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

Question

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.

Answer 1

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.