucleus, the lower the energy