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 (one of the natron lakes in Egypt, in limestone): : He recognized this as the reverse of the familiar reaction : 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. 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. == Reaction kinetics ==

For the reversible reaction A⇌B, the forward step A→B has a rate constant k_1 and the backwards step B→A has a rate constant k_{-1}. The concentration of A obeys the following differential equation: {{NumBlk|:|\frac{d[\mathrm{A}]}{dt}=-k_\text{1}[\mathrm{A}]+k_\text{-1}[\mathrm{B}].|}} 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 t, we can set up the following equation: {{NumBlk|:|[\mathrm{B}]=[\mathrm{A}]_\text{0}-[\mathrm{A}].|}} Combining and , we can write :\frac{d[\mathrm{A}]}{dt}=-k_\text{1}[\mathrm{A}]+k_\text{-1}([\mathrm{A}]_\text{0}-[\mathrm{A}]). Separation of variables is possible and using an initial value [\mathrm{A}](t=0) = [\mathrm{A}]_0, we obtain: :C=\frac{{-\ln}(-k_\text{1}[\mathrm{A}]_\text{0})}{k_\text{1}+k_\text{-1}} and after some algebra we arrive at the final kinetic expression: :[\mathrm{A}]=\frac{k_\text{-1}[\mathrm{A}]_\text{0}}{k_\text{1}+k_\text{-1}}+\frac{k_\text{1}[\mathrm{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: :[\mathrm{A}]_\infty=\frac{k_\text{-1}[\mathrm{A}]_\text{0}}{k_\text{1}+k_\text{-1}} :[\mathrm{B}]_\infty=[\mathrm{A}]_\text{0}-[\mathrm{A}]_\infty=[\mathrm{A}]_\text{0}-\frac{k_\text{-1}[\mathrm{A}]_\text{0}}{k_\text{1} +k_\text{-1}} :\frac{[\mathrm{B}]_\infty}{[\mathrm{A}]_\infty}=\frac{k_\text{1}}{k_\text{-1}}=K_\text{eq} :[\mathrm{A}]=[\mathrm{A}]_\infty+([\mathrm{A}]_\text{0}-[\mathrm{A}]_\infty)\exp(-k_\text{1}+k_\text{-1})t Thus, the formula can be linearized in order to determine k_1+k_{-1}: :\ln([\mathrm{A}]-[\mathrm{A}]_\infty)=\ln([\mathrm{A}]_\text{0}-[\mathrm{A}]_\infty)-(k_\text{1}+k_\text{-1})t To find the individual constants k_1 and k_{-1}, the following formula is required: :K_\text{eq}=\frac{k_\text{1}}{k_\text{-1}}=\frac{[\mathrm{B}]_\infty}{[\mathrm{A}]_\infty} ==See also==