Editing Reversible reaction (section)

== Reaction kinetics ==
For the reversible reaction A⇌B, the forward step A→B has a rate constant <math>k_1</math> and the backwards step B→A has a rate constant <math>k_{-1}</math>. The concentration of A obeys the following differential equation:

{{NumBlk|:|<math>\frac{d[A]}{dt}=-k_\text{1}[A]+k_\text{-1}[B]</math>.|{{EquationRef|1}}}}

If we consider that the concentration of product B at anytime is equal to the concentration of reactants at time zero minus the concentration of reactants at time <math>t</math>, we can set up the following equation:

{{NumBlk|:|<math>[B]=[A]_\text{0}-[A]</math>.|{{EquationRef|2}}}}

Combining {{EquationNote|1}} and {{EquationNote|2}}, we can write

:<math>\frac{d[A]}{dt}=-k_\text{1}[A]+k_\text{-1}([A]_\text{0}-[A])</math>.

Separation of variables is possible and using an initial value <math>[A](t=0) = [A]_0</math>, we obtain:

:<math>C=\frac{{-\ln}(-k_\text{1}[A]_\text{0})}{k_\text{1}+k_\text{-1}}</math>

and after some algebra we arrive at the final kinetic expression:

:<math>[A]=\frac{k_\text{-1}[A]_\text{0}}{k_\text{1}+k_\text{-1}}+\frac{k_\text{1}[A]_\text{0}}{k_\text{1}+k_\text{-1}}\exp{{(-k_\text{1}+k_\text{-1}})t}</math>.

The concentration of A and B at infinite time has a behavior as follows:

:<math>[A]_\infty=\frac{k_\text{-1}[A]_\text{0}}{k_\text{1}+k_\text{-1}}</math>

:<math>[B]_\infty=[A]_\text{0}-[A]_\infty=[A]_\text{0}-\frac{k_\text{-1}[A]_\text{0}}{k_\text{1} +k_\text{-1}}</math>

:<math>\frac{[B]_\infty}{[A]_\infty}=\frac{k_\text{1}}{k_\text{-1}}=K_\text{eq}</math>

:<math>[A]=[A]_\infty+([A]_\text{0}-[A]_\infty)\exp(-k_\text{1}+k_\text{-1})t</math>

Thus, the formula can be linearized in order to determine <math>k_1+k_{-1}</math>:

:<math>\ln([A]-[A]_\infty)=\ln([A]_\text{0}-[A]_\infty)-(k_\text{1}+k_\text{-1})t</math>

To find the individual constants <math>k_1</math> and <math>k_{-1}</math>, the following formula is required:

:<math>K_\text{eq}=\frac{k_\text{1}}{k_\text{-1}}=\frac{[B]_\infty}{[A]_\infty}</math>