Download One more thing: radial probability distributions Angular functions:

109
One more thing: radial probability distributions
There are two different ways to view the radial probability of the electron in the hydrogen atom.
In the first, we view the distribution in one dimension. Let’s take the 1s function as an example:
ψ1s(r) 2 =
1 −2r / a 0
e
πa 30
This function has a maximum at the origin (r = 0), and decreases exponentially with increasing
distance.
In one dimension, the probability of finding the electron in an element of length dr at distance r
from the origin is given as
ψ1s(r) 2 dr
In three dimensions, the element of length dr sweeps out a spherical shell as θ sweeps out its full
range from 0 to π, and φ varies from 0 to 2π. The probability of finding the electron in that shell,
with volume dV = 4πr2dr is just
ψ1s(r) 2 dV = ψ1s(r) 2 4πr2dr
For the one-dimensional case, dr does not vary as a function of r. However, dV = 4πr2dr does
vary with r. The shell volume goes to zero as r → 0.
1-D
3-D
Angular functions:
When the l quantum number is non-zero, the orbitals have an angular dependence. We will
learn that this angular dependence ultimately determines the shapes of molecules
110
l = 1: these are p-orbitals. Possible values of m l are –1, 0, +1
The p-orbitals look like the following:
Notice that the three p-orbitals only differ in spatial orientation. These orbitals point along the
coordinate axes. Alternate lobes have opposite signs. Each orbital has one nodal plane.
l = 2: these are d-orbitals. Possible values of m l are –2, –1, 0, +1, +2
Notice that the five d-orbitals (almost) only differ in spatial orientation. Alternate lobes have
opposite signs. Each orbital has two nodal planes.
Finally, it is possible to have orbitals with l = 3: these are f-orbitals. Possible values of m l
are –3, –2, –1, 0, +1, +2, +3
111
Notice that the seven f-orbitals (almost) only differ in spatial orientation. Alternate lobes have
opposite signs. Each orbital has three nodal planes.
Remember that in the one-electron atom, there is no dependence of the energy on the angular
momentum quantum number.
In many-electron atoms, the energy of the atom does depend on the l quantum number.
Many-Electron Atoms
When we begin to examine the structures of atoms with more than one electron, it is important to
understand how the electrons interact in such an atom. First of al, consider the potential energy
of the system. Being able to formulate this in the Bohr theory was impossible. In the wave
mechanics picture, we can account for all interactions.
112
-e
r12
•
• -e
r2
•
r1
+Ze
Schematic diagram of Coulomb
interactions for the potential energy
The diagram, for two electrons, shows how we can account for the attractions of electrons 1 and
2 ( • ) with the nucleus ( • ), and the electron-electron repulsion
Potential energy = −
Ze 2
Ze 2
e2
−
+
4πε 0 r1 4πε 0 r2 4πε 0 r12
The last term makes exact calculations of the energy impossible, but the Self Consistent Field
(SCF) model is a good approximate method and says that electron 1 interacts with the
spherically-averaged field of electron 2. More generally, if the atom has N electrons, the SCF
model says that any given electron interacts with the spherically averaged field of the remaining
N-1 electrons.
So, the presence of additional electrons changes the orbital energy structure. The biggest change
is that for a given principal quantum number,
Ens < Enp < End < Enf
The diagram below illustrates the relationships for the n = 3 orbitals.
3p
3s
3d
113
A similar situation holds for the 2s and 2p orbitals. For many-electron atoms, we want to place
the electrons in orbitals according to energy as outlined above, but there is one additional idea
that we must incorporate. Electrons are observed to have spin, and there are two possible values
that the spin can have:
spin “up” or the ms quantum number is assigned a value of + ½
spin “down” or the ms quantum number is assigned a value of -½
The Pauli Exclusion Principle states that no two electrons on the same atom can have the
same four quantum numbers (n, l , m l , ms)
We can build up electron configurations by adding electrons to orbitals in accordance with
energy considerations and the Pauli Principle. This procedure is called the Aufbau Principle.
