Foundations — Transition state theory — activated complex (intro)

This page assumes you have seen nothing. Before you can read the parent topic comfortably, we build every symbol, word, and picture it leans on — one brick at a time, each brick resting on the last.

0. The picture the whole topic lives inside

Every idea below is a label on one drawing: energy going up the page, the progress of the reaction going across the page. Look at this first and keep it in your mind.

The curve is a hill. The left flat land is where we start (reactants), the right flat land is where we finish (products), and the single bump in the middle is the moment of transformation. That is the entire stage. Now we name its parts.

1. "Energy" — the height of the hill

Plain words: think of a ball on a slope. High up = lots of stored energy (it could roll down). Low down = little stored energy. Atoms are the same: some arrangements of atoms store more energy than others.

The picture: the vertical axis of the drawing above. Up = more energy = less stable. Down = less energy = more stable.

Why the topic needs it: the whole theory is a statement about which arrangement of atoms sits high and which sits low. Without a height axis there is no hill, no peak, no barrier.

2. "Reaction coordinate" — the across-axis

Plain words: as bonds stretch, break, and re-form, hundreds of atomic positions change at once. We squash all that complexity into one left-to-right measure of "how done are we?"

The picture: the horizontal axis. Far left = reactants untouched. Far right = products finished. Middle = halfway through the rearrangement.

Why the topic needs it: a reaction is a process, not a snapshot. We need an axis for "progress" so we can plot how energy rises and falls as the reaction happens. See Reaction coordinate diagrams and Potential energy surfaces for where this axis really comes from — it is the minimum-energy path across a many-dimensional surface.

Look at the red path in the figure: out of a whole mountainous landscape (many atomic positions), the reaction chooses the easiest route through the lowest pass. Flattening that red path onto a page gives us the simple hill from Section 0.

3. The three key spots on the hill

Now we name the three heights that every formula in the parent note refers to.

The picture: three dots on the curve — left valley, right valley, top of the bump.

Why the topic needs it: every arrow and every subtraction in the parent note is a difference of two of these three heights. Learn these three and the formulas become obvious.

4. $E_a$ — the activation energy (the "toll")

$E_a^{\text{forward}}=E_{\text{TS}}-E_{\text{reactants}}$

Reading the symbol out loud: "$E$ sub $a$" — the letter $E$ means energy, the little $a$ means activation. The minus sign means subtract the lower height from the higher height to get the size of the climb.

The picture: the vertical gap from the left valley up to the peak — an upward arrow in the figure below.

Why the topic needs it: this is the toll every reactant pair must pay to react. A tall toll = slow reaction; a short toll = fast reaction. This is the exact quantity Arrhenius plugs into his rate law.

In the figure the red upward arrow is $E_a$ (forward). Notice there is a different upward arrow on the right side: climbing from the product valley back up to the same peak is $E_a^{\text{reverse}}$.

$E_a^{\text{reverse}}=E_{\text{TS}}-E_{\text{products}}$

5. $\Delta$ — the "change in" symbol and $\Delta H$

Plain words: $\Delta (\text{anything})=(\text{value at end})-(\text{value at start})$. It is a subtraction, nothing more.

The picture: the vertical gap between the two flat lands — ignore the peak entirely.

• If the product valley is lower, $\Delta H$ is negative → energy released → exothermic.
• If the product valley is higher, $\Delta H$ is positive → energy absorbed → endothermic.

Why the topic needs it: a huge parent-note idea is that the toll ($E_a$, involves the peak) and the drop ($\Delta H$, ignores the peak) are independent. You cannot read one off the other.

Why this is true (from the definitions, no new ideas):

The two $E_{\text{TS}}$ cancel (the peak is shared by both climbs), leaving

WHAT we did: subtracted two definitions. WHY: to show the peak's absolute height drops out. WHAT IT LOOKS LIKE: in the figure, the difference of the two red arrows is exactly the height gap between the two valleys.

6. The double-dagger symbol $‡$

Plain words: it is just a flag that says "this quantity is measured at the top of the hill." You will see it in the Eyring equation, which turns the peak's height into an actual reaction rate.

Why the topic needs it: it distinguishes peak quantities from valley quantities at a glance.

7. Maximum vs. minimum — why the transition state is a peak

The picture: a valley is a bowl (ball rests). A pass is a saddle: walk across the ridge and you're at a high point along the trail, yet you're at the lowest place to cross the ridge.

Why the topic needs it: an intermediate sits in a minimum (a real dip, can be caught and studied). The activated complex sits at a maximum along the path (a saddle) — it cannot be caught because there is no dip to hold it. This single geometric difference is the heart of the parent note's biggest warning; see SN1 vs SN2 mechanisms for where real intermediates (SN1) versus pure transition states (SN2) appear.

The red dot marks the saddle: highest along the black reaction path, but a low pass compared to the walls on either side.

8. Partial bonds and $\delta$

The picture: a dashed or dotted line instead of a solid line between two atoms — a bond in the middle of being made or broken.

Why the topic needs it: the activated complex is defined by partial bonds — old bonds not fully gone, new bonds not fully there. The $\delta$ notation is how we draw "in-between."

9. How the foundations feed the topic

Each foundation on the left is a single label on the hill; together they are transition state theory.

Equipment checklist

Cover the right side and answer each before reading the parent note.