1.4 Popular Scattering Model Potentials
29
The value of the cross section πd
2 is the area of a circle or equivalently the maximum
circumference of sphere with radius d.
It is also worth noting that this extremely simplified model of the interaction still
allows one to derive the constant velocity Arrhenius rate coefficient k(T ). This model
of the chemical reaction based on the rigid sphere potential assumes that collisions
leading to contact of the two particles are reactive if the translational energy is greater
than a threshold value E a . So nonzero contributions to the integral come only from
E a (which can be taken as the lower limit of the integral), where σ(E tr ) = πd
2 .
Using an analytical approximation to the rate constant Eq. (1.27), one obtains
10 the
approximate solution
11
k(T ) =
8k B T
πμ
1/2
πd
2 e
−E a /k B T
= AT
1/2 e
−E a /k B T
(1.59)
where A =
8k B π
μ
1/2
d
2 . This formulation of k(T ) coincides with that of the kinetic
theory of gases that expresses it as the product of the mean velocity of a particle
(8k B T /πμ)
1/2 times the cross-sectional area of the particle (πd
2 ) associated with
the fraction of effective collisions (e
−E a /kT ).
Equation 1.59 can be improved by multiplying A by a steric factor g. The factor g
is a constant that takes into account the fact that molecules and related interactions do
not, in general, have spherical symmetry and that, as a consequence, reactivity varies
with the angle of collision. In this case, you can give g an angular dependence having
a closed form that is simple, analytical, and integrable (for example, an ellipse as is
often done for the study of liquids). The model can also take into account the fact
that for molecules the collision involves other N degrees of freedom. In this case, if
the interaction is expressed as V (r, ξ 1 , ξ 2 , ξ 3 , ..., ξ N ), the integral in Eq. 1.44 has the
form
∞
a
dr
b 1
a 1
dξ 1
b 2
a 2
dξ 2 ...
b N
a N
dξ N
b
r 2
1 − b 2 /r 2 − V (r, ξ 1 , ξ 2 , ξ 3 , ..., ξ N )/E
1/2 ,
where a i and b i are the turning points of the variable ξ i .
1.4.2 The Repulsive Coulomb Potential
The Coulomb potential is a particular case of the family of the repulsive potentials
V (r ) = Br
−δ , where δ is set equal to 1 (see Fig. 1.12). Typically, the part of the
repulsive potential of a diatomic molecule is described by an index δ that varies
between 9 and 15.
10
xe −x dx = −(e −x + xe −x ).
11 e −Ea /k B T +
Ea
k B T e −Ea /k B T e −Ea /k B T provided 1 >> E a /k B T .
29
The value of the cross section πd
2 is the area of a circle or equivalently the maximum
circumference of sphere with radius d.
It is also worth noting that this extremely simplified model of the interaction still
allows one to derive the constant velocity Arrhenius rate coefficient k(T ). This model
of the chemical reaction based on the rigid sphere potential assumes that collisions
leading to contact of the two particles are reactive if the translational energy is greater
than a threshold value E a . So nonzero contributions to the integral come only from
E a (which can be taken as the lower limit of the integral), where σ(E tr ) = πd
2 .
Using an analytical approximation to the rate constant Eq. (1.27), one obtains
10 the
approximate solution
11
k(T ) =
8k B T
πμ
1/2
πd
2 e
−E a /k B T
= AT
1/2 e
−E a /k B T
(1.59)
where A =
8k B π
μ
1/2
d
2 . This formulation of k(T ) coincides with that of the kinetic
theory of gases that expresses it as the product of the mean velocity of a particle
(8k B T /πμ)
1/2 times the cross-sectional area of the particle (πd
2 ) associated with
the fraction of effective collisions (e
−E a /kT ).
Equation 1.59 can be improved by multiplying A by a steric factor g. The factor g
is a constant that takes into account the fact that molecules and related interactions do
not, in general, have spherical symmetry and that, as a consequence, reactivity varies
with the angle of collision. In this case, you can give g an angular dependence having
a closed form that is simple, analytical, and integrable (for example, an ellipse as is
often done for the study of liquids). The model can also take into account the fact
that for molecules the collision involves other N degrees of freedom. In this case, if
the interaction is expressed as V (r, ξ 1 , ξ 2 , ξ 3 , ..., ξ N ), the integral in Eq. 1.44 has the
form
∞
a
dr
b 1
a 1
dξ 1
b 2
a 2
dξ 2 ...
b N
a N
dξ N
b
r 2
1 − b 2 /r 2 − V (r, ξ 1 , ξ 2 , ξ 3 , ..., ξ N )/E
1/2 ,
where a i and b i are the turning points of the variable ξ i .
1.4.2 The Repulsive Coulomb Potential
The Coulomb potential is a particular case of the family of the repulsive potentials
V (r ) = Br
−δ , where δ is set equal to 1 (see Fig. 1.12). Typically, the part of the
repulsive potential of a diatomic molecule is described by an index δ that varies
between 9 and 15.
10
xe −x dx = −(e −x + xe −x ).
11 e −Ea /k B T +
Ea
k B T e −Ea /k B T e −Ea /k B T provided 1 >> E a /k B T .