So, the ground state of the hydrogen atom corresponds to the electron configuration (1s)1
The four quantum numbers are (100½). The second electron, to make the helium atom, has
quantum numbers (100– ½), and we indicate the electron configuration as (1s)2. The Pauli
Principle does not allow a third 1s electron, so we must go to the n = 2 level.
We can create the following atoms: Li
Be
(1s)2(2s)1
(1s)2(2s)2
Since the 2s level is now filled, we must go to the 2p level, which lies higher in energy than the
2s level. More on this in a moment.
All three of the 2p levels are degenerate, so we can choose only one for the first 2p electron. We
construct boron in this way:
B
(1s)2(2s)2(2px)1
2p
2s
The next atom is the carbon atom. Where to put the electron? Hund’s Rule tells us that the
minimum energy configuration is achieved by placing the second 2p electron in a different 2p
orbital with the same spin as the first electron.
C
2p
2s
(1s)2(2s)2(2px)1(2py)1
114
Hund’s Rule is based on the idea that two electrons with the same spin are more effective in
minimizing their repulsive energy – they “avoid each other” more effectively.
Now, let’s add the seventh electron to make nitrogen:
(1s)2(2s)2(2px)1(2py)1(2pz)1
N
2p
2s
This half-filled subshell is especially stable. The energy to remove an electron (the ionization
potential) is particular high for species like this. Half-filled subshells and filled subshells are
spherically symmetric.So, now the next electron, to form O, must pair spins.
O
(1s)2(2s)2(2px)2(2py)1(2pz)1
2p
2s
Similarly, the electron configurations for F and Ne result:
2p
2p
2s
2s
F
Ne
Note that the neon atom has now filled the n = 2 shell.
Sometimes, you will see the electron configuration (1s)2 denoted [He], and the electron
configuration [He](2s)2(2p)6 denoted [Ne]. These configurations are referred to as the “helium
core” and the neon core” respectively.
Screening and the Sodium Atom
The sodium atom has the electron configuration [Ne](3s)1. Now, the sodium atom has a nuclear
charge of +11, and 10 electrons in the neon core. Suppose the 3s electron in sodium was always
outside of the electron cloud of the 10 neon electrons, as in the following diagram:
115
Nucleus, charge = +11
Core electron
distribution, -10 e
+11
Valence electron
Note that the valence electron in this diagram never penetrates the core. In real atoms, thtis not
the case. The following diagram shows that the penetration of the core is greratest for the 3s
electron, less for the 3p, smaller yet for the 3d. This is why
Ens < Enp < End < Enf
Penetration by the 3p electron
Penetration by the 3s electron
116
This diagram indicates the nature of the “penetration” in many-electron atoms.
Perfect screening
3p
3d
3s
We can continue the Aufbau principle to the n = 4
levels. The diagram at the left is a use mnemonic
device for determining which orbital is lower in
energy.
Finally, the next diagram illustrates the basic
structure of the Periodic Table.
117
The Periodic Table and Periodic Law
Mendelev was the first to make correlations of trends in chemical reactivity. The Periodic Law
states that “Chemical properties of the elements are a periodic function of atomic number.” The
structure of the table as interpreted by electron configurations tells us why.
The next four plots show a number of properties of the elements as a function of their positions
in the Periodic table:
Ionization energy:
Across a given row of the Periodic Table, the increase in ionization potential is largely a function
of increased nuclear charge. The fine structure arises from special stabilities of filled and halffilled subshells.
118
Electron affinities:
The electron affinity is defined (in the real world, not necessarily the world of the “Zumster”) by
the follwioing relationship:
A + e- → A-
EA = -∆H
So, the vertical scale of the following plot should be reversed in sign to be consistent with
essentially everyone else in the entire world:
119
Atomic size: