Chapter 5 - Electronic Structure of Atoms and Ions

17/10/26

what is r (radius)?

the distance between an electron and the nucleus

what is coulomb’s law

V_(r) ∝ (q₁q₂) / r, where

• V_(r) is the potential energy

• q₁ is the charge of particle one (likely an electron or number of protons)

• q₂ is the charge of particle two (likely an electron or number of protons)

• r is the distance b/w the particles

can also be written as V(r) ∝ - Z / r,

this form is made for atomic units, where the charge of particle one is that of an electron (i.e. amu makes the charge = -1) and charge 2 (the only visible one) is that of the nucleus (Z)

we understand more of the s equation now!!!

the s equation for an atom contains terms representing the kinetic energy (what we looked at with the particle in a box) and the potential energy term describes the attraction between the electron to the nucleus. additionally this part of the s equation depends on three coordinates because the electron moves in three-dimensional space while being subject to the coulombic attraction to the nucleus

solving the s equation

it can only be solved analytically for one-electron atoms and ions. analytic solution are obtained by solving a system of equations on paper.

for atoms/ions with more than one electron, scientists use computers to determine very good approximations.

in either vase, since V depends on r, atoms have a spherical shapes, ergo we use spherical polar coordinates

solutions for one-electron-species, Ψ, that satisfy the schrodinger equation are called the atomic orbitals

wavefunctions as atomic orbitals

wavefunctions that satisfy the schrodinger equation for one-electron-species can be written in polar coordinates as a product of two parts:

Ψ(r, θ, φ) = R(r) Y(θ, φ)

the radial part:

• R(r) depends only on r and describes the size of the orbital

the angular part:

• Y(θ, φ) is a function of only the angles θ and φ, and corresponds to the shape and orientation of the orbital

atomic orbitals as spherical polar coordinates

r = distance from centre (nucleus)

θ = is the angle of radius line, but on the 3-D vertical Z-plane. it ranges from 0 to π, where 0 is straight upwards, and π is straight down

φ = the angle of radius line but on the horizontal x-y plane. it ranges from 0 to 2π in a flat circle.

quantum numbers

the solutions for the s equation for the one-dimensional particle in a box could be described with only one letter, n. but for one-electron atoms and ions, three quantum numbers (n, ℓ, m_ℓ) are needed to label each wavefunction. each set of three quantum numbers describes one solution of the s equation. they have restrictions to their values though

what does n mean?

n = principal quantum number

• allowed numbers: integers 1- infinity

• also referred to as a shell

• describes:

  • the relative energy of the particle

  • the radius of the orbital

  • the number of nodes. still (n-1) in total

what does ℓ mean?

ℓ = the angular momentum quantum number

• allowed values: integer 0 to (n-1)

• also referred to as a subshell

• describes:

  • the 3D shape of orbitals

  • because it determines the number of angular nodes in an orbital

what does m_ℓ mean?

m_ℓ = magnetic quantum number

• allowed values: integers -ℓ to ℓ

• describes:

  • specific orbital in a subshell

  • number of orbitals in a subshell

what are the possible values for the quantum number n?

the principle quantum number can be any positive (non-zero) integer, just like it was for particle in a box

what the fuck is a node?

just like in chapter four, a node is a place where the electron will not exist/never be. they come from points around the nucleus where the wavefunction will equal 0.

there are two types: radial and angular

radial nodes are nodes that result from the r coordinate/value of the orbital

angular nodes are nodes that result from the θ or φ coordinate/value of the orbital

for any given wavefunction:

• total # of nodes = n-1

• # of angular nodes = ℓ or (n-1)

• # of radial nodes = (n-1) - ℓ

how to find nodes based on the wavefunction of an orbital?

1. look at the function

2. identify where the variables are: (r, θ, φ)

3. do things to them that’ll make this part of the func = 0

   1. typically the angular parts will be attached to a cos or sin, so think about what values would make these = 0

   2. but remember the value limits on θ

   3. radial parts will be attached to a constant as part of σ = Zr / a_o , so it’ll usually just have to be r = 0????

what are the possible values for the quantum number ℓ?

for a given n, the angular momentum quantum number takes on all interger values b/w 0 and n-1

ex: n=5, ℓ= [0, 4]

what are the possible values for the quantum number m_ℓ?

for a given ℓ, the magnetic quantum number takes on all integer values between ℓ and -ℓ

ex: n = 5, ℓ could be [0, 4], therefore m could be [-4, 4]

subshells and ℓ

• ℓ = 0 means subshell s

• ℓ = 1 means subshell p

• ℓ = 2 means subshell d

• ℓ = 3 means subshell f

because ℓ values are associated with subshells, what do they tell us?

