Chapter 18: Reaction Dynamics

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Okay, let's unpack this.

We are embarking on a deep dive into, well, the microscopic core of chemical change reaction dynamics.

That's right.

We're moving far beyond just measuring how fast reactions go, you know, classic kinetics.

Instead, we're focusing on the actual choreography, sort of moment by moment action when chemical bonds break in new ones form.

Yeah, the nitty gritty, it's dense stuff, definitely.

We're talking energy changes comparable to breaking bonds entirely.

But the insights you get are just incredible, right?

Absolutely.

What we're covering today is basically the theoretical toolkit physical chemists use to understand these elementary reaction steps.

So calculating exact rates from scratch is hard.

Oh, incredibly hard.

Yeah, from first principles.

Very tough.

But we have these models, these frameworks that help us grasp the broad features, the main patterns.

And that's our plan today.

Walk through those frameworks.

Exactly.

We'll move systematically from the simpler gas phase models right up to the more sophisticated quantum approaches.

Got it.

So our mission is to give you a kind of structural shortcut through Focus 18 from Atkins covering five key topics.

Collision theory, diffusion and solution, transition state theory, molecular dynamics, and finally, electron transfer.

Let's jump right in.

Gas phase first.

Let's start there.

Simplest case.

Okay, collision theory.

This is our foundation, right?

Treats molecules like tiny billiard balls, basically.

Pretty much hard spheres flying around.

It's the simplest quantitative picture we have, mainly for simple species reacting in the gas phase.

So for a reaction, say A plus B goes to products,

what needs to happen according to this model?

Three things.

Three key things, yes.

First, obvious one, collision.

They actually have to physically hit each other.

Makes sense.

And the rate of hitting the collision density depends on their size.

That's the collision cross -section sigma and how fast they're moving towards each other, the relative speed.

And that speed depends on temperature.

It does, yeah.

Proportional to the square root of T.

So you already see a temperature link there, even before

Okay, so step one.

Yeah.

They collide.

What's next?

Step two, the energy threshold.

Just bumping into each other isn't enough.

The kinetic energy they have along the line connecting their centers when they hit needs to be greater than some minimum value, epsilon.

Which we usually connect to the activation energy.

Exactly.

We use that Boltzmann factor, E to the minus EA over RT to account for this.

It basically tells you what fraction of all the collisions actually have enough energy, enough oomph to potentially react.

Okay, collision energy.

And the third one, this is where the billiard ball idea breaks down a bit.

Right, the steric requirement.

Even if they hit hard enough, they usually have to hit in the right orientation.

Like puzzle pieces.

Sort of, yeah.

Think about complex molecules.

Maybe only hitting a specific atom or functional group works.

This need for correct alignment is wrapped up in the steric factor, P.

And P is usually small.

For complex molecules, yes.

Often much less than one.

Meaning only a small fraction of sufficiently energetic collisions actually lead to products because the orientation requirement is strict.

But,

here's the weird part you mentioned.

The potassium plus bromine reaction.

K plus Br2.

There, P is bigger than one.

Yeah, significantly bigger.

How does that even work?

More reactions than collisions predict?

Seems impossible.

It does seem impossible if you stick to the hard sphere idea.

But this is where the harpoon mechanism comes in.

It's brilliant.

Okay, harpoon.

So, the K atom has an electron that's relatively easy to remove as it gets close to the Br2 molecule but still quite far away in terms of physical contact.

At a certain critical distance, our star, that electron just jumps.

That's the harpoon from the K atom over to the Br2.

Creating ions.

K plus and Br2.

Exactly.

And now you have oppositely charged ions.

Which attract each other strongly.

Coulomb force.

Right.

And that attraction pulls them together from much further away than they would have just bumped into each other based on size alone.

Ah, so the effective collision cross -section, sigma star, is much bigger than the physical size sigma.

Precisely.

The electron transfer defines the interaction distance, not just bumping.

That's why P can be greater than one.

It's electrostatics driving it.

Wow.

Okay, that makes sense.

Now, just quickly, you also mentioned the RRK model for unimolecular reactions.

One molecule falling apart.

Size matters there too.

It does, but differently.

RRK theory deals with how energy gets distributed within a single energized molecule.

Okay.

