# Law of mass action

The law of mass action (abbreviation "MWG") states that in a reversible reaction in chemical equilibrium the quotient from

has a fixed value that is characteristic of the reaction under consideration, the equilibrium or mass action constant .

The reason is that in equilibrium, the back and forth reactions are equally fast (see below). ${\ displaystyle v _ {\ text {hin}} = v _ {\ text {back}}}$

The size of the equilibrium constant follows from the thermodynamic equilibrium condition of maximum entropy , i.e. H. minimum Gibbs energy . It depends on external conditions such as temperature , concentration and pressure .

## history

Cato Maximilian Guldberg (left) and Peter Waage in 1891

The law of mass action was determined experimentally by the Norwegian chemists Cato Maximilian Guldberg and Peter Waage and first published in Norwegian in 1864 and in French in 1867 (with their experimental data and a modified law). They had derived the law of mass action from the so-called “active mass” (an outdated expression for activity) instead of from concentration. Their publication did not attract much attention for a long time. Irishman John Hewitt Jellett came to similar conclusions in 1873 and Jacobus Henricus van 't Hoff also in 1877. Especially after the publications of Van' t Hoff (but also by others like August Friedrich Horstmann ), Guldberg and Waage had the impression that their work was not enough was known and in 1879 they published a more detailed explanation in German in the Journal für Praktische Chemie. Van 't Hoff then recognized their priority.

## Exact formulation for reactions in chemical equilibrium

The law of mass action applies to chemical reactions, for example the following:

${\ displaystyle | \ nu _ {\ mathrm {A}} | \, \ mathrm {A} + | \ nu _ {\ mathrm {B}} | \, \ mathrm {B} \ rightleftharpoons \ nu _ {\ mathrm {C}} \, \ mathrm {C} + \ nu _ {\ mathrm {D}} \, \ mathrm {D}}$

If a reaction equation is formulated with stoichiometric numbers, the absolute value of the stoichiometric numbers is used, since in a reaction equation the numbers are always positive. If no number is given for a reaction partner, the factor 1 is meant.

The general formulation of the chemical equilibrium constants using the law of mass action is:

${\ displaystyle K = \ prod _ {i} a_ {i} ^ {{\ nu} _ {i}} = \ exp \ left ({\ frac {\ Delta G | _ {GG} (T, p, N_ {i}, \ dots) - \ Delta G ^ {\ circ} (T, p, N_ {i}, \ dots)} {k _ {\ mathrm {B}} T}} \ right),}$

where the subscript GG means evaluation of the change in energy due to the reaction in chemical equilibrium.

The negative stoichiometric numbers of the educts lead to negative exponents when setting up the equation and ensure that the educts appear in the denominator of the fraction:

${\ displaystyle K = {\ frac {{a _ {\ text {C}}} ^ {| \ nu _ {\ text {C}} |} \ cdot {a _ {\ text {D}}} ^ {| \ nu _ {\ text {D}} |}} {{a _ {\ text {A}}} ^ {| \ nu _ {\ text {A}} |} \ cdot {a _ {\ text {B}}} ^ {| \ nu _ {\ text {B}} |}}}}$

Here are:

• Π the product mark ,
• ${\ displaystyle a_ {i} = \ exp \ left ({\ frac {\ mu _ {i} - \ mu _ {i} ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right )}$the relative activity of the i th species in a reaction equation (Man the mass action law may establish with absolute activities, but then slightly different relationships apply)
• ${\ displaystyle \ nu _ {i}}$the stoichiometric number of the i th particle type, being used for starting materials (left half of a reaction equation) and negative sign for products (right half of a reaction equation) positive sign for the numbers,
• K is the equilibrium constant of the given reaction equation.
• ${\ displaystyle \ Delta G = G _ {\ text {final}} - G _ {\ text {begin}}}$is the change in Gibbs energy due to the reaction.
• ${\ displaystyle k _ {\ mathrm {B}}}$the Boltzmann constant and T the temperature in Kelvin.

The equilibrium constant K indicates the position of the equilibrium, i.e. it describes how many product molecules there are on one reactant molecule. An exact derivation of the law, which is independent of the reaction path, takes place in thermodynamics with the help of the chemical potential .