because a quantum particle can have multiple ℓ values, we know which subshells that particle has.

a particle will always have n many subshells.

as well, there are always n many values of ℓ,

and for each value of ℓ, there are 2ℓ + 1 values of m_ℓ

how many orbitals are in a shell?

there are n² orbitals in the n^th shell. so the 2nd shell would have 4 orbitals innit

what are shells and subshells?

each value of n is a shell is referred to by chemists as a shell

and each value of ℓ is a subshell (remember 0 = s, 1 = p, d = 2, 3 = f, and 4 = g and so on and so forth)

the principal quantum number and angular momentum quantum number together represent a specific type of orbital (subshell, like 1s, 2s, 2p, 3d, and so on)

mid-point summary

• each orbital is described by a set of quantum numbers, n, ℓ, m_ℓ

• the principle quantum number, n, specifies the shell and size of the orbital. the possible values of n are 1, 2, 3… infinity

• there are n orbitals in n^th

• an orbital has (n-1) total nodes (# total nodes = # angular nodes + # radial nodes)

• the possible values of ℓ are ℓ = 0, 1, 2, 3, … (n-1)

• for each value of ℓ, there are 2ℓ + 1 orbitals

  • that is, p orbitals come in sets of three, d orbitals come in sets of five, f orbitals in sets of seven

  • within these sets, each orbital has a unique value of m_ℓ

• the magnetic quantum number m_ℓ, specifies the orientation of the orbital and possible values range from -ℓ to +ℓ

• the probability of finding an electron at a particular location is proportional Ψ², and is zero at a node

• the phase of the wavefunction changes across every node

what the fucking fuck are these graphs bro?

NOTE: all of these graphs have distance for the nucleus, or radius, as their x-axis, and these are all for the 1s orbital, and are showing things about the RADIAL

the top one is the radial portion of the wavefunction for the 1s orbital

the middle is the square of that function, so the probability distribution of finding an electron at a point in space.

then, the third is that radial probability distribution multiplied by the area of a sphere, as the 1s orbital is spherical. what it demonstrates is the radial probability distribution of finding an electron in the spherical area around the nucleus (the first shell), based on the electron’s distance from the nucleus

s orbitals and probability distributions

1. all s orbitals have ℓ = 0 and therefore have no angular nodes

2. basically, we know all s orbitals to just be spheres, which means that they don’t have angular nodes, just radial nodes

3. the probability distribution of s orbitals varies with r only, which means they are spherical in shape.

• based on these graphs, there are a few trends:

• as n increases, the distribution shifts to larger r, and the most probable values of r increases. WHICH MEANS, that increasing shell number means a greater probability of finding the electron further from the nucleus. however, this isnt always true, look at the little peaks.

• the number of raidal nodes is n - 1, since there are no angular nodes in s orbitals. we dont count r = 0 as a node just as we didnt count the ends as nodes for wave-in-box. nodes increase as n does too.

• the mathematical equations for all the wavefuncs have an exponential function that causes the probability curves to approach 0 exponentially forever. therefore, this means the probability extends to infinity, but the likelihood of finding something infinitely far from the nucleus is effectively zero. not that this doesn’t count as a node either

trends for s orbitals

• based on these graphs, there are a few trends:

• as n increases, the distribution shifts to larger r, and the most probable values of r increases. WHICH MEANS, that increasing shell number means a greater probability of finding the electron further from the nucleus. however, this isnt always true, look at the little peaks.

• the number of raidal nodes is n - 1, since there are no angular nodes in s orbitals. we dont count r = 0 as a node just as we didnt count the ends as nodes for wave-in-box. nodes increase as n does too.

• the mathematical equations for all the wavefuncs have an exponential function that causes the probability curves to approach 0 exponentially forever. therefore, this means the probability extends to infinity, but the likelihood of finding something infinitely far from the nucleus is effectively zero. not that this doesn’t count as a node either

ANGULAR probability graphs for s orbitals, based on different elements

s orbitals are incapable of having angular nodes

note how as Z increases (Z = atomic number i.e. # of protons), the probability decays faster with r. this means that the more protons in the nucleus, the more likely it is to find the electron near it (cuz of the attraction b/w them (coulomb’s law) in s orbitals

drawing orbitals

to make them graphically, we find the value of r at which the area under the probability distribution curve = 90% of the total.

from there you can make it into a 3D representation of the boundary surface (the outer edge) or a 2D one (a cross-section of the 3D one)

the 2-D ones are capable of showing nodes inside of the shell because its a cross section