3. Collision Theory
kinetic theory of ideal gases
consider first a pure system
assumptions
1) particles of mass m and radius r ; ceaseless random motion
2) dilute gas: r ¿ l , l = mean free path = average
distance a particle travels between collisions
3) no interactions between particles, except perfectly
0
elastic collisions; particles are hard spheres (E tr = E tr,
E tr = 12 m~
v 2 translational kinetic energy, 0 = after collision)
velocity ~
v = (v x , v y , v z )
~
v 2 = v x2 + v y2 + v z2
speed:
v = |~
v| =
q
v x2 + v y2 + v z2
number density: N =
CHEM 6114
N
V
3.1
For a system in equilibrium, the probability P i of finding a molecule with energy ²i is given by the Boltzmann distribution
Pi =
Ni
∝ g i exp(−²i /kT )
N
where g i is the degeneracy, i.e., the number of different states of a molecule having the same energy
²i
properties of a probability: P i ≤ 1, normalization:
P
j
Pj = 1
g i exp(−²i /kT )
Pi = P
j g j exp(−² j /kT )
apply to distribution of speeds in an ideal gas, ²i =
²(v) = 21 mv 2,
4πv 2dv · exp(−mv 2/2kT )
dP (v) = F (v)dv = R ∞
0
4πv 2dv · exp(−mv 2/2kT )
dP (v) = the probability of finding a molecule with a
speed between v and v +dv , i.e., a molecule lying in a
CHEM 6114
3.2
spherical shell of radius v and thickness dv in velocity
space
F (v)dv = the fraction of molecules with speed between v and v + dv
4πv 2dv = volume of a spherical shell of radius v and
thickness dv = g (dv)
[Figure: spherical shell in velocity space, Atkins 9th ed., Fig. 20.5]
CHEM 6114
3.3
Maxwell distribution of speeds
m
dP (v) = F (v)dv = 4π
2πkT
µ
¶3/2
Ã
2
!
mv
v exp −
dv
2kT
2
The Maxwell distribution is of course properly normalized,
∞
Z
F (v)dv = 1
0
CHEM 6114
3.4
[Figure: Maxwell distribution of speeds, Atkins 9th ed., Fig. 20.3]
average speed
∞
Z
c = ⟨v⟩ =
0
µ
8kT
vF (v)dv =
πm
¶1/2
mean-square speed
­
v
®
2
Z
=
∞
v 2F (v)dv =
0
3kT
m
elastic collisions
determine how often a specific particle collides with
other particles in the gas
since this is a representative particle, we can assume
that it moves with the average speed c
we can replace the moving collision partners by stationary particles, if we replace the average speed c
by the relative average speed c rel:
µ
c=
8kT
πm
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¶1/2
µ
,
c rel =
8kT
πµ
¶1/2
3.5
1
1
1
+
,
≡
µ mA mB
µ = reduced mass
for identical particles
1 2
m
=
=⇒ µ =
µ m
2
p
c rel = 2c
[Figure: collision tube, Atkins 9th ed., Fig. 20.8]
number of stationary particles inside the collision
2
tube: N σ0 · c rel∆t , σ0 = πd = elastic collision crosssection
for collisions between identical particles, d = 2r
collision frequency z = average number of collisions
CHEM 6114
3.6
of a particle per unit time
z=
N σ0 · c rel∆t
∆t
p
z = 2N σ0c
p N
z = 2 σ0c
V
p P
z= 2
σ0c;
kT
= N σ0c rel
N
(PV = nRT =
RT = N kT )
NA
mean free path l = the average distance a molecule
travels between two successive collisions
1
c
1
kT
l =c· = p
=p
=p
z
2N σ0c
2(N /V )σ0
2P σ0
total number of collisions per unit volume per unit
time for identical particles:
1
σ0c
4kT
ZAA = z NA = p NA2 = σ0
2
πm A
2
µ
¶1/2
NA 2
for mixture of gases A and B, total number of collisions per unit volume per unit time for dissimilar parCHEM 6114
3.7
ticles:
ZAB = σ0
µ
8kT
πµ
¶1/2
NANB
2
for dissimilar particles: σ0 = πd with d = r A + r B
hard-sphere collision theory
if all collisions were reactive, then the kinetic equation for the number density of A for the reaction
A + B −→ P
would be
dNA
= −ZAB
dt
and for the molarity
d[A]
= −ZAB/N A = −r
dt
where N A is Avogadro’s number
CHEM 6114
3.8
so we obtain
r = k AB[A][B] = σ0
k AB = σ0
µ
8kT
πµ
µ
8kT
πµ
¶1/2
N A [A][B]
¶1/2
NA
most collisions will not be reactive, since collisions
must occur with sufficient energy in the gas phase to
give rise to a reactive event
before molecules can get close enough to react, they
must overcome an energy barrier ²a
only molecules with sufficient kinetic energy along