In chemical equilibrium, at constant temperature and constant pressure, the following applies . Hence the equilibrium constant is given by . If the change in the standard reaction Gibbs energy of a reaction is known, the equilibrium constant can be calculated from this (see also reaction quotient ) ${\ displaystyle \ Delta G | _ {GG} = \ sum _ {i} \ nu _ {i} \ mu _ {i} = 0}$${\ displaystyle K = \ exp (- {\ frac {\ Delta G ^ {\ circ}} {k _ {\ mathrm {B}} T}})}$ ${\ displaystyle \ Delta _ {r} G ^ {\ circ}}$

The equilibrium constant depends on:

1. the temperature
2. and in the liquid phase also from the pressure in the system :, where is the molar reaction volume .${\ displaystyle \ left ({\ frac {\ partial \ ln K} {\ partial p}} \ right) _ {T} = - {\ frac {\ Delta _ {R} {\ overline {V}} _ { \ text {fl}} ^ {0}} {RT}}}$${\ displaystyle {\ overline {V}} _ {\ text {fl}} ^ {0}}$

The temperature dependence is described by the van-'t-Hoff'sche reaction isobars (if the pressure in the system is constant) or by the van-'t-Hoff'sche reaction isochore (if the volume in the system is constant).

### variants

Not only relative activities can be inserted into the law of mass action. Instead of having the activity of the law of mass action is often the molar concentration (Reactions in solution ), the partial pressure (reactions in a gas phase) or the amount of substance placed, whereby in general the numerical value of K changes. The law of mass action can also be expressed by a combination of these quantities (pressure, concentration, ...). To distinguish it, add to the index of K the specification of the quantity with which K was calculated ( K c for concentration, K p for partial pressure, K x for the amount of substance). The various constants can be converted into one another using simple relationships.

## Derivation via the reaction kinetics

The speed of a chemical reaction is directly proportional to the relative activity of the starting materials, high their stoichiometric coefficient:

${\ displaystyle v \, \ sim \, a ({\ text {Edukte}}) ^ {| \ nu _ {i} |}}$

The higher the activity of the starting materials, the faster the reaction takes place.

In the course of an equilibrium reaction, the activity of the starting materials steadily decreases. This also reduces the speed of the forward reaction . At the same time, the activity of the products is constantly increasing. This increases the speed of the reverse reaction. Finally, if both reaction rates are the same, the same amount of product as starting material is formed in the same time span; i.e. that equilibrium has been reached.

In the reaction equation , the equilibrium arrow is used to describe:

${\ displaystyle a \; {\ text {A}} + b \; {\ text {B}} \ \ rightleftharpoons \ c \; {\ text {C}} + d \; {\ text {D}}}$

The speed of the forward chemical reaction or the reverse chemical reaction is: ${\ displaystyle v _ {\ mathrm {hin}}}$${\ displaystyle v _ {\ text {back}}}$

${\ displaystyle v _ {\ mathrm {hin}} = k _ {\ mathrm {hin}} \ cdot a ^ {a} ({\ text {A}}) \ cdot a ^ {b} ({\ text {B} })}$
${\ displaystyle v _ {\ text {back}} = k _ {\ text {back}} \ cdot a ^ {c} ({\ text {C}}) \ cdot a ^ {d} ({\ text {D} })}$

Here, k hin is the rate constant of the forward reaction and k back is the rate constant of the reverse reaction.

In the state of equilibrium, the speeds of the forward and backward reactions are the same (compare Detailed Balance ):

${\ displaystyle v _ {\ mathrm {hin}} = v _ {\ text {back}}}$

From this it follows for the equilibrium constant (also mass action constant ): ${\ displaystyle K}$

${\ displaystyle K = {\ frac {k _ {\ mathrm {hin}}} {k _ {\ text {back}}}} = {\ frac {a ^ {c} ({\ text {C}}) \ cdot a ^ {d} ({\ text {D}})} {a ^ {a} ({\ text {A}}) \ cdot a ^ {b} ({\ text {B}})}}}$.

