Classical Approximation / Limit
[1-D (One-Dimensional) Translation]
Motion of a particle (mass ) in a 1-D box of
length
:
eigenvalue:
(
= 1, 2, 3, ...) (2.1.1)
density of states:
(2.1.2)
[3-D Translation]
Motion of a particle (mass ) in a 3-D box
(
)
[Independent in ,
, and
directions]
eigenvalue:
(
) (2.1.3)
density of states:
(2.1.4)
= volume:
density of states per unit volume:
(2.1.5)
[3-D Relative Translation]
Relative translation of particle A and B:
(mass)
(reduced mass)
(2.1.6)
(2.1.7)
note:
1-D translation activated complex in the
transition-state theory
2-D translation diffusion on a surface of
an adsorbed molecule
3-D translation reaction producing two
molecules
Problem-2.1 1) Derive the expression for the density of states per unit area of 2-D translation. 2) Derive the 3-D translational energy distribution for canonical ensemble (Maxwell-Boltzmann distribution) at temperature T from equation (2.1.5). |
[Rotational Motion]
moment of inertia:
(2.2.1)
(: mass of
th atom,
:
distance of
th atom from rotation axis)
rotational constant:
(2.2.2)
<Rotational symmetry number>
= number of indistinctive
rotational conformation
(same atoms are indistinctive)
ex.) (N2)=2,
(HCl)=1,
(CO2)=2,
(H2O)=2,
(NH3)=3,
(C2H4)=4,
(CH4)=12
<Rigid rotor approximation>
is constant (i.e.,
independent on the speed of rotation, the vibrational states, or etc.)
[1-D Rotator]
e.g. intramolecular rotations
[H3C-CC-CH3]
eigenvalue:
(
), degeneracy:
(2.2.3)
(
)
except for
(2.2.4)
density of states:
(2.2.5)
[2-D Rotator]
e.g. linear molecules [N2, CO2]
eigenvalue:
(
),
degeneracy:
(2.2.6)
density of states:
(2.2.7)
[3-D Rotator]
e.g. non-linear molecules
a) Spherical top ...
[CH4, SF6]:
eigenvalue:
(
),
degeneracy:
(2.2.8)
density of states:
(2.2.9)
b) Symmetric top ... or
[NH3,
CH3]: (approximation)
Asymmetric top ...
(H2CO, HCO):
(approximation)
density of states: ,
(2.2.10)
[Summary]
rotational degree of freedom (dimension of rotation)
,
(2.2.11)
[Vibrational Motion]
(
: force constant,
: reduced mass)
(2.3.1)
<Harmonic Oscillator Approximation>
is constant
(i.e., independent of deviation
)
Solution to the classical mechanics:
,
(
: frequency [
])
(2.3.2)
[One Vibrator]
eigenvalue: ,
(2.3.3)
density of states:
(2.3.4)
[ Vibrators]
Number of vibrators:
(non-linear molecule),
(linear molecule)
two vibrators:
three vibrators:
.....
vibrators:
(2.3.5)
note:
1) Density of states is crude approximation (fig. 2.1) for vibration
since
200-3000
.
2) in eq. 2.3.5, is the
energy from the classical origin (see fig. 2.2).
3) ,
Fig. 2.1a Comparison of by eq.
2.3.5 (solid line) and exact
(dots) for H2O.
Fig. 2.1b Comparison of by eq.
2.3.5 (solid line) and exact
(dots) for CH3OH.
Fig. 2.2
Problem-2.2 Estimate the vibrational density of states for ethane [ ![]() ![]() ![]() 2954, 1388, 995, 289, 2896, 1379, 2969(2), 1468(2), 1190(2), 2985(2), 1469(2), 822(2) Note that the ![]() |