F t ¼
2pmkT
ð
Þ
3=2
h 3
Á RT:
ð2:2:16bÞ
for any species. Here
– m is a species mass,
– V is the volume in which the gas is enclosed,
– p is pressure.
The (2.2.16a) is valid for a single molecule in volume V, the presence of other
species interacting weakly with it (an ideal gas) does not affect. The (2.2.16b) is
valid for one mole of an ideal gas (pV = RT). For species/cm
3 units:
F t ¼
2pmkT
ð
Þ
3=2
h 3
ð2:2:16cÞ
These partition functions can be obtained from the solution of the wave equation
of a species moving in a potential box of volume V [3], p. 26. As follows from the
solution of this wave equation, the translational motion of a species in a box is
quantized, although the energy levels are extremely dense: e n = n
2 h
2 /8 ml, where
l is the side of the box, n is a quantum number, 0 or integer, and h
2 /
8 m = (6:63 Á 10
À27 erg s)
2 /8 Á 16 Á 1:67 Á 10
À24 g for an oxygen atom, for example,
which is 2:05 Á 10
À31 erg cm = 1:03 Á 10
À15 cm
À1 for l = 1 cm (1 cm
−1 is 1:98546 Á
10
À16 erg/species). Consequently, the average energy corresponding to 300 K,
240 cm
−1 , with l = 1 cm, corresponds
n ¼
ffiffiffiffiffiffiffi ffi
240
p
1:03 Á 10 À15 ¼ 4:8 Á 10
8
states. Huge value! It follows that the translational partition function, equal to
F t ¼
P
j exp e j =kT
À
Á
is also very large, for m = 16 amu (oxygen atom) and V = 1
cm
3
F t = 6:2 Á 10
25 . For iodine atoms, m I = 127 amu, F t = 1:39 Á 10
27 .
The electronic partition sum of an atom is equal to its electronic statistical
weight (see (2.2.11)). The electronic statistical weight is equal to the degeneracy of
the state under consideration; the latter for the angular momentum of (any) S, L, J,
I … is equal to 2S + 1, 2L + 1, etc. Consequently, for the I(
2 P 3/2 ) state (the
spin-orbital degeneracy is removed, and we consider the J = 3/2 degeneracy):
F e = 2 Á 3=2 þ 1 ¼ 4. The I(
2
P 1/2 ) state at T % 300 K population is exp(−0.94/
0.03) = 2:47 Á 10
À14 . Since the dissociation of the iodine molecule takes place only
at high temperatures, for each T, it is necessary to consider the partition function
(2.2.11).
The nuclear partition sum of an atom is F n = 2I + 1 = 2∙5/2 + 1 = 6; (I = 5/2 is
the nuclear spin of
127 I isotope). As one will see later, the nuclear partition functions
are reduced.
20
2 General Kinetic Rules for Chemical Reactions, Collisional …
2pmkT
ð
Þ
3=2
h 3
Á RT:
ð2:2:16bÞ
for any species. Here
– m is a species mass,
– V is the volume in which the gas is enclosed,
– p is pressure.
The (2.2.16a) is valid for a single molecule in volume V, the presence of other
species interacting weakly with it (an ideal gas) does not affect. The (2.2.16b) is
valid for one mole of an ideal gas (pV = RT). For species/cm
3 units:
F t ¼
2pmkT
ð
Þ
3=2
h 3
ð2:2:16cÞ
These partition functions can be obtained from the solution of the wave equation
of a species moving in a potential box of volume V [3], p. 26. As follows from the
solution of this wave equation, the translational motion of a species in a box is
quantized, although the energy levels are extremely dense: e n = n
2 h
2 /8 ml, where
l is the side of the box, n is a quantum number, 0 or integer, and h
2 /
8 m = (6:63 Á 10
À27 erg s)
2 /8 Á 16 Á 1:67 Á 10
À24 g for an oxygen atom, for example,
which is 2:05 Á 10
À31 erg cm = 1:03 Á 10
À15 cm
À1 for l = 1 cm (1 cm
−1 is 1:98546 Á
10
À16 erg/species). Consequently, the average energy corresponding to 300 K,
240 cm
−1 , with l = 1 cm, corresponds
n ¼
ffiffiffiffiffiffiffi ffi
240
p
1:03 Á 10 À15 ¼ 4:8 Á 10
8
states. Huge value! It follows that the translational partition function, equal to
F t ¼
P
j exp e j =kT
À
Á
is also very large, for m = 16 amu (oxygen atom) and V = 1
cm
3
F t = 6:2 Á 10
25 . For iodine atoms, m I = 127 amu, F t = 1:39 Á 10
27 .
The electronic partition sum of an atom is equal to its electronic statistical
weight (see (2.2.11)). The electronic statistical weight is equal to the degeneracy of
the state under consideration; the latter for the angular momentum of (any) S, L, J,
I … is equal to 2S + 1, 2L + 1, etc. Consequently, for the I(
2 P 3/2 ) state (the
spin-orbital degeneracy is removed, and we consider the J = 3/2 degeneracy):
F e = 2 Á 3=2 þ 1 ¼ 4. The I(
2
P 1/2 ) state at T % 300 K population is exp(−0.94/
0.03) = 2:47 Á 10
À14 . Since the dissociation of the iodine molecule takes place only
at high temperatures, for each T, it is necessary to consider the partition function
(2.2.11).
The nuclear partition sum of an atom is F n = 2I + 1 = 2∙5/2 + 1 = 6; (I = 5/2 is
the nuclear spin of
127 I isotope). As one will see later, the nuclear partition functions
are reduced.
20
2 General Kinetic Rules for Chemical Reactions, Collisional …