How do I write the deterministic rate equation for bimolecular reaction with similar particles?

I was wondering how to write down the deterministic rate equation for a bimolecular reaction with similar particles.

e.g.

$$\ce{A ->[k_+] 2B}$$

and

$$\ce{2B ->[k_-] A}$$

Now the rate equations for the above reactions are:

$$\mathrm{\frac{\delta A}{\delta t}= - k_{+} A + k_{-} B^2}$$

$$\mathrm{\frac{\delta B}{\delta t}= 2 k_{+} A - 2 k_{-} B^2}$$

Now I have included a factor of 2 in the second equation to conserve $\mathrm{A + \frac{B}{2}}$.

But I don't understand the reasoning for this factor of 2 in the second equation.

1 Answer

There is really only one independent equation here because

$\frac{dB}{dt} = -2\frac{dA}{dt}$

The "second equation" in the OP is just -2 times the first equation.

Every molecule of A destroyed creates 2 molecules of B.

Every 2 molecules of B destroyed creates 1 molecule of A.

• I understand that. but chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/… They mention the rate of the bimolecular reaction with similar species without a factor of 2. – nitin Nov 19 '14 at 16:17
• I don't see anything at that site that contradicts what you did. Certainly you could replace "2k" with a different constant "c" and still have a true equation. – DavePhD Nov 19 '14 at 16:43