Reversible reaction

The concept of a reversible reaction was introduced by Claude Louis Berthollet in 1803, after he had observed the formation of sodium carbonate crystals at the edge of a salt lake[3] (one of the natron lakes in Egypt, in limestone):

2NaCl + CaCO₃ → Na₂CO₃ + CaCl₂

He recognized this as the reverse of the familiar reaction

Na₂CO₃ + CaCl₂→ 2NaCl + CaCO₃

Until then, chemical reactions were thought to always proceed in one direction. Berthollet reasoned that the excess of salt in the lake helped push the "reverse" reaction towards the formation of sodium carbonate.[4]

In 1864, Peter Waage and Cato Maximilian Guldberg formulated their law of mass action which quantified Berthollet's observation. Between 1884 and 1888, Le Chatelier and Braun formulated Le Chatelier's principle, which extended the same idea to a more general statement on the effects of factors other than concentration on the position of the equilibrium.

For the reversible reaction A⇌B, the forward step A→B has a rate constant ${\displaystyle k_{1}}$ and the backwards step B→A has a rate constant ${\displaystyle k_{-1}}$. The concentration of A obeys the following differential equation:

${\displaystyle {\frac {d[A]}{dt}}=-k_{\text{1}}[A]+k_{\text{-1}}[B]}$.

(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 ${\displaystyle t}$, we can set up the following equation:

${\displaystyle [B]=[A]_{\text{0}}-[A]}$.

(2)

Combining 1 and 2, we can write

${\displaystyle {\frac {d[A]}{dt}}=-k_{\text{1}}[A]+k_{\text{-1}}([A]_{\text{0}}-[A])}$.

Separation of variables is possible and using an initial value ${\displaystyle [A](t=0)=[A]_{0}}$, we obtain:

${\displaystyle C={\frac {{-\ln }(-k_{\text{1}}[A]_{\text{0}})}{k_{\text{1}}+k_{\text{-1}}}}}$

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

${\displaystyle [A]={\frac {k_{\text{-1}}[A]_{\text{0}}}{k_{\text{1}}+k_{\text{-1}}}}+{\frac {k_{\text{1}}[A]_{\text{0}}}{k_{\text{1}}+k_{\text{-1}}}}\exp {{(-k_{\text{1}}+k_{\text{-1}}})t}}$.

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

${\displaystyle [A]_{\infty }={\frac {k_{\text{-1}}[A]_{\text{0}}}{k_{\text{1}}+k_{\text{-1}}}}}$

${\displaystyle [B]_{\infty }=[A]_{\text{0}}-[A]_{\infty }=[A]_{\text{0}}-{\frac {k_{\text{-1}}[A]_{\text{0}}}{k_{\text{1}}+k_{\text{-1}}}}}$

${\displaystyle {\frac {[B]_{\infty }}{[A]_{\infty }}}={\frac {k_{\text{1}}}{k_{\text{-1}}}}=K_{\text{eq}}}$

${\displaystyle [A]=[A]_{\infty }+([A]_{\text{0}}-[A]_{\infty })\exp(-k_{\text{1}}+k_{\text{-1}})t}$

Thus, the formula can be linearized in order to determine ${\displaystyle k_{1}+k_{-1}}$:

${\displaystyle \ln([A]-[A]_{\infty })=\ln([A]_{\text{0}}-[A]_{\infty })-(k_{\text{1}}+k_{\text{-1}})t}$

To find the individual constants ${\displaystyle k_{1}}$ and ${\displaystyle k_{-1}}$, the following formula is required:

${\displaystyle K_{\text{eq}}={\frac {k_{\text{1}}}{k_{\text{-1}}}}={\frac {[B]_{\infty }}{[A]_{\infty }}}}$

• Dynamic equilibrium
• Chemical equilibrium
• Irreversibility
• Microscopic reversibility
• Static equilibrium