Editing Chemical equilibrium (section)

== Historical introduction ==

The [[Concept learning|concept]] of chemical equilibrium was developed in 1803, after [[Claude Louis Berthollet|Berthollet]] found that some [[chemical reaction]]s are [[Reversible reaction|reversible]].<ref>{{cite book|last1=Berthollet|first1=C.L.|title=Essai de statique chimique|trans-title=Essay on chemical statics|date=1803|publisher=Firmin Didot|location=Paris, France|url=https://archive.org/details/essaidestatiquec01bert|language=fr}} On pp. 404–407, Berthellot mentions that when he accompanied Napoleon on his expedition to Egypt, he (Berthellot) visited Lake Natron and found sodium carbonate along its shores.  He realized that this was a product of the reverse of the usual reaction Na<sub>2</sub>CO<sub>3</sub> + CaCl<sub>2</sub> → 2NaCl + CaCO<sub>3</sub>↓ and therefore that the final state of a reaction was a state of equilibrium between two opposing processes.  From p. 405: ''" ... la décomposition du muriate de soude continue donc jusqu'à ce qu'il se soit formé assez de muriate de chaux, parce que l'acide muriatique devant se partager entre les deux bases en raison de leur action, il arrive un terme où leurs forces se balancent."'' ( ... the [[Chemical decomposition|decomposition]] of the sodium chloride thus continues until enough calcium chloride is formed, because the hydrochloric acid must be shared between the two bases in the ratio of their action [i.e., capacity to react]; it reaches an end [point] at which their forces are balanced.)</ref> For any reaction mixture to exist at equilibrium, the [[reaction rate|rates]] of the forward and backward (reverse) reactions must be equal. In the following [[chemical equation]], arrows point both ways to indicate equilibrium.<ref>The notation {{eqm}} was proposed in 1884 by the Dutch chemist [[Jacobus Henricus van 't Hoff]]. See: {{cite book|last1=van 't Hoff|first1=J.H.|title=Études de Dynamique Chemique|trans-title=Studies of chemical dynamics|date=1884|publisher=Frederik Muller & Co.|location=Amsterdam, Netherlands|pages=4–5|url=https://archive.org/stream/etudesdedynamiqu00hoff#page/4/mode/2up|language=fr}} Van 't Hoff called reactions that didn't proceed to completion "limited reactions".  From pp. 4–5:  ''"Or M. Pfaundler a relié ces deux phénomênes ... s'accomplit en même temps dans deux sens opposés."'' (Now Mr. Pfaundler has joined these two phenomena in a single concept by considering the observed limit as the result of two opposing reactions, driving the one in the example cited to the formation of sea salt [i.e., NaCl] and nitric acid, [and] the other to hydrochloric acid and sodium nitrate.  This consideration, which experiment validates, justifies the expression "chemical equilibrium", which is used to characterize the final state of limited reactions.  I would propose to translate this expression by the following symbol:  
:HCl + NO<sub>3</sub> Na {{eqm}} NO<sub>3</sub> H + Cl Na . 
I thus replace, in this case, the = sign in the chemical equation by the sign {{eqm}}, which in reality doesn't express just equality but shows also the direction of the reaction.  This clearly expresses that a chemical action occurs simultaneously in two opposing directions.)</ref> A and B are [[reactant]] chemical species, S and T are product species, and [[Alpha (letter)|''α'']], [[Beta (letter)|''β'']], [[sigma|''σ'']], and [[Tau|''τ'']] are the [[stoichiometric coefficient]]s of the respective reactants and products:
:''α''&nbsp;A + ''β''&nbsp;B {{eqm}} ''σ''&nbsp;S + ''τ''&nbsp;T

The equilibrium concentration position of a reaction is said to lie "far to the right" if, at equilibrium, nearly all the reactants are consumed. Conversely the equilibrium position is said to be "far to the left" if hardly any product is formed from the reactants.

[[Cato Maximilian Guldberg|Guldberg]] and [[Peter Waage|Waage]] (1865), building on Berthollet's ideas, proposed the [[law of mass action]]:

:<math chem>\begin{align}
\text{forward reaction rate} &= k_{+} \ce{A}^\alpha\ce{B}^\beta \\
\text{backward reaction rate} &= k_{-} \ce{S}^\sigma\ce{T}^\tau
\end{align}</math>

where A, B, S and T are [[activity (chemistry)|active masses]] and ''k''<sub>+</sub> and ''k''<sub>−</sub> are [[rate constant]]s. Since at equilibrium forward and backward rates are equal:

:<math chem> k_+ \left\{ \ce A \right\}^\alpha \left\{\ce B \right\}^\beta = k_{-} \left\{\ce S \right\}^\sigma\left\{\ce T \right\}^\tau</math>

and the ratio of the rate constants is also a constant, now known as an [[equilibrium constant]].
:<math chem>K_c=\frac{k_+}{k_-}=\frac{\{\ce S\}^\sigma \{\ce T\}^\tau } {\{\ce A\}^\alpha \{\ce B\}^\beta}</math>

By convention, the products form the [[numerator]].
However, the [[law of mass action]] is valid only for concerted one-step reactions that proceed through a single [[transition state]] and is '''not valid in general''' because [[reaction rate#Rate equation|rate equations]] do not, in general, follow the [[stoichiometry]] of the reaction as Guldberg and Waage had proposed (see, for example, [[nucleophilic aliphatic substitution]] by S<sub>N</sub>1 or reaction of [[hydrogen]] and [[bromine]] to form [[hydrogen bromide]]). Equality of forward and backward reaction rates, however, is a [[Necessary and sufficient conditions|necessary condition]] for chemical equilibrium, though it is not [[Necessary and sufficient conditions|sufficient]] to explain why equilibrium occurs.