The rates and are themselves concentration-dependent, since the chemical potentials are concentration-dependent. In accordance with the theory of the transition state , we write: ${\ displaystyle k _ {\ mathrm {hin}}}$${\ displaystyle k _ {\ text {back}}}$

${\ displaystyle k _ {\ text {hin}} = \ kappa {\ frac {k _ {\ mathrm {B}} T} {h}} \ exp \ left (- {\ frac {{G \ ddagger} ^ {\ circ} -G _ {\ text {reactants}} ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right)}$
${\ displaystyle k _ {\ text {back}} = \ kappa {\ frac {k _ {\ mathrm {B}} T} {h}} \ exp \ left (- {\ frac {{G \ ddagger} ^ {\ circ} -G _ {\ text {Products}} ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right),}$

wherein the Plank's constant, is the so-called transmission coefficient in the transition state theory, the standard free energy of a reaction-specific transition state , and the standard free energy are of the reactants. This results (with ): ${\ displaystyle h}$${\ displaystyle \ kappa}$${\ displaystyle {G \ ddagger} ^ {\ circ}}$${\ displaystyle G _ {\ text {Reactants}} ^ {\ circ} = \ sum _ {i} \ mu _ {i} ^ {\ circ} N_ {i}}$${\ displaystyle \ Delta N_ {i} = \ nu _ {i}}$

${\ displaystyle K = {\ frac {k _ {\ mathrm {hin}}} {k _ {\ text {back}}}} = \ exp \ left (- {\ frac {\ Delta G _ {\ text {reactants}} ^ {\ circ} - \ Delta G _ {\ text {Products}} ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right) = \ exp \ left (- {\ frac {\ Delta G ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right) = \ exp \ left (- {\ frac {\ sum _ {i} \ nu _ {i} \ mu _ {i } ^ {\ circ}} {k _ {\ mathrm {B}} T}} \ right),}$

with , and${\ displaystyle \ Delta G _ {\ text {Reactants}} ^ {\ circ} = {G \ ddagger} ^ {\ circ} -G _ {\ text {Reactants}} ^ {\ circ}}$${\ displaystyle \ Delta G _ {\ text {Products}} ^ {\ circ} = {G \ ddagger} ^ {\ circ} -G _ {\ text {Products}} ^ {\ circ}}$${\ displaystyle \ Delta G ^ {\ circ} = G _ {\ text {Products}} ^ {\ circ} -G _ {\ text {Reactants}} ^ {\ circ}}$

## Homogeneous solution equilibria

For the description of the reactions in dilute solution, the respective concentration is normally used as a parameter. In the case of more concentrated solutions, however, the activity coefficient can deviate significantly from 1, so that this approximation must be used with caution.

The law of mass action is used for example for the reaction

${\ displaystyle \ mathrm {\ alpha \, A + \ beta \, B \ \ rightleftharpoons \ \ gamma \, C + \ delta \, D}}$

formulated as follows:

${\ displaystyle K_ {c} = {\ frac {c ^ {\ mathrm {\ gamma}} (\ mathrm {C}) \ cdot c ^ {\ mathrm {\ delta}} (\ mathrm {D})} { c ^ {\ mathrm {\ alpha}} (\ mathrm {A}) \ cdot c ^ {\ mathrm {\ beta}} (\ mathrm {B})}}}$

Here, c (A), c (B), c (C), c (D) are the molar equilibrium concentrations of the starting materials (in the denominator) or products (in the numerator). They are also often briefly noted as [A], [B], [C], and [D]. The exponent contains the stoichiometric number, i.e. the number of particles of this type that are required for a formula conversion , whereby their signs are omitted without noting any extra signs.

The equilibrium constant can have dimensions, but it is always dimensionless. ${\ displaystyle K_ {c}}$${\ displaystyle K}$

## Homogeneous gas equilibria

In the case of homogeneous gas equilibria, the following applies to the constant of mass action K p with the partial pressures p i :

${\ displaystyle K_ {p} = \ prod _ {i} p_ {i} ^ {{\ nu} _ {i}}}$

In the formation of hydrogen iodide from elemental hydrogen and iodine

${\ displaystyle \ mathrm {{H_ {2}} _ {(g)} + {I_ {2}} _ {(g)} \ rightleftharpoons 2 \ {HI} _ {(g)}}}$ the balance arises
${\ displaystyle K_ {p} = {\ frac {p ^ {2} \ mathrm {(HI)}} {p \ mathrm {(H_ {2})} \ cdot p \ mathrm {(I_ {2})} }}}$ a.

