Equation 1 · Chemical Dynamics and Catalysis in Practice: An Advanced Technical Guide
What does this equation mean?
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.
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.
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Symbol X_RC,i
C,i is part of the quantity the equation computes from the expression on the right.
Symbol k_i
the rate constant of step i , and the partial derivative is taken holding the equilibrium constant of that step fixed (so that only kinetics, not thermodynamics, is perturbed) and all other rate constants constant.
Symbol K_i
is an input to the expression that computes the quantity on the left.
Symbol k_j neq i
neq i is an input to the expression that computes the quantity on the left.
=
The expressions on both sides represent the same quantity under the stated assumptions.
See an illustrated explanation →derivative
This notation tracks how one quantity changes with another. The indicated variable tells which change is being measured.
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.
Denominator: partial ln k_i
The complete quantity below the fraction bar; it must be nonzero for this division.
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
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…
Read the full surrounding passage
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 reaction rate, is the rate constant of step i , and the partial derivative is taken holding the equilibrium constant of that step fixed (so that only kinetics, not thermodynamics, is perturbed) and all other rate constants constant. This equation matters because it is the formal object the rest of this section’s isotope analysis is trying to estimate indirectly — DRC values are rarely measured directly; they are inferred from how observables like isotope ratios and apparent activation energies respond to controlled perturbations.
Sources cited in the surrounding passage
These citations give research context. Read each source to check which claims it supports.
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