The Lindemann Theory of Unimolecular Reactions

Bimolecular reactions make sense. Two molecules collide, bonds break, new bonds form. Simple.

But unimolecular reactions? A single molecule just… reacts? Where does the energy come from? This puzzle bothered chemists for years until F.A. Lindemann proposed an elegant solution in 1922.

The Problem with Unimolecular Reactions

Collision theory explains bimolecular reactions beautifully.

When two molecules A and B collide, their relative kinetic energy exceeds the threshold energy. The collision breaks existing bonds and forms new ones.

But what about a reaction like \( A \longrightarrow P \)? Just one molecule turning into product?

If molecule A needs activation energy, it must get that energy from somewhere. The obvious answer: collision with another molecule. But that would make it a second-order reaction. And experimentally, many unimolecular gas reactions follow first-order kinetics.

Something didn’t add up.

Lindemann’s Mechanism

Frederick Lindemann proposed a clever two-step mechanism in 1922. Here’s how it works.

For a unimolecular reaction \( A \longrightarrow P \), Lindemann suggested:

Step 1 (Activation): \( A + A \rightleftharpoons A^* + A \)

The forward rate constant is \( k_f \) and the backward rate constant is \( k_b \).

Step 2 (Reaction): \( A^* \longrightarrow P \) with rate constant \( k_{f_2} \)

The \( A^* \) is an energized molecule. It’s picked up enough vibrational energy to react. Think of it as a molecule that’s been “wound up” and ready to spring.

Important distinction: \( A^* \) is not an activated complex. It’s just a regular molecule sitting in a high vibrational energy state. The energy exceeds the threshold needed for reaction, but the molecule hasn’t committed to reacting yet.

Where the Energy Comes From

In Step 1, two A molecules collide. Kinetic energy from the second molecule transfers into vibrational energy of the first. That’s your energized \( A^* \).

The collision partner doesn’t have to be another A molecule. It could be a product molecule, an inert gas, or anything else present in the system. It just doesn’t appear in the overall stoichiometry \( A \longrightarrow P \).

The Time Lag

Here’s the key insight. There’s a delay between energization and reaction.

Once \( A^* \) forms, two things can happen:

It can collide with another molecule and lose its energy (de-energization back to A). The vibrational energy converts back to kinetic energy of the collision partner.

Or it can react. The excess vibrational energy breaks the right bonds and forms product P.

This competition between de-energization and reaction is what makes the Lindemann mechanism work.

The Math

We’ll use the steady-state approximation here. Since \( A^* \) is short-lived and reactive, its concentration stays roughly constant. Rate of formation equals rate of destruction.

Rate of \( A^* \) formation: \( k_f [A]^2 \)

Rate of \( A^* \) destruction: \( k_b [A][A^*] + k_{f_2}[A^*] \)

Setting these equal (steady-state):

$$\frac{d[A^*]}{dt} = k_f [A]^2 – k_b [A][A^*] – k_{f_2}[A^*] = 0 \quad \text{…(1)}$$

Solving for \( [A^*] \):

$$[A^*] = \frac{k_f [A]^2}{k_b [A] + k_{f_2}} \quad \text{…(2)}$$

The reaction rate is:

$$r = -\frac{d[A]}{dt} = k_{f_2}[A^*] \quad \text{…(3)}$$

Substituting equation 2 into equation 3:

$$r = \frac{k_f k_{f_2} [A]^2}{k_b [A] + k_{f_2}} \quad \text{…(4)}$$

This rate law has no definite order. But we can look at two limiting cases.

High Pressure Limit

When \( k_b[A] \gg k_{f_2} \), the \( k_{f_2} \) term in the denominator becomes negligible:

$$r = \frac{k_f k_{f_2}}{k_b}[A] \quad \text{…(5)}$$

First-order kinetics. At high pressures, \( [A] \) is large, so collisions are frequent. The energized molecules get de-energized quickly, and only a small fraction react. The rate-determining step is the unimolecular decomposition of \( A^* \).

Low Pressure Limit

When \( k_{f_2} \gg k_b[A] \), the \( k_b[A] \) term becomes negligible:

$$r = k_f [A]^2 \quad \text{…(6)}$$

Second-order kinetics. At low pressures, collisions are rare. Once \( A^* \) forms, it’s more likely to react than to collide again and lose energy. The rate-determining step is now the bimolecular activation.

The Unimolecular Rate Constant

We define the experimental rate as:

$$r = k_{uni}[A] \quad \text{…(7)}$$

Comparing equations 4 and 7:

$$k_{uni} = \frac{k_f k_{f_2} [A]}{k_b[A] + k_{f_2}}$$

Or equivalently:

$$k_{uni} = \frac{k_f k_{f_2}}{k_b + k_{f_2}/[A]}$$

This shows how \( k_{uni} \) varies with pressure (through \( [A] \)). At high \( [A] \), it approaches a constant. At low \( [A] \), it decreases. This pressure dependence is a key experimental test of the Lindemann mechanism.

Limitations of the Lindemann Theory

The Lindemann mechanism was a breakthrough, but it’s not perfect.

The theory predicts that \( k_{uni} \) should decrease linearly with \( 1/[A] \) at low pressures. Experimentally, the falloff is more gradual. The simple two-step mechanism doesn’t capture all the physics.

Later refinements by Hinshelwood, Rice, Ramsperger, and Kassel (the RRKM theory) accounted for the distribution of energy among vibrational modes. But Lindemann’s core insight, that activation happens through collisions and there’s a time lag before reaction, remains the foundation.

Video Lecture

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