K p and K c or p i and c i can be linked to one another using the ideal gas equation :

${\ displaystyle p_ {i} V = n_ {i} RT}$
${\ displaystyle \ Leftrightarrow p_ {i} = {\ frac {n_ {i}} {V}} RT = c_ {i} RT}$

For the constant of mass action K p during the formation of hydrogen iodide we get:

${\ displaystyle K_ {p} = {\ frac {[c \ mathrm {(HI)} RT] ^ {2}} {c \ mathrm {(H_ {2})} RT \ cdot \ c \ mathrm {(I_ {2})} RT}} = {\ frac {c ^ {2} \ mathrm {(HI)}} {c \ mathrm {(H_ {2})} \ cdot c \ mathrm {(I_ {2}) }}} = K_ {c}}$

If the number of particles in the products in a gas phase equilibrium is equal to the number of particles in the starting materials, RT is canceled out in the law of mass action. However, if one looks at the reaction of sulfur dioxide and oxygen to form sulfur trioxide

${\ displaystyle \ mathrm {2 \ {SO_ {2}} _ {(g)} + {O_ {2}} _ {(g)} \ rightleftharpoons 2 \ {SO_ {3}} _ {(g)}} }$ With ${\ displaystyle K_ {p} = {\ frac {p ^ {2} \ mathrm {\ mathrm {(SO_ {3})}}} {p ^ {2} \ mathrm {(SO_ {2})} \ cdot p \ mathrm {(O_ {2})}}}}$

and replacing the pressures with concentrations results in:

${\ displaystyle K_ {p} = {\ frac {[c \ mathrm {(SO_ {3})} RT] ^ {2}} {[c \ mathrm {(SO_ {2})} RT] ^ {2} \ cdot c \ mathrm {(O_ {2})} RT}} = {\ frac {c ^ {2} \ mathrm {\ mathrm {(SO_ {3})}}} {c ^ {2} \ mathrm { (SO_ {2})} \ cdot c \ mathrm {(O_ {2})}}} \ cdot {\ frac {1} {RT}} = K_ {c} \ cdot {\ frac {1} {RT} }}$

The number of particles decreases during the reaction and a factor 1 / ( RT ) remains in the law of mass action.

If you consider the formation of ammonia in the Haber-Bosch process, the following results:

${\ displaystyle \ mathrm {{3 \ H_ {2}} _ {(g)} + {N_ {2}} _ {(g)} \ rightleftharpoons {2 \ NH_ {3}} _ {(g)}} }$
${\ displaystyle K_ {p} = {\ frac {p ^ {2} \ mathrm {(NH_ {3})}} {p \ mathrm {(N_ {2})} \ cdot p ^ {3} \ mathrm { (H_ {2})}}}}$
${\ displaystyle K_ {p} = {\ frac {[c \ mathrm {(NH_ {3})} RT] ^ {2}} {c \ mathrm {(N_ {2})} RT \ cdot [c \ mathrm {(H_ {2})} RT] ^ {3}}} = {\ frac {c ^ {2} \ mathrm {(NH_ {3})}} {c \ mathrm {(N_ {2})} \ cdot c ^ {3} \ mathrm {(H_ {2})}}} \ cdot {\ frac {1} {(RT) ^ {2}}} = K_ {c} \ cdot {\ frac {1} { (RT) ^ {2}}}}$

In general, the law of mass action of a gas phase equilibrium can be expressed as:

${\ displaystyle K_ {p} = K_ {c} \ cdot {(RT) ^ {\ sum \ nu _ {i}}}}$

Here Σ ν i is the sum of the stoichiometric numbers of the reaction under consideration. In the case of the formation of HI from the elements, Σ ν i = −1–1 + 2 = 0 and in the Haber-Bosch method Σ ν i = −3–1 + 2 = −2.