The basic idea is that for the molecule to break apart, enough energy has to concentrate in one specific bond, the one that's going to break.

Right.

If the molecule is large, meaning it has many vibrational modes, many ways to vibrate a high value that energy can spread out, get temporarily stored in other vibrations.

So it takes longer for enough energy to pool in the right place.

Exactly.

It delays the accumulation of energy in the critical bond, so the overall rate of decomposition is slower for larger molecules.

More places for the energy to hide, essentially.

Okay, that covers gases.

Now let's dive into the messier world of liquids.

Adding a solvent.

That changes everything, right?

What's the biggest difference?

Oh, huge difference.

The cage effect.

That's the key concept.

The cage effect.

Yeah.

In a gas, molecules collide and fly apart.

In a liquid, they're surrounded by solvent molecules, like being in a crowded room.

So they get trapped.

Sort of.

When reactant molecules A and B happen to meet, the solvent molecules hem them in, preventing them from quickly separating again.

They linger near each other.

Forming an encounter pair.

Exactly.

We call it the encounter pair, AB.

They have multiple chances to bump into each other within this solvent cage before they might diffuse apart.

And this lingering changes what controls the overall reaction speed.

It does.

It leads to two main categories.

Two limiting cases for reactions and solution.

Okay, what are they?

First is diffusion controlled.

In this case, the actual chemical reaction step, once A and B are together in the cage, is super fast.

Essentially zero activation barrier.

So as soon as they meet, they react.

Pretty much instantly, yeah.

The bottleneck, the slow step, is simply the process of A and B diffusing through the solvent until they encounter each other in the first place.

The rate is controlled by $80.

The diffusion rate, constant.

What kind of reactions are like this?

Often things like radical recombination.

Radicals are usually very reactive.

Make sense.

And the other extreme.

That's activation controlled.

Here, A and B meet frequently.

Forming the encounter pair is relatively fast.

$80 is large.

But the reaction itself, the transformation AD right arrow has a significant energy barrier.

So they meet, hang out in the cage, but don't necessarily react right away.

Right.

They need to acquire enough energy while they're in that encounter pair to get over the activation hump.

The chemical step is much slower than the diffusion steps.

The rate depends on activating the pair.

Let's go back to the diffusion limit.

Can we calculate that rate?

We can model it, yes.

It depends on the rate at which A and B diffuse together to within a certain critical distance, r star.

The derived equation shows key dollars is proportional to that distance r star and the sum of their individual diffusion coefficients, dA plus dV, how fast they move through the solvent.

And those diffusion coefficients, they relate to the solvent itself.

They do.

And here's the really neat part, especially from a practical standpoint.

If you use the Stokes -Einstein relation, which links diffusion to viscosity,

you find that T to R becomes inversely proportional to the solvent viscosity eta.

And it also depends on temperature T.

So for a purely diffusion controlled reaction, the rate depends mainly on how thick the solvent is.

Basically, yes.

And temperature, of course.

But viscosity is key.

If you want to speed up a diffusion controlled reaction, you don't necessarily need a catalyst or drastically higher T.

You just use a less viscous solvent, thin it out.

Exactly.

Lower the viscosity, they diffuse faster, they meet faster, reaction goes faster.

It's a beautiful example of how the medium, the environment, directly controls the kinetics in solution.

Okay, moving on.

Now for what you call the centerpiece.

Transition state theory, TST.

This gives us a more detailed picture than just collisions, right?

Much more detailed.

It focuses on the highest energy point along the reaction pathway.

The top of the hill.

Precisely.

We imagine the reaction proceeding along a reaction coordinate that's the minimum energy path involving specific atomic movements over a potential energy barrier.

And the very peak of that barrier.

That specific fleeting arrangement of atoms at the exact peak is the transition state.

It's the critical configuration.

The cluster of atoms around that peak is called the activated complex denoted Sienager.

And TST makes a big assumption about this activated complex, doesn't it?

Something about equilibrium.

Yes, a crucial assumption.

It assumes there's a rapid pre -equilibrium setup between the reactants, A and B, and this activated complex.

So reactants quickly form the complex and it quickly falls back to reactants.

Exactly.

But some small fraction of the activated complex also goes forward to form products.

