Upload
trinhcong
View
220
Download
5
Embed Size (px)
Citation preview
1
Lecture 35 Chapt 28, Sections 1-4
Bimolecular reactions in the gas phase
Anouncements: Exam tomorrow 2:00 is the primary time. vdW 237I have gotten several suggestions for lecture ideas, thanks and keep them coming.
Outline:
hard sphere collisionenergy dependenceimpact parameterinternal energy distribution
Review
Enzyme catalysis
reaction rate linear at low [S]
saturates at high [S] (saturating rate called vmax)
Michaelis-Menten mechanism
excess substrate, so d[ES]/dt = 0
initial rate, so [P] = 0
then
where
is the Michaelis constant.
It represents the apparent dissociation constant of ES to E and S.
Often find these things with a Lineweaver-Burke plot
2
Crossed molecular beam studies allowed researchers to gain tremendous
insight with incredible detail into how reactions occur in the gas phase. This was
an area of tremendous development in the 70s and 80s and is still very much active
today. We will just scratch the surface of a couple of reactions over the next couple
days and get you familiar with development of some simple gas-phase reaction
theory and some terminology.
Hard-Sphere Collision Theory
If we have the simple reaction
we expect the rate to be given by?
As with all theoretical development, we will start with the simplest possible
description – that every collision between A and B yields a successful chemical
reaction.
If this is the case, then we should be able to predict the rate of reaction from
our kinetic theory of gases.
v = collision rate = ZAB = σAB <ur> ρA*ρB
*
Concentration and number density are the same thing within some unit
conversions, so we can see that the reaction constant is just the collision cross-
section times the relative velocities.
k = σAB < ur >
For instance, in the reaction H2 + C2H4 Y C2H6
σAB = πd2AB = π{½(270 pm + 430 pm)}2 = 3.85 × 10-19 m2
(the diameters come from way back in table 25.3)
and
3
Plugging in for kB and T (298 K) gives <ur> = 1.83 × 103 m/s
Finally, k = (1000 dm3/m3)(6.022×1023 mol-1)(3.85 × 10-19 m2)(1.83 × 103 m/s)
= 4.24 × 1011 dm3/(mol @s)
The experimental value is 3.49 × 10-26 dm3/(mol @s), so we aren’t
looking too good with our model – off by over 35 orders of magnitude. But, this
really is no surprise since we assumed every single collision lead to a reaction.
More complex models of reaction cross-section
We want to make an improvement, but again we should try to start simple.
We really want σ to be about when a reaction occurs, not just when a collision
occurs. It makes sense, then for the reaction cross-section to depend on the relative
speeds of the molecules.
k(ur) = σ(ur) +ur,
or
This integral we did before in several forms way back in chapter 25 (except
for the σ bit). But, now we would are more interested in energy than speed.
4
Relative kinetic energy: Er = ½µ ur2
(Big Equation!!!)
This is the equation we will be using for the rest of the chapter. It
allows us to test various models for the reaction cross-section against experimental
rate constants.
Energy threshold for F
Lets assume that all collisions with relative kinetic energy above a cutoff lead
to reactions and all below lead to nothing. So, a simple step function:
Evaluating the above integral with this function gives:
Notice that we now have an Arrhenius-looking expression with E0 playing
the role of the activation energy.
If we go back to the H2 + C2H4 Y C2H6 example, we can set this all equal to
the experimental k and solve for E0. We then get E0 = 223 kJ/mol. Experiment is
180 kJ/mol. Doesn’t look too terrible, certainly much better than our first model,
but remember that k depends strongly on E0, so the T-dependence of the rate would
be pretty poor.
5
Impact Parameter
Let’s consider a little bit more complexity. Think of two molecules hitting
head on, or just grazing each other. The amount of collision energy available for
reaction is very different in these two cases. We can define an Impact Parameter
to help describe the difference in these two cases.
So, if b (the impact parameter) is big, no collision or glancing collision.
What is b for head on collision? b approaches 0.
In the Line of Centers model (loc), we assume that only kinetic energy that
lies along the line of centers contributes to the reaction. This is a complicated bit of
geometry, but the solution is:
What is the σ behavior? if Er just equals E0, we get σ =0. As Er gets really
big then σ approaches πd2 – every collision leads to reaction.
Plugging this into our trusty k integral equation gives:
6
Note that this looks even more like the Arrhenius equation. In fact, in this
equation, Ea = E0 + ½kBT and A = +ur,σABe½
(there is some T dependence in ur, so Ea is not as simple as E0.)
We have also done a better job comparing to experiment. However, the
shape of our cross-section function does not really compare very well to
experimental measurements and we are still getting rate constants that are too
high by a couple/few orders of magnitude:
Reaction Expmnt A (L @mol-1@s-1) Calculated A
NO +O3 6 NO2 + O2 7.94 × 108 5.01 × 1010
2ClO 6 Cl2 + O2 6.31 × 107 2.50 × 1010
H2 + C2H4 6 C2H6 1.24 × 106 7.30 × 1011
So, we are still making it too easy for molecules to react. One factor to
consider is the orientation of the molecules when they are colliding. This is surely
important, but your book doesn’t really treat it in detail, so we won’t either. It turns
out this is still not enough.
Internal Energy Distribution
One thing we will consider in detail is the way energy is distributed internally
in the reactants. Look at how the cross section depends on the total energy of
reactants along with the H2+ vibrational state.
H2+ + He Y HeH+ + H
(See Fig 28.4 from McQaurrie and Simon below)
7
The total energy includes the
translational and vibrational (and
rotational) energies of the
reactants.
- vibrational levels 0-3 have
an energy less than E0, so
additional transitional energy
is needed to induce reaction.
That’s why we see a
threshold.
- vibrational levels > 4
already have more than E0
energy so they react even
with no additional energy.
Thus, it isn’t just the total energy that matters, it matters how that energy is
distributed in the molecule. Lots of vibrational energy means more reactive.