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CH225.2
Today:
•Atomic structure
•Quantum numbers and orbitals
•Shapes of atomic orbitals
Chemists are molecular architects & builders.
Atoms are the building blocks – the ‘bricks’ - of chemistry.
Electrons hold the atoms bonded like glue or mortar.
nuclei + core electrons
valence electrons
Chemists pay a lot of attention to the electrons in molecules.
We follow the electrons to understand and control formation or cleavage of
chemical bonds a chemical reaction.
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Hydrogen atom
The Bohr model
A model of the hydrogen atom was proposed by the Danish physicist Niels Bohr in
1913. He suggested that the atom of hydrogen consists of a negative electron
revolving around the positive nucleus - proton.
me
r
-central nucleus of mass mn
-electron of mass me
r -distance between the electron and nucleus
orbit
mn
The total energy E of the electron is a function of the distance between the electron
and nucleus, r :
2
E=-
e
8r
constants:
e -electron charge
 -permittivity of vacuum
(Note the –ve sign of E, meaning energy is released when e approaches the nucleus)
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The principal quantum number, n
Bohr had to place an important constraint upon his model. To make it agree with the
experimental properties of atomic hydrogen, he suggested that only certain distances
r (orbits) were allowed for the electron. Hence, the electron energy E is ‘quantized’.
h2
ao =
mee2
r = aon2
E=
-mee4
82h2n2
ao = 1 a.u. ≈ 0.53 Å
n = the principal quantum number = 1, 2, 3....
The relative sizes of orbits:
n
n2
r, Å
1
1
0.53
-0.00022
2
4
2.12
-0.00005
3
9
4.76
-0.00002
E, J
n=3
n=2
n=1
1
4
9 n2
Under normal conditions, the electron occupies the lowest orbit (n = 1) in atomic
hydrogen, at the distance of ca. 0.53 Å from the nucleus.
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Particle or wave?
Electrons and other atomic particles should not be regarded as little hard lumps of
matter like small billiard-balls.
Electrons have properties of both particles and waves. For example, they can be
weighed (a property of a particle) and they can be diffracted (a property of a wave).
The particle-wave duality was expressed by de Broglie. He has postulated that all
matter possesses characteristics of both waves and particles; the wavelength can be
expressed as a function of mass and velocity:
The wave properties become significant only for very small particles. Consider a
cricket ball, mass m = 0.15 kg moving at v =160 km/h = 44 m/s. Substituting into the
above equation gives the wavelength of the associated wave as ca. 10-34 m. For
comparison, the diameter of an atom is ca 10-10 m. A distance as short as 10-34 m is
meaningless.
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From orbits to orbitals.
The uncertainty principle formulated by Heisenberg tells us that the exact position of
the electron in a hydrogen atom can not be accurately determined.
Only, the probability (likelihood) of finding the electron somewhere around the
nucleus can be calculated.
If the probability at some point is high, then the electron density is said to be high at
that point.
The electron density representation describes an orbital, the term related to an orbit
in the Bohr model.
Orbital
80 % probability
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Schrödinger's wave function, ψ
The novel quantity ψ is a wave function that can be obtained for a particle, such as
electron, by solving Schrödinger's equation. For the special case of a particle moving
in one direction (along x) it has the form of a 2nd order differential equation:
h2 d2
+ (EV) = 0
2
2
 m dx
where E is the total energy (constant), V is the potential energy.
The probability of finding the particle in a region between x and x + dx is ψ2dx. The
total probability of finding the particle moving along x must be exactly 1 and can be
determined by integrating over all x:


2
dx  1

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Schrödinger's wave equation
Schrödinger’s equation was solved for the hydrogen atom in 1926 . The mathematical
details are not easy, and Schrödinger himself needed advice from colleagues before
he could obtain the solution. The following qualitative features are important:
In the case of the hydrogen atom, ψ is a function of three Cartesian coordinates x, y,
and z, or alternatively, three polar coordinates r, Θ, φ. As a consequence, the
solution of the Schrödinger's equation for hydrogen involves 3 quantum numbers:
n, the principal quantum number, can take the values 1, 2, 3...
l, the angular momentum quantum number, can take the values 0, 1, 2... up to n – 1
ml, the magnetic quantum number, can take the values -l, -(l-1),..., 0, ..., (l-1), l
The electron energy depends only on n, and is given by the same equation as in the
theory of Bohr :
E=
-mee4
82h2n2
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Quantum numbers and orbital names
Any valid combination of n, l, and ml defines a unique electron orbital. The orbitals are
named after the quantum numbers. The first part of the name is the principal quantum
number n. The second is linked to the angular momentum quantum number l:
1st shell
2nd shell
3rd shell
The principal quantum number n describes a shell.
There is one orbital in the first shell, four in the second, and nine in the third.
Each set of orbitals within a shell for which l is the same is termed a subshell.
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Electron spin
Solutions of the Schrödinger equation are 'exact' in the mathematical sense. They
are not quite 'correct', however. The errors are very small for the hydrogen atom, but
become more significant for heavier atoms. Partly, this is due to the assumption that
only the electron moves, the nucleus being stationary.
In 1928, Dirac developed an equation which combined the wave theory with
relativity. For atoms with small nuclear charges, it gives results which are numerically
very similar to those from Schrödinger equation. Importantly, Dirac demonstrated that
electron has an angular momentum and resembles a tiny bar magnet. This property,
called spin, was known experimentally in 1928 but was unaccounted in the
Schrödinger equation. The spin of electron is I = ½, and there are two spin states
corresponding to the quantum numbers I = -1/2 and +1/2.
Therefore, the state of any electron in an atom is described by four quantum
numbers: n, l, ml, and I, which combinations should be different for all electrons in an
atom. This is the Pauli exclusion principle.
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Shapes of atomic orbitals
Atomic orbitals are typically shown as boundary surfaces. They include the regions
where the electron is very likely (e.g. 90% probability) to be found. Different
shading/colors can be used if the wave function changes its sign.
All s orbitals are spherical:
1s
The three 2p orbitals all have the same shape; they are directional and orthogonal:
2px
2py
2pz
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