TST focuses on the rate of that forward step.

And this leads to the airing equation.

Yes.

The airing equation connects the overall rate constant, bill or dollars, to properties of this transition state.

It says Toto dollars is proportional to the equilibrium constant for forming the activated complex, Toto DAG.

Using equilibrium ideas for a rate.

That's the power of it.

Toto DAG tells us the concentration of the activated complex relative to reactants.

Then we multiply that by the frequency at which the complex vibrates along the reaction coordinate and falls apart into products.

Okay.

And the really useful part is that we can express that equilibrium constant using thermodynamics, right?

Like Gibbs energy.

Exactly.

We relate Toto DAG to the Gibbs energy of activation, Delta DAG or GA.

And just like regular Gibbs energy, we can break this down into an enthalpy of activation, Delta DAG or H, and an entropy of activation.

Delta DAG or T.

That's the one.

And this is where TST provides a much deeper understanding than collision theory.

Remember the steric factor, P.

Yeah, the orientation requirement.

P was usually small.

Well, the entropy of activation, Delta DAG or R, gives us the physical reason why.

How so?

Think about forming the activated complex.

If the reactants have to come together in a very specific, highly ordered, rigid arrangement to reach that transition state.

They lose a lot of freedom, rotational translational freedom.

Precisely.

That loss of freedom, that increase in order means a negative change in entropy.

Delta DAG or DAG will be negative, possibly quite large and negative.

And a negative Delta DAG or SAE makes Delta DAG or GA larger, more positive.

Which makes stugger smaller, and therefore makes the rate constant telidor smaller.

Ah, so a large negative entropy of activation corresponds directly to a small steric factor, P.

It's about the loss of molecular freedom needed to reach the reactive configuration.

You've got it.

It quantifies the structural price of reaching the transition state.

TST also helps with reactions involving ions in solution, right?

The kinetic salt effect.

Yes.

This deals with how changing the overall ionic strength, I, of the solution affects the rate constant for reactions between ions.

Ionic strength.

That's like the total concentration of ions.

Sort of, yeah, accounting for their charges.

The effect depends crucially on the product of the charges of the reacting ions.

Z8 times ZAB.

Okay.

So if two positive ions react,

VAZAB is positive.

Right.

In that case, increasing the ionic strength increases the reaction rate.

Why?

Because adding more inert ions creates an ionic atmosphere around the reacting ions.

This atmosphere tends to stabilize the highly charged activated complex, which has a charge, ZA plus ZAB, more than it stabilizes the individual reactant ions.

Lowering the energy of the activated complex speeds up the reaction.

And if the ions have opposite charges, ZAZAB is negative.

Then increasing ionic strength decreases the rate.

The stabilization effects work differently.

Interesting.

Okay.

One more TST related tool.

The kinetic isotope effect, KIE.

You swap an atom for its heavier isotope, like hydrogen per deuterium.

Exactly.

And you measure the change in reaction rates.

A fantastic diagnostic tool for mechanisms.

And usually the reaction slows down with the heavier isotope.

Often, yes, especially if the bond to that atom is being broken or significantly weakened in the rate determining step.

For breaking a CH versus CD bond, the KE, KH, KD is typically around seven at room temperature.

Why such a big difference?

Just a tiny mass change.

It comes down to quantum mechanics, specifically zero point vibrational energy, ZPE.

Okay.

Even at absolute zero, molecules vibrate.

A bond like CH has a certain minimum vibrational energy, ZPE.

Because deuterium is heavier, the CD bond vibrates more slowly and its ZPE is lower.

It sits deeper in the potential energy well.

Right.

Now think about the transition state where this bond is breaking.

That specific vibration is essentially gone or very weak.

So the ZPE difference between CH and CD largely disappears at the transition state.

Ah, so the CD bond starts off lower in energy due to its lower ZPE.

But has to reach roughly the same energy peak as CH at the transition state.

Eating the net energy input needed, the activation energy is effectively higher for breaking the CD bond compared to the CH bond.

Higher barrier, slower rate.

And a CH, significantly different from one, tells you that bond is involved in the slow step.

Exactly.

It's powerful evidence for mechanism determination.

Okay.

Now we're zooming in even further.

Molecular dynamics.

