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Reaction rate for multiple steps 
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#1
Apr214, 03:57 PM

P: 295

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 E_{A}) 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! 


#2
Apr214, 04:48 PM

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P: 1,393

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.



#3
Apr714, 07:42 PM

P: 140

Coupled differential equations. Most likely only soluble numerically. Note that "activation energy" is a scalar quantity, and we KNOW that reactions progress through a highdimensional 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.



#4
Apr714, 11:17 PM

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PF Gold
P: 5,259

Reaction rate for multiple steps
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 


#5
Apr814, 05:33 AM

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P: 1,987

If you can assume the reaction gets into a steady state you can solve the linear equations algebraically (or also graphtheoretically).
If all the reactions are first order or pseudofirst 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. 


#6
Apr814, 07:24 AM

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PF Gold
P: 5,259

For more information on solving large sets of nonlinear 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 


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