Integrating Reaction Rates

huitzilopochtli

Hi,
I'm having trouble integrating the rate of reaction for a consecutive reaction.

Suppose A->B->C

rate of removal of A will be -k1a (smaller letter denotes concentration, numbers should be subscripts)
rate of formation of B will be k1a - k2b
rate of formation of C will be k2b

Now the first one is easy to integrate
da/dt = -k1a, a=a0 at time zero
a=a0exp[-k1t]

However, I have no idea how to integrate the second one which will look something like: db/dt= k1a0exp[-k1t] -k2b
b=0 at time zero
Assuming I've got that right, of course.

Now I have the answer to this integration I just don't know how to arrive at it. The books says its a standard form of an integration equation but I don't know how to do it.

Any help?

Squashie

If your situation can be treated as analogous to a radioactive decay chain (which is seems it can so I assume so) with A -> B -> C, and decay constants λa and λb (equivalent to your k's). Then expressions for the number of atoms N of type B (or concentration B in your case) after time t can be derived like so:

(Rate of change of Nb with respect to time.)
(dNb/dt) = -λb.Nb + λa.Na

(Rearranging and multiplying across by common factor.)
(dNb/dt).[exp(λb.t)] + λb.Nb.[exp(λb.t)] =λa.Na.[exp(λb.t)]

(Left hand side of the equation is equal to the differential product rule of..)
d/dt.[Nb.exp(λb.t)] = λa.Na.[exp(λb.t)]

(You know an equation for Na in terms of No. Substitute into above.)
Na = No[exp(-λa.t)]
d/dt.[Nb.exp(λb.t)] = λa. No.[exp(-λa.t)] .[exp(λb.t)]

(Sort out your powers on the right side of the equation and simplify.)
d/dt.[Nb.exp(λb.t)] = λa.No.[ exp((λb-λa).t) ]

(Integrate both sides with respect to time dt.)
[Nb.exp(λb.t)] = Integrate{λa.No.[ exp((λb-λa).t) ] }dt
[Nb.exp(λb.t)] = (λa.No). Integrate { exp((λb-λa).t) } dt
[Nb.exp(λb.t)] = (λa.No.).[1/(λb-λa)].exp((λb-λa).t) + Constant
[Nb.exp(λb.t)] = No.[λa/(λb-λa)].exp((λb-λa).t) + Constant

(At t = 0, Nb = 0. Substitute these values into the above and calculate integration constant.)
Constant = -[λa/(λb-λa)].No

(Substitute back in your constant. Multiply equation across by exp(-λb.t). Rearrange and simplify.)
Nb = No.[λa/(λb-λa)].[exp(-λa.t) - exp(-λa.t)]


You have an expression for #atoms (or your concentrations) entirely in terms of the #atoms at the beginning (your original concentration) and decay constants (reaction constants).