I am reading a textbook by Keith J. Laidler titled 'Chemical Kinetics' (3rd ed.). Two different differential forms are given for the reaction (pp30, pp38): $ 2A \leftrightarrow B $ with forward rate $k_1$ and backward rate $k_2$.
- $\frac{dx}{dt} = k_1(a_0 - 2x)^2 - k_2x $
- $\frac{dx}{dt} = k_1(a_0 - x) - k_2 (x/2) $
The second formulation is intuitive to me i.e. A will lose 1 molecule to produce 1/2 molecule of B. In the first, I am bit confused about $(a_0 - 2x)^2$ term, shouldn't it be $(a_0-x)^2$?
I was hoping to get the same solution trajectories for both equations, but I am not. With values $a_0=1$, $k_1 = 2$ and $k_2 = 1$, solution at $t=10$ for the first equation is approximately 0.3048 while for second, it is 0.8.
In the reaction $2A\to B$, one considers that two molecules $A$ should be in contact to become a molecule $B$. This event is typically assumed to happen during a time interval of length $dt$ with a probability proportional to the product of the concentrations of the molecules involved in the meeting hence, in your case, with a probability proportional to $[A]\cdot[A]=[A]^2$. And each of these events leads to the creation of one molecule $B$ and to the disappearance of two molecules $A$.
On the other hand, the reaction $B\to2A$ happens at a rate proportional to $[B]$ and leads to the creation of two molecules $A$ and to the disappearance of one molecule $B$.
Introducing the concentrations $a(t)$ and $b(t)$ of $A$ and $B$ at time $t$, these considerations lead to $$a'(t)=-2k_1a(t)^2+2k_2b(t)$$ and $$b'(t)=k_1a(t)^2-k_2b(t)$$ The dynamics above implies the conservation law one was expecting from the start, namely, $$a'(t)+2b'(t)=0$$ which implies $$a(t)+2b(t)=a_0+2b_0$$ Now, your book might be assuming that $b_0=0$ and might define $$x(t)=b(t)$$ in which case $$x'(t)=k_1(a_0-2x(t))^2-k_2x(t)$$ that is, the first formulation in your post. The second formulation in your post, on the other hand, seems to violate everything we know about the dynamics of chemical reactions and it produces very different graphs of the function $t\mapsto x(t)$.