Statistical Molecular Thermodynamicspollux.chem.umn.edu/4501/Lectures/ThermoVid_1_05.pdf ·...
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Statistical Molecular Thermodynamics
Christopher J. Cramer
Video 1.5
Atomic Energy Levels
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How is energy stored in an atom?
ATOMS
Electrons: Electronic energy. Changes in the kinetic and potential energy of one or more electrons associated with the nucleus.
Nuclear motion: Translational energy. The atom can move (translate) in space — kinetic energy only.
Connecting macroscopic thermodynamics to a molecular understanding requires that we understand how energy is
distributed on a microscopic scale.
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Electronic Energy levels��The Hydrogen Atom
–1.36x10-19 J
EN
ER
GY
Ground state 1=n
2=n
3=n4=n∞=n
HYDROGEN ATOM
0 J
–2.42x10-19 J
–5.44x10-19 J
–2.18x10-18 J
+ ( )∞=rSeries limit (n=∞), electron and proton are infinitely separated, i.e., there is no interaction.
+
Ground state (n=1), average distance between electron and proton is r = 5.3x10–11 m (1 bohr).
< r > = 5.3x10–11 m
M
n nε–
–
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Electronic Energy levels��The Hydrogen Atom
–1.36x10-19 J
EN
ER
GY
Ground state 1=n
2=n
3=n4=n∞=n
HYDROGEN ATOM
0 J
–2.42x10-19 J
–5.44x10-19 J
–2.18x10-18 J
n nε
€
εn = −2.17869 ×10−18
n2J = −109680
n2cm-1
integer values (n = 1, 2, 3, …)
convert J to cm-1
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Wave Functions and Degeneracy��The Hydrogen Atom
EN
ER
GY
Ground state 1=n
2=n
3=n4=n∞=n
n nε
Wave functions are atomic orbitals
The number of wave functions, or states, with the same energy is called the degeneracy, gn.
–1.36x10-19 J
HYDROGEN ATOM
0 J
–2.42x10-19 J
–5.44x10-19 J
–2.18x10-18 J
n = 1, gn = 1
n = 2, gn = 4
n = 3, gn = 9
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Electronic Energy levels��Many-Electron Atoms
There is no simple formula for the electronic energy levels of atoms having more than one electron. Instead, we can refer to tabulated data:
For electronic energy levels, there is usually a very large gap from the ground state to the first excited state. As a result, we will see that, at “reasonable” temperatures, we will seldom need to consider any states above the ground state when computing “partition functions”.
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Translational energy levels In addition to electronic energy, atoms have translational energy.
To find the allowed translational energies we solve the Schrödinger equation for a particle of mass m in a box of variable side lengths.
L,3,2,18 2
22
== nmahn
nε
The allowed states are non-degenerate, i.e., gn = 1 for all values of n.
In 3D,
⎟⎟⎠
⎞⎜⎜⎝
⎛++= 2
2
2
2
2
22
8 cn
bn
an
mh zyx
nnn zyxε
L
L
L
,3,2,1
,3,2,1,3,2,1
=
=
=
z
y
x
nnn
States can be degenerate. E.g., if a = c then the two different states (nx = 1, ny = 1, nz = 2) and (nx = 2, ny = 1, nz = 1) have the same energy.
x y
z
a b
c
In 1D, motion is along the x dimension and the particle is constrained to the interval ax ≤≤0
x