Alternatively, it is often useful to specify the composition of the gas phase using mole fractions (molar proportions χ i ):

${\ displaystyle K_ {x} = \ prod _ {i} x_ {i} ^ {{\ nu} _ {i}}}$ With ${\ displaystyle x_ {i} = {\ frac {p_ {i}} {p}}}$

In general, the following applies here:

${\ displaystyle K_ {x} = K_ {c} \ cdot {\ left ({\ dfrac {RT} {p}} \ right) ^ {\ sum \ nu _ {i}}}}$

## understanding

Here are some of the points to keep in mind:

• The MWG applies to each individual partial reaction. Overall, it often appears as if a reaction consists of only one reaction step, but actually consists of many individual steps with more species than those that appear in the reaction equation. These must also be taken into account (e.g. all chain reactions ).
• The MWG only describes thermodynamically controlled processes. A high activation energy can mean that the actual state of equilibrium is not reached ( diamond is only a metastable modification of carbon under normal conditions , but the activation energy for a rearrangement to graphite is so high that the reaction generally does not take place or only takes place immeasurably slowly ).
• All activities used are equilibrium activities that are often difficult to measure or calculate.
• For a given reaction, the equilibrium constant K depends on the temperature and the pressure. The pressure dependence is very weak in the condensed phase and is often neglected.

## Examples

${\ displaystyle \ mathrm {2 \ {CO} _ {(g)} \ rightleftharpoons {CO_ {2}} _ {(g)} + {C} _ {(s)}} \ quad \ quad K_ {p} = {\ frac {p \ mathrm {(CO_ {2})}} {p ^ {2} \ mathrm {(CO)}}}}$
${\ displaystyle \ mathrm {CH_ {3} COOH _ {(aq)} + H_ {2} O _ {(aq)} \ rightleftharpoons H_ {3} O _ {(aq)} ^ {+} + CH_ {3} COO_ { (aq)} ^ {-}} \ quad \ quad K _ {\ mathrm {s}} = {\ frac {c (\ mathrm {H} _ {3} \ mathrm {O} ^ {+}) \ cdot c (\ mathrm {CH_ {3} COO} ^ {-})} {c (\ mathrm {CH_ {3} COOH})}} = K \ cdot c (\ mathrm {H_ {2} O})}$

Synthesis of potassium hexacyanidoferrate (II)

${\ displaystyle \ mathrm {Fe ^ {2 +} + 6 \ CN ^ {-} \ rightleftharpoons [Fe (CN) _ {6}] ^ {4-}} \ qquad K_ {A} = \ mathrm {\ frac {c ([Fe (CN) _ {6}] ^ {4 -})} {c (Fe ^ {2 +}) \ cdot c ^ {6} (CN ^ {-})}}}$

## Related sizes and principles

 K L Solubility product K S Acid constant K B Base constant K W Ion product of water buffer

## The MWG in semiconductor electronics

The law of mass action states that in semiconductors in thermal equilibrium the product of the charge carrier densities from the valence and conduction band is constant.

${\ displaystyle n_ {0} p_ {0} = {n_ {i}} ^ {2}}$

With the intrinsic charge carrier density and the densities of free electrons and holes in thermal equilibrium . The law of mass action applies in intrinsic, ie undoped, as well as in doped semiconductors. ${\ displaystyle n_ {i}}$${\ displaystyle n_ {0}, p_ {0}}$

## literature

• Charles E. Mortimer: Chemistry. 7th edition. Georg Thieme Verlag, Stuttgart 2001, ISBN 3-13-484307-2 .
• Peter W. Atkins : Physical Chemistry. Wiley-VCH, ISBN 3-527-30236-0 .
• Ostwald's Classic of Exact Sciences No. 139. Thermodynamic Treatises on Molecular Theory and Chemical Equilibria. Three essays from the years 1867, 1868, 1870 and 1872 by CM Guldberg . Translated from Norwegian and edited by R. Abegg . Leipzig: Wilh. Engelmann, 1903.

Wiktionary: Law of mass action  - explanations of meanings, word origins, synonyms, translations

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