# Reaction rate for multiple steps

## Main Question or Discussion Point

Just a curious question.

How would the rate for a net reaction involving multiple steps of all similar (perhaps exact - i know it's highly unlikely) activation energies be calculated? I've been told to just look at the slowest step (highest EA) in all cases, even if they are similar in speed, but wouldn't the rate be significantly different if we don't take into account the other slow steps, even though they may not be the slowest?

Any insight and explanations would be very helpful! Thank you!

Ygggdrasil
Gold Member
2019 Award
For a reaction composed of N identical, irreversible steps each occurring with first order kinetics at a rate k, the waiting times for the appearance of the final product will follow an Erlang distribution, so the plot of [product] versus time will look like the cumulative distribution function of the Erlang distribution.

Coupled differential equations. Most likely only soluble numerically. Note that "activation energy" is a scalar quantity, and we KNOW that reactions progress through a high-dimensional space of both configuration, velocity, and internal degrees of freedom (vibrations, rotations, bond energies, all sorts of stuff). What I am saying is that the "big boys and girls" have moved beyond simple, crude and inaccurate (unless you consider calculated rates being within a "couple" of orders of magnitude to be "good" accuracy) models.

Chestermiller
Mentor
Good point. Once "stiff" ODE solvers became available in the 1970s, it became feasible to solve kinetics systems involving dozens of chemical species and hundreds of reactions without difficulty. No need to decide which reactions are very fast, and which are slow any more.

Chet

epenguin
Homework Helper
Gold Member
If you can assume the reaction gets into a steady state you can solve the linear equations algebraically (or also graph-theoretically).

If all the reactions are first order or pseudo-first order (and conditions are often set up so that they are) then you can get a solvable set of linear differential equations.

Just play around with something like

A ⇔ B → C , A ⇔ B ⇔ C → D

and the like and you will begin to see how it works out when rate constants are similar and when they are very different.

You get such linear equations in e.g. 'temperature jump' experiments where the system is suddenly though only slightly changed quickly and the 'relaxation' to new equilibrium followed.

Maybe a student will not be able to use the advanced methods of the last two posts. In fact a lead to where to find more on this material could be useful. To be able to solve hundreds of equations is one thing, but to be able to relate the solutions of such a mess to anything experimentally measured must be another, so I would be curious to know more of what these big boys and girls do.

Chestermiller
Mentor
If you can assume the reaction gets into a steady state you can solve the linear equations algebraically (or also graph-theoretically).

If all the reactions are first order or pseudo-first order (and conditions are often set up so that they are) then you can get a solvable set of linear differential equations.

Just play around with something like

A ⇔ B → C , A ⇔ B ⇔ C → D

and the like and you will begin to see how it works out when rate constants are similar and when they are very different.

You get such linear equations in e.g. 'temperature jump' experiments where the system is suddenly though only slightly changed quickly and the 'relaxation' to new equilibrium followed.

Maybe a student will not be able to use the advanced methods of the last two posts. In fact a lead to where to find more on this material could be useful. To be able to solve hundreds of equations is one thing, but to be able to relate the solutions of such a mess to anything experimentally measured must be another, so I would be curious to know more of what these big boys and girls do.
In models of atmospheric chemistry and transport, there are dozens of chemical species involved, participating in hundreds of reactions. These models are used to predict the effect of man-made pollutants, such as CFCs, on the changes in concentration of species (such as ozone) in the atmosphere. In many industrial chemical reactors to produce intermediates for making other desired products (e.g., polymers), there are also many species and reactions involved. We use reaction kinetic modeling to design and improve the operation of these reactors. In such applications, we are not trying to experimentally study the kinetics of any one reaction. We are trying to apply what was determined from simpler kinetics experiments in the laboratory to solve a much more complex practical problem.

For more information on solving large sets of non-linear coupled ordinary differential equations, some of which have very high rate constants and some of which have low rate constants, Google "stiff ordinary differential equations" or "C.W. Gear" (the pioneering mathematician who developed the clever techniques and original software for handling such problems). Software for solving stiff systems is now available for free online.

Chet