the line of centers AB to surmount this energy barrier
will react
line-of-centers theory
impact parameter, b , is the closest perpendicular distance between the centers of the molecules
CHEM 6114
3.9
rA
b
rB
for hard spheres, collisions are only possible if b ≤ d =
rA + rB
b = 0 head-on collision
consider molecule A approaching molecule B with relative velocity ~
v
vlc
d
B
b
α
v
A
only the relative velocity v lc along the line of centers
can be used to overcome the energy barrier ²a
v lc = v cos α
b
sin α = ,
d
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cos2 α + sin2 α = 1
3.10
s
v lc = v cos α = v
d 2 − b2
d2
2
total translational energy ² = 12 µv ; ² ≥ ²a , otherwise
even a head-on collision will not lead to a reactive
event
energy along the line of centers
²lc = ²
d 2 − b2
d2
as b increases, ²lc decreases; ²lc has its largest value
for b = 0, head-on collision
since a reactive collision occurs only for ²lc ≥ ²a , then
reaction only occurs for impact parameters b ≤ b max,
where
²a = ²
2
d 2 − b max
d2
or
2
b max
= d2
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µ
²a
1−
²
¶
² ≥ ²a
3.11
b max increases as the total energy ² increases
for a given ², the reactive collision cross-section is
µ
¶
µ
¶
²
²
a
a
2
σ(²) = πb max
= πd 2 1 −
= σ0 1 −
²
²
for ² ≥ ²a
σ(²) = 0 for ² < ²a
a molecule A moving with relative velocity v (relative
1/2
kinetic energy ², v = (2²/µ) ) through NB B molecules per unit volume strikes vσ(²)NB per unit time
with sufficient directed energy to react =⇒
r N = NA
∞
Z
0
(2²/µ)1/2σ(²)NBF (²)d²
where F (²) is the Maxwell distribution
µ
1
F (²)d² = 2π
πkT
¶3/2
²1/2 exp(−²/kT )d²
and we find
r N = σ0
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µ
8kT
πµ
¶1/2
exp(−²a /kT )NANB
3.12
with E a = N A ²a , we obtain for the rate constant
k AB = σ0
µ
8kT
πµ
¶1/2
N A exp(−E a /RT )
the predicted rate constant follows the Arrhenius law
comparing the form of k AB predicted by collision theory with the Arrhenius law, the pre-exponential factor
A is given by
A th = σ0
µ
8kT
πµ
¶1/2
NA
collision theory predicts that the pre-exponential factor is weakly temperature dependent
p
A th ∝ T
for many reactions, this temperature dependence is
swamped by the strong temperature dependence of
the exponential term
CHEM 6114
3.13
the following Table (from M. J. Pilling and P. W. Seakins,
Reaction Kinetics, Oxford University Press, Oxford,
1995) compares the predicted and experimental values of A for some reactions, the quantity P is defined
as
P=
A exp
A th
Rxn
T
1
2
3
4
600
300
470
800
Ea
10−11 A exp
10−11 A th
P
0
0
102
180
10
0.24
0.094
1.24 × 10−5
2.1
1.1
0.59
7.3
4.8
0.22
0.16
1.7 × 10−6
T is in K, E a in kJ mol−1, A in L mol−1 s−1
1: K + Br2 −→ KBr + Br
2: CH3 + CH3 −→ C2H6
3: 2 NOCl −→ 2 NO + Cl2
4: H2 + C2H4 −→ C2H6
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3.14
except for Rxn 1, the theoretical values are too large,
5
for Rxn 4 by more than a factor of 10
P < 1 indicates that the relative orientation of the molecules is important in reactive collisions; P is known
as the steric factor and is generally several orders of
magnitude smaller than 1
k AB = P σ0
µ
8kT
πµ
¶1/2
N A exp(−E a /RT )
orientation does not explain values of P > 1, which
would seem to imply that the molecules react faster
than they collide
advantages of collision theory: (i) simple model;
(ii) provides good picture of bimolecular reactions;
(iii) predicts qualitatively the form of the temperature
dependence of the rate constant
shortcomings: (i) hard sphere assumption neglects
structure of the molecules, steric factor P is an ad
hoc way of including conformational effects, however no method for calculating P ; (ii) assumes that
CHEM 6114
3.15
molecules react instantaneously; (iii) neglects intermolecular forces; however long range attractive interactions are important in explaining reactions where
P >1
CHEM 6114
3.16