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Published equation contexts

XRC,i=(∂ln⁡r∂ln⁡ki)Ki,kj≠iX_{RC,i} = \left(\frac{\partial \ln r}{\partial \ln k_i}\right)_{K_i, k_{j \neq i}}

Why this formula appears here

The rigorous move is Charles Campbell’s degree of rate control (DRC) framework, which asks, for each elementary step, how much the overall rate would change if that step’s rate constant were perturbed while holding all others fixed [ 1 ] . A step with a DRC near one is rate-controlling; a step with a DRC near zero is kinetically irrelevant even if it is thermodynamically significant. This is not a qualitative heuristic — it is a derivative of the log of the rate with respect to the log of a single rate constant, evaluated at the reaction conditions of interest, and it can be computed from a microkinetic model built on density-functional or experimental barriers. Here r is the overall…

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XRC,iX_{RC,i}

Symbol X_RC,i

XRX_RC,i is part of the quantity the equation computes from the expression on the right.

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kik_i

Symbol k_i

the rate constant of step i , and the partial derivative is taken holding the equilibrium constant KiK_i of that step fixed (so that only kinetics, not thermodynamics, is perturbed) and all other rate constants constant.

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kj≠ik_{j \neq i}

Symbol k_j neq i

kjk_j neq i is an input to the expression that computes the quantity on the left.

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∂ln⁡ki\partial \ln k_i

Denominator: partial ln k_i

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

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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.

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Published contexts (1)

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XRC,i=(∂ln⁡r∂ln⁡ki)Ki,kj≠iX_{RC,i} = \left(\frac{\partial \ln r}{\partial \ln k_i}\right)_{K_i, k_{j \neq i}}

Equation 1 · Chemistry

Chemical Dynamics and Catalysis in Practice: An Advanced Technical Guide

This equation states an equality: the expressions on both sides have the same value under the article’s assumptions.

The rigorous move is Charles Campbell’s degree of rate control (DRC) framework, which asks, for each elementary step, how much the overall rate would change if that step’s rate constant were perturbed while holding all others fixed [ 1 ] . A step with a DRC near one is rate-controlling; a step with a DRC near zero is kinetically irrelevant even if it is thermodynamically significant. This is not a qualitative heuristic — it is a derivative of the log of the rate with respect to the log of a single rate constant, evaluated at the reaction conditions of interest, and it can be computed from a microkinetic model built on density-functional or experimental barriers. Here r is the overall…

Meanings in this article

  • rr: the overall reaction rate.
  • kik_i: the rate constant of step i , and the partial derivative is taken holding the equilibrium constant KiK_i of that step fixed (so that only kinetics, not thermodynamics, is perturbed) and all other rate constants constant.
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