Despite the limitations of this derivation, the equilibrium constant for a reaction is indeed a constant, independent of the activities of the various species involved, though it does depend on temperature as observed by the [[van 't Hoff equation]]. Adding a [[catalyst]] will affect both the forward reaction and the reverse reaction in the same way and will not have an effect on the equilibrium constant. The catalyst will speed up both reactions thereby increasing the speed at which equilibrium is reached.<ref name=aj>{{cite book|last1=Atkins |first1=Peter W. |last2=Jones |first2=Loretta  |title=Chemical Principles: The Quest for Insight |edition=2nd |isbn=978-0-7167-9903-0|year=2008 }}</ref><ref>{{cite book|title=Chemistry: Matter and Its Changes |first=James E. |last=Brady |publisher=Fred Senese |edition=4th |isbn=0-471-21517-1|date=2004-02-04 }}</ref>

Although the [[macroscopic]] equilibrium concentrations are constant in time, reactions do occur at the molecular level. For example, in the case of [[acetic acid]] dissolved in water and forming [[acetate]] and [[hydronium]] ions,
:{{chem2|CH3CO2H + H2O <-> CH3CO2- + H3O+}}
a proton may hop from one molecule of acetic acid onto a water molecule and then onto an acetate anion to form another molecule of acetic acid and leaving the number of acetic acid molecules unchanged. This is an example of [[dynamic equilibrium]]. Equilibria, like the rest of thermodynamics, are statistical phenomena, averages of microscopic behavior.

'''[[Le Châtelier's principle]]''' (1884) predicts the behavior of an equilibrium system when changes to its reaction conditions occur. ''If a dynamic equilibrium is disturbed by changing the conditions, the position of equilibrium moves to partially reverse the change''. For example, adding more S (to the chemical reaction above) from the outside will cause an excess of products, and the system will try to counteract this by increasing the reverse reaction and pushing the equilibrium point backward (though the equilibrium constant will stay the same).

If [[mineral acid]] is added to the acetic acid mixture, increasing the concentration of hydronium ion, the amount of dissociation must decrease as the reaction is driven to the left in accordance with this principle. This can also be deduced from the equilibrium constant expression for the reaction:
:<math chem>K=\frac {\{\ce{CH3CO2-}\}\{\ce{H3O+}\}} \ce{\{CH3CO2H\}}</math>
If {H<sub>3</sub>O<sup>+</sup>} increases {CH<sub>3</sub>CO<sub>2</sub>H} must increase and {{chem2|CH3CO2-}} must decrease. The H<sub>2</sub>O is left out, as it is the solvent and its concentration remains high and nearly constant.

[[Josiah Willard Gibbs|J. W. Gibbs]] suggested in 1873 that equilibrium is attained when the "available energy" (now known as [[Gibbs free energy]] or Gibbs energy) of the system is at its minimum value, assuming the reaction is carried out at a constant temperature and pressure. What this means is that the derivative of the Gibbs energy with respect to [[reaction coordinate]] (a measure of the [[extent of reaction]] that has occurred, ranging from [[zero]] for all reactants to a maximum for all products) vanishes (because dG = 0), signaling a [[stationary point]]. This derivative is called the reaction Gibbs energy (or energy change) and corresponds to the difference between the [[chemical potential]]s of reactants and products at the composition of the reaction mixture.<ref name=Atkins/> This criterion is both necessary and sufficient. If a mixture is not at equilibrium, the liberation of the excess Gibbs energy (or [[Helmholtz energy]] at constant volume reactions) is the "driving force" for the composition of the mixture to change until equilibrium is reached. The equilibrium constant can be related to the standard [[Gibbs energy|Gibbs free energy]] change for the reaction by the equation

:<math>\Delta_rG^\ominus = -RT \ln K_\mathrm{eq}</math>

where ''R'' is the [[universal gas constant]] and ''T'' the [[temperature]].

When the reactants are [[Solution (chemistry)|dissolved]] in a medium of high [[ionic strength]] the quotient of [[activity coefficient]]s may be taken to be constant. In that case the '''concentration quotient''', ''K''<sub>c</sub>,
:<math chem>K_\ce{c}=\frac{[\ce S]^\sigma [\ce T]^\tau } {[\ce A]^\alpha [\ce B]^\beta}</math>
where [A] is the [[concentration]] of A, etc., is independent of the [[analytical concentration]] of the reactants. For this reason, equilibrium constants for [[Solution (chemistry)|solution]]s are usually [[Determination of equilibrium constants|determined]] in media of high ionic strength. ''K<sub>c</sub>'' varies with [[ionic strength]], temperature and pressure (or volume). Likewise ''K<sub>p</sub>'' for gases depends on [[partial pressure]]. These constants are easier to measure and encountered in high-school chemistry courses.