Transition State Theory — 2D Reaction-Coordinate Landscape

Activation Parameters

Activation enthalpy ΔH‡ 60 kJ/mol

Activation entropy ΔS‡ -20 J/mol·K

Temperature T 298 K

Live Readouts

Gibbs energy of activation ΔG‡

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Eyring rate constant k

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Half-life t₁⁄₂

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Net crossings

Reactant side

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Product side

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Reactant well (x < 0)

Near dividing surface

Product well (x > 0)

How it works

This is the 2D companion to the 3D transition-state-theory landscape simulator. Instead of animating a scripted per-particle coin-flip on a rendered 3D mesh, it integrates a genuine overdamped Langevin equation for each particle's reaction coordinate x — the same stochastic-dynamics method used in real molecular-dynamics free-energy calculations:

γ·dx/dt = −dG(x)/dx + √(2·γ·k_BT)·ξ(t)
G(x) = ΔG‡·exp(−x²/2w²) + confinement wells at x = ±x₀

The free-energy profile's barrier height is exactly ΔG‡ = ΔH‡ − T·ΔS‡ from the Eyring equation shown in the readouts — moving a slider reshapes the actual barrier every particle diffuses across, it is not a separate cosmetic animation. The reaction rate here is measured directly as the flux through the dividing surface x = 0 (net left-to-right crossings per second), the literal definition transition-state theory is built on — a more direct way to see the rate than a scripted success probability.

ΔH‡ — raises the barrier's underlying enthalpic height; a taller barrier means far fewer thermal fluctuations are large enough to cross it.
ΔS‡ — folded into ΔG‡ exactly as in the Eyring equation; a very negative ΔS‡ raises ΔG‡ and slows crossings, a positive ΔS‡ can even make the barrier vanish (downhill diffusion).
T — raises both the thermal noise amplitude √(k_BT) driving the random walk and softens the Boltzmann suppression of the barrier, so crossings speed up in both ways at once — exactly the same direction as the real Eyring exponential.

Because the real Eyring rate constant spans dozens of orders of magnitude across the slider range (nanoseconds to geological ages), the barrier height fed into the Langevin integrator is uniformly divided by a fixed softening constant so crossings stay watchable — the same compression trick the 3D simulator uses on its per-attempt probability — while the ΔG‡, k and half-life numbers above are always the exact, unsoftened physical values.