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Chemical Equilibrium. Revised edition

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 Chemical Equilibrium 

Ricardo Hidalgo

Escuela de Quimica, Universidad de Costa Rica, San Pedro de Montes de Oca, San Jose, Costa Rica

2012 January

     In this worksheet we introduce various quantities pertaining to chemical equilibrium treated at a level of general chemistry.  After explaining the reaction quotient and equilibrium quotient, we treat prototypical systems in aqueous solution.  The treatment is intended to be illustrative rather than comprehensive.  This worksheet operates satisfactorily with Maple releases 12 - 15.


     A state of equilibrium denotes a condition of a system in which all acting influences are cancelled by others, resulting in a stable, balanced or constant system. In nature, one can identify equilibria of three basic types -- mechanical equilibrium, which characterizes the state of a body or a physical system at rest or in unaccelerated motion in which the resultant of all forces acting thereon is zero and the sum of all torques about any axis is zero; thermal equilibrium, which characterizes the state in which two physical systems in contact have zero net exchange of energy so that their temperatures are equal, and chemical equilibrium, which characterizes a chemical system in which the concentrations of all species present -- reactants and products -- remain constant. Chemical equilibrium is dynamic:  although the individual concentrations are invariant, on a molecular scale there is great activity such that reactants are converted into products, and vice versa, but these two processes occur at the same rate so that there is no net change in the system.

  Reaction quotient and equilibrium quotient


     For a reversible chemical reaction ocurring in solution,

a A + b B <--> c C + d D

chemeq := a*A + b*B = c*C + d*D;

the reaction quotient Q, at any given extent of reaction, is defined in the following expression,

Q = (a[C]^c * a[D]^d) / (a[A]^a * a[B]^b);

in which a denotes the activity of a specific species at any given time. The relation between activity a and concentration c is

a = gamma*c;

in which gamma or gamma denotes an activity coefficient.  When the concentrations of solutes in the solution are sufficiently small, corresponding to a dilute solution, that their behaviour approximates that of an ideal mixture, the activity coefficients of the species tend to one. Taking that limiting condition as a pragmatic assumption, we express the reaction quotient for the reaction displayed above in terms of concentrations as follows:

Q = ([C]^c*[D]^d)/([A]^a*[B]^b);

When the reaction under consideration occurs in the gaseous phase, fugacities replace activities; provided that the mixture is ideal and hence obeys Dalton's law, the reaction quotient is expressed in terms of partial pressures instead of fugacities.

      Let us take for example the reaction between dihydrogen and dinitrogen to produce ammonia, which became an important industrial process associated with Haber and Bosch:

3 H[2] (g)  +  N[2] (g)  <-->   2 NH[3] (g)

The reaction quotient in this case is expressed as

Q = p[NH3]^2/(p[H2]^3*p[N2]);

According to an assumption of ideal behaviour, for the gaseous compounds at particular partial pressures and after a sufficient period at 723 K, the reaction quotient of this system attains a value 4.07 10^(-5) bar^(-2); if at constant volume the partial pressures of the three gases remain constant, the reaction attains what is called chemical equilibrium. At this stage, at which the partial pressures of the three gases remain constant in a dynamic equilibrium, the reaction quotient becomes an equilibrium quotient, denoted K[p] because the concentrations are expressed as partial pressures.. This value is roughly constant at a given pressure and temperature regardless of the initial partial pressures of the reactants and the products.

K[p] := p[NH3]^2/(p[H2]^3*p[N2]) = evalf(4.07*10^(-5), 3)/bar^2;


  example 1

  Chemical equilibrium in aqueous solution

    Many chemical reactions are implemented in solution; such a medium provides a convenient control of the process, primarily to assure homogeneous conditions. Water is the most abundant and commonly used solvent on earth; it is not toxic and many substances of practical importance are soluble therein. We classify the substances that are soluble in water according to two types -- electrolytes and non-electrolytes:  the former, such as salts of metallic elements, dissociate into ions when dissolved, resulting in a solution that conducts electric current, and the latter, such as sucrose, do not dissociate when dissolved.  Among electrolytes, there are strong electrolytes and weak electrolytes:  with water as solvent, a strong electrolyte dissociates fully into ions when dissolved, whereas a weak electrolyte dissociates partially.  For solute MX as a weak electrolyte; when dissolved in water, three species are hence present in solution -- MX (aq), X^`-`(aq) and M^`+`(aq).

MX (aq)   <-->   X^`-`(aq)  +  M^`+`(aq)

As for the gaseous system involving ammonia discussed above, we write direcly an expression for the equilibrium quotient.

K[eq]  =  [X^`-`*`(aq)`]*[M^`+`*`(aq)`]/[MX(aq)] 

Deficiencies in the formating of equations limit the nature of the displayed expressions, but the meaning of each expression should be clear from the context.

  Acid-base equilibria  


     According to a definition of acids and bases by Bronsted and Lowry, an acid is a proton donor and a base is a proton acceptor; for any substance to behave as an acid, there must be a proton acceptor, and vice versa.  Another important definition is that by Arrhenius according to which an acid is a chemical substance that, when dissolved in water, gives hydrogen ions, represented as H^`+`, and a base gives hydroxyl ions, OH^`-`.

     For example, let us analyze the dissociation of HCl, a strong electrolyte, on dissolution in water:

 HCl (g)  +  H[2]O (l)  --> H^`+`(aq)  +  Cl^`-`(aq)

This chemical equation indicates that HCl is indeed an Arrhenius acid because it yields H^`+` ions on dissolution in water. There is no such thing as a free proton in water; a proton is a small chemical entity (its diameter is about 10^(-5)  that of Li^`+`) and has a large density of positive charge, and thus a great affinity for an electronic cloud. A superior representation of the proton in aqueous system is a hydronium ion in forms H[3]O^`+`,  H[5]O[2]^`+`,  H[7]O[3]^`+`  and  H[9]O[4]^`+` , according to which the proton accepts the electronic cloud of one or more water molecules; although the latter species is likely the most abundant, we typically write the hydrated proton as  H[3]O^`+` (aq).  A common expression of the above chemical equation is, hence,

 HCl (g) +   H[2]O (l)  -->  H[3]O^`+` (aq)  +  Cl^`-` (aq)

The behaviour according to Bronsted and Lowry is clearly seen in this example:  HCl is an acