← Back to article

Equation 3 · Lowering the Barrier: What a Catalyst Actually Does

What does this equation mean?

k=κ kBThexp⁡(−ΔG‡RT),k = \kappa\,\frac{k_{\mathrm B}T}{h}\exp\left(-\frac{\Delta G^{\ddagger}}{RT}\right),

Read the formula alongside the article passage below. Each part has a deeper page with its role in the equation, the supporting passage and nearby citations.

Start withk_mathrm BT
Divide byh
This relates tok
How to read the two sides of this formula. Follow the article passage for the meaning of each quantity.

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

Read it piece by piece

kk

Symbol k

k occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Understand this part →

κ\kappa

Symbol kappa

kappa is one of the signed contributions combined to compute the quantity on the left.

Understand this part →

kBk_{\mathrm B}

Symbol k_mathrm B

kmk_mathrm B occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Understand this part →

TT

Symbol T

T is one of the signed contributions combined to compute the quantity on the left.

Understand this part →

hh

Symbol h

h is one of the signed contributions combined to compute the quantity on the left.

Understand this part →

ΔG‡\Delta G^{\ddagger}

Symbol Δ G^ddagger

Δ GdG^ddagger occurs above the fraction bar. The numerator is divided by the entire denominator below it.

Understand this part →

RR

Symbol R

R occurs below the fraction bar. The quantity above the bar is divided by this expression; zero is excluded as a denominator.

Understand this part →

=

=

The expressions on both sides represent the same quantity under the stated assumptions.

Understand this part →

See an illustrated explanation →
fraction

fraction

Divide the expression above the line by the one below it.

Understand this part →

See an illustrated explanation →
change

change

Capital delta attached to a quantity marks a difference between two values of that quantity; the article’s sign convention determines the order.

Understand this part →

subscript

subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

Understand this part →

superscript

superscript

A raised number can be a power. When it is a label or bound, it selects a case or the upper limit of a sum; the formula’s structure distinguishes these uses.

Understand this part →

See an illustrated explanation →
kBTk_{\mathrm B}T

Numerator: k_mathrm BT

The complete quantity above the fraction bar.

Understand this part →

RTRT

Denominator: RT

The complete quantity below the fraction bar; it must be nonzero for this division.

Understand this part →

How to interpret it

With a fixed numerator, increasing a nonzero denominator reduces the fraction. Read it with the definitions, units, and assumptions supplied by the article.

What the article says around this equation

Transition state theory recasts this in thermodynamic language. Treating the activated complex as in quasi-equilibrium with the reactants gives k=κ kBThexp⁡(−ΔG‡RT)k = \kappa\,\frac{k_{\mathrm B}T}{h}\exp\left(-\frac{\Delta G^{\ddagger}}{RT}\right). in which the universal frequency factor sets the timescale, the Gibbs energy of activation sets the barrier, and the transmission coefficient absorbs the failures of the picture — recrossing of the dividing surface, tunnelling, and the fact that the chosen reaction coordinate is rarely perfect. Splitting the activation Gibbs energy into enthalpic and entropic parts is the point of the exercise. It makes explicit that a catalyst can help either by lowering the enthalpic cost of bond reorganisation or by paying the entropic price…
Read the full surrounding passage
Transition state theory recasts this in thermodynamic language. Treating the activated complex as in quasi-equilibrium with the reactants gives k=κ kBThexp⁡(−ΔG‡RT)k = \kappa\,\frac{k_{\mathrm B}T}{h}\exp\left(-\frac{\Delta G^{\ddagger}}{RT}\right). in which the universal frequency factor sets the timescale, the Gibbs energy of activation sets the barrier, and the transmission coefficient absorbs the failures of the picture — recrossing of the dividing surface, tunnelling, and the fact that the chosen reaction coordinate is rarely perfect. Splitting the activation Gibbs energy into enthalpic and entropic parts is the point of the exercise. It makes explicit that a catalyst can help either by lowering the enthalpic cost of bond reorganisation or by paying the entropic price of bringing partners together and holding them in the right geometry. Enzymes exploit both routes; heterogeneous catalysts lean heavily on the second, because a surface that has already adsorbed both partners has removed most of their translational and rotational freedom before the reaction begins.

Read the equation in its article →

For background, read the article’s source list.

Return to Lowering the Barrier: What a Catalyst Actually Does

See this formula across 1 published context →

Browse the mathematical compendium →