Trying to actually see the atoms move during a reaction.

Experimentally, this involves molecular beams.

Yes.

That's the key experimental technique here.

Why beams?

Why in a vacuum.

Because you want to study single collision events isolated from everything else.

No solvent, no collisions with background gas.

Pure conditions.

Extremely pure.

You create these narrow collimated beams of molecules, often selecting them so they are in specific known energy states, like a particular vibrational or rotational level.

So you control the input state very precise.

Very precisely.

Then you cross these beams, let them collide and detect the products, often also measuring their energy states.

Vibration, rotation, speed, angle.

Wow.

So you get state -to -state information.

Reactant state A goes to product state B.

Exactly.

It gives you the state -to -state cross -section sigma n prime.

It's an incredibly detailed picture of exactly how energy flows during that single reactive encounter.

Far beyond the average rates you get from bulk experiments.

And the theoretical counterpart to this is the potential energy surface.

The P, yes.

That's the map, yes.

Imagine plotting the total potential energy of the system as a function of the positions of all the atoms involved.

That sounds multi -dimensional.

Oh, it is.

For even a simple A plus B C reaction, it's 3D, often shown as 2D contour plots like a topographical map.

The valleys are stable molecules, reactants, products, and the pass between them goes over a mountain pass.

The saddle point.

The saddle point, which represents the transition state.

The path of lowest energy connecting reactants to products through that saddle point is the reaction coordinate.

Okay, but the shape of this surface around the saddle point is crucial, right?

You mentioned attractive versus repulsive surfaces.

Absolutely crucial.

It tells you what kind of energy is most effective at promoting the reaction.

Let's break that down.

What's an attractive surface?

On an attractive surface, the saddle point, the energy barrier, occurs early in the molecules, say A and B C, are still relatively far apart, just starting to interact strongly.

Early barrier.

So what kind of energy helps most?

Translational kinetic energy.

You basically need to slam the reactants together with enough speed to get over that initial hump.

Think of rolling a ball at the hill that starts rising immediately.

Okay.

And what about the products?

Because the energy release happens as the new bond forms after the barrier, while the products are separating, that energy often gets channeled into the internal modes of the products.

So they often emerge vibrationally excited.

Makes sense.

Now the opposite, a repulsive surface.

Here, the saddle point occurs late in the reaction coordinate.

The barrier is encountered when the reactant atom A is already very close to B and the old B -C bond is significantly stretched, almost broken.

Late barrier, close encounter needed.

Right.

To get over this late barrier, you need energy already stored within the reactant molecule, specifically in the bond about to break.

So high vibrational energy in the B -C reactant is most effective.

Just slamming them together harder, translation, doesn't help as much if the B -C bond isn't already vibrating strongly.

So you need to shake the reactant molecule first.

Kind of, yeah.

Excite its vibrations.

And the products.

Since the energies are released primarily as the A and B atoms push away from C, they fly apart fast.

Exactly.

The products often emerge with high translational kinetic energy.

So understanding the PE shape isn't just academic.

It tells you whether to heat the whole system up or maybe use something like a laser to specifically excite reactant vibrations.

Precisely.

It moves us towards rational control of reactivity rather than just using brute force temperature increases.

You can strategically supply the right kind of energy.

Amazing.

Okay, final section.

Applying some of these ideas, especially TST and quantum mechanics, to a fundamental process, electron transfer, like in redox reactions or even biology.

Yes.

Marcus Siri is the cornerstone here.

It describes how electrons move from a donor, D, to an acceptor, A.

And the electron itself.

It doesn't just flow like water, does it?

Quantum effects.

Big time.

The electron transfer itself happens via quantum mechanical tunneling?

Tunneling.

Through the space between D and A.

Effectively, yes.

It doesn't need to overcome the space like a classical particle.

The probability of tunneling drops off exponentially with the distance, D, between the donor and acceptor edges.

Something like E to the minus beta D.

So distance is critical.

But tunneling isn't the only factor controlling the rate.

There are constraints from the atoms, the nuclei.

Correct.

Two major constraints related to the nuclear framework.

First is the Frank Condon principle.

I remember that from spectroscopy.

Electrons move much faster than nuclei.

Exactly.

Electron transfer is essentially instantaneous on the scale of nuclear motion.

The atoms are effectively phrased in during the electron jump.

Okay.

So what does that mean for the transfer?

It means the transfer can only happen efficiently when the nuclear arrangement is such that the energy of the system before the jump, DNA, is momentarily equal to the energy after the jump, D plus an A, within that frozen nuclear configuration.

The potential energy surfaces have to intersect.

Precisely.

The system has to distort itself, rearrange its atoms and the surrounding solvent molecules to reach a specific geometry, Q double dagger, where those reactant and product electronic states have the same energy.

Only then can the electron jump according to Frank Condon.

And getting to that specific geometry,

that costs energy.

Yes.

That's the second major factor.

The reorganization energy, lambda or sometimes written as delta.

Reorganization energy.

It's the energy penalty required to distort the reactants and the solvent from their normal equilibrium geometries to that special intersection geometry, Q double dagger, where the electron transfer can actually occur.

You have to pay this energy price before the electron jumps.

So you need tunneling probability.

You need the system to rearrange, costing delta.

And it has to happen at the intersection point.

That's the core framework of Marcus theory.

And this leads to probably the most famous and maybe most unintuitive prediction in all of reaction dynamics.

The inverted region.

Ah, yes.

The inverted region.

Truly a landmark prediction and discovery.

It completely breaks from classical thinking.

So Marcus theory predicts how the activation energy, delta dagger G, changes as you make the reaction more thermodynamically favorable, right?

As the overall Gibbs energy change, delta G becomes more negative.

Yes.

It predicts a parabolic relationship between the activation energy and the driving force.

Okay.

So initially as a reaction gets more exergonic, more negative delta G circ, the activation barrier gets smaller and the rate increases.

That makes sense.

That's the normal region.

Faster reaction with greater driving force.

Classical intuition holds there.

But the parabola eventually turns over.

It does.

The minimum activation energy occurs when the driving force exactly matches the reorganization energy.

Delta G circ of delta.

If you make the reaction even more exergonic, beyond this point.

So delta G circuit comes greater than delta E, right?

Yes.

Now you're in the inverted region.

And paradoxically, the activation energy starts to increase again.

Right.

Making the reaction more thermodynamically favorable makes it slower.

Exactly.

It's because the potential energy surfaces, parabolas in the simple model, are now so offset vertically that their intersection point starts moving up the reactant surface again, increasing the barrier to reach it.

That's wild.

Impossible to explain with just Arrhenius or simple collision ideas?

Absolutely impossible.

It's experimental confirmation was a huge triumph for Marcus theory and our understanding of electron transfer.

Hashtag tag tag outro.

Wow.

Okay.

That was a serious deep dive.

We've covered a huge amount of ground.

We started with simple collisions in gases, saw how the steric factor led to the harpoon mechanism.

The electron jump.

Then we moved into solutions, the cage effect and how viscosity can control diffusion limited rates.

Right.

The medium matters.

Then transition state theory gave us the activated complex and crucially linked the entropy of activation to those steric requirements.

Delta dagger serolust, explaining P.

We looked at molecular dynamics, molecular beams seeing state to state details and how the PES shape, attractive or repulsive, tells us whether to use translational or vibrational energy.

Strategic energy input.

And finally, Marcus theory for electron transfer, bringing in tunneling reorganization energy and that mind bending inverted region.

Where more driving force means a slower reaction.

Yeah.

The level of understanding and potential control this gives us is just phenomenal.

It really is.

And thinking about that connection between PES shape and energy type leads to a final thought for you, the listener.

Okay.

Let's say you're a synthetic chemist and detailed studies, maybe KIE, maybe computational modeling strongly suggests your key reaction step proceeds over a repulsive potential energy surface.

You know you need vibrational energy in a specific reactant bond.

Right.

Not just general heating.

Exactly.

Instead of just cranking up the Bunsen burner, what practical technique might you use in the lab to selectively pump energy into those reactant misbrations and drive the reaction more efficiently?

Think about ways to excite specific molecular motions.

Something to ponder.

A great place to leave it.

Thank you so much for walking us through all of that.

My pleasure.

It's fascinating stuff.

And thank you all for joining us on this deep dive into the very heart of chemical change reaction dynamics.