Topic 3: Atomic orbitals and quantum numbers

Question 1

How is it reasoned for electrons to have discrete energy levels without having discrete orbits?

Answer 1

Electrons have wave behaviors, but as electrostatic attraction from the nucleus bound them to a restricted space/motion, the electrons can only have discrete energy levels.
Just like a guitar string can only have discrete frequencies/wavelengths.

Question 2

What is the lowest energy level for standing wave, like guitar string or electron wavefunction, called? What confines the electron wave in an atom?

Answer 2

Fundamental
Electron wave is confined by the electrostatic attraction to the nucleus.

Question 3

What is electron wavefunction?

Answer 3

Electron wavefunction is the amplitude of the electron’s matter wave.

Question 4

How did Born inteprete the meaning of wavefunction?

Answer 4

Square of the wavefunction gives us the probability of finding electron at a particular point in space at any time or the probability density.

Question 5

How does the number of nodes related to the energy of an electron wavefunction?

Answer 5

More nodes means shorter wavelength, longer frequency and thus higher energy

Question 6

Explain the differences between probability density and radial probability

Answer 6

• Probability density refers to the electron density at any given time or the probability of finding an electrons in a point in space
• Radial probability refers to the distribution of electron at all points of a sphere (s orbital) with given radius

Question 7

Why do all wavefunctions and electron density decay exponentially towards 0 as going further away from nucleus?

Answer 7

With the electrostatic attration of the nucleus, most electrons are bound near to the nucleus and as going further away from the nucleus, electrons are less bound, thus have more energy.
Most are bound near the nucleus –> electron density decays towards 0
Electrons are not bound within a perimeter –> wavefunction decays towards 0

Question 8

Explain why orbital size increases with energy

Answer 8

At higher energy level or higher n, the electrons are less bound to the nucleus by the electrostatic attration, thus they have more free energy and higher average distance from the nucleus.
Atoms with higher energy thus have more electrons spreading out –> orbital size increases

Question 9

What is the principal quantum number and why does it have to be integer?

Answer 9

Indicated the number of energy levels an atom/orbital has
Because the energy levels of an atom or orbital is in discrete levels, the quantum number has to be integers.

Question 10

Identify and describe types of nodes

Answer 10

• Linear/planar node: dividing the space/drum heads 2D representation into planes with opposite amplitude
• Circular/spherical node: concentric circles with opposite amplitude

Question 11

Explain in details what angular quantum number is, how it is calculated and what angular momentum number is for which orbital

Answer 11

Electron density in an orbital can be concentrated along preferred axes or a certain angular momentum. Angular quantum number correlated with the number of preferred axes in an orbital, thus gives the shape of electron distribution for that orbital.
l can take any value from 0 to n-1
l = 0 –> s
l = 1 –> p
l = 2 –> d
l = 3 –> f

Question 12

How does energy level, n, restrict the shape of orbital?

Answer 12

Lower energy level means the electrons space is more compact due to larger effect of electrostatic attraction of the nucleus. Thus, the distribution of electrons is more retricted and the shape is more confined.

Question 13

Explain in details what magnetic quantum number is, what limits it and how it is calculated

Answer 13

Objects with preferred axes have directionality/orientation or directional dependence which is also quantized. So, magnetic quantum number gives the orientation or number of different orbitals of a type per energy level.

The number of preferred axes limits the number of possible orientations.

m can take any integer value between -l and l

Question 14

How to calculate the number of nodes in general, and number of planar vs spherical node?

Answer 14

Number of all nodes is n-1
Number of planar nodes is l number of that orbital type, and the rest is spherical nodes.

Question 15

How to call quantum states or wavefunction with equal energy level? Give example

Answer 15

Quantum states or wavefunction with equal energy level is called degenerate.
2px, 2py, and 2pz orbitals are degenerate.

Question 16

Suggest lobe representation for 2p, 3p and 4p orbital

Answer 16

2p: 2 lobes
3p: 2 lobes with 2 spheres within
4p: like 3p but other 2 spheres within

Question 17

Apart from indicating the energy level, what does principal quantum number tell about number of possible orbital types?

Answer 17

n is the exactly the number of orbital types that can reach that energy level

Question 18

Explain in details what spin quantum number is

Answer 18

Whe a beam of atoms is ejected so that it goes through a magnetic field, some atoms are deflected in one way while some are deflected the other way. Because magnetic field is associated with spinning electric charge, electrons now have spin property.
ms = 1/2 or -1/2

Question 19

Define Pauli Exclusion Principle

Answer 19

No two electrons in the same atom can have the same quantum state or the same quantum numbers n,l,ml,ms

Question 20

Define Aufbau Principle

Answer 20

Electrons in an atoms and molecules are not usually excited, but rather stay at lowest possible energy state or ground state.

Question 21

For multi-electron model, how does energy level affected by interactions between electrons and how does Hund’s rule address this problem?

Answer 21

With more than 1 electron, electrons in the same orbital repel each other which can increase energy level of an atom. So, keeping them furthest apart by placing them in different orbitals of the same orbital type can keep the atom at lowest energy configuration.

The Hund’s rule states that to keep the lowest energy configuration in orbitals with same energy level is the one with the highest possible number of unpaired electrons with parallel spins.
Which means all orbitals of the same orbital type must be filled with parallel spins of electrons before the second electron can be put in.

Question 22

Explain orbital shielding and give example of its effects through 2s and 2p orbital

Answer 22

Orbital shielding means electrons near the nucleus shield the effect electrostatic attraction from electrons that are further away or at higher energy level.

2s orbital has 2 spherical nodes which means there’s a spherical space of its near the nucleus. 2p has planar node, so the average distance of electrons in this orbital is further, thus is better shielded by the 1s thant the 2s –> 2p has higher energy level.

Question 23

What is valence electrons and how can it be determined?
Where are atoms with the same valence electron in the periodic table and how are they similar?

Answer 23

Classical model of atom has the concept of electrons residing in “shells” and the outmost electrons, outside the noble gas core, are more easily accessible in reactions and called valence electrons.
Atoms with the same valence electron are in the same group in the periodic table and have similar chemical behavior as well as forming compounds with same stoichiometry.

Question 24

How do filled electron shells tell about activity of an atom?

Answer 24

Atoms with filled electron shells are unreactive.