Consider an equilibrium
[ce]
A <=> B
[/ce]
Assuming both the forward and reverse reactions are elementary, the rates of reaction can be modeled by the differential equations:
[ce]
\frac{d}{dt} = p[A] - q
[/ce]
[ce]
\frac{d[A]}{dt} = q - p[A]
[/ce]
Where [ce]q[/ce] is the rate constant for the forward reaction and [ce]p[/ce] is the rate constant for the reverse reaction. The general solution of this system is:
[ce]
= \frac{p_0-q[A]_0}{p+q}e^{-pt-qt}+q\left(\frac{[A]_0+_0}{p+q}\right)
[/ce]
[ce]
[A] = \frac{q[A]_0-p_0}{p+q}e^{-pt-qt}+p\left(\frac{[A]_0+_0}{p+q}\right)
[/ce]
By taking the limit as [ce]t[/ce] goes to infinity, it can be shown that
[ce]
\frac{q}{p} = K
[/ce]
where [ce]K[/ce] is the equilibrium constant. [ce]K[/ce] can be derived from the Gibbs energy of reaction by the equation
[ce]
\Delta G = -RT \ln{K}
[/ce]
Are there any errors in this line of reasoning? If there aren't, is it possible to predict values for [ce]p[/ce] and [ce]q[/ce] theoretically (i.e. without actually performing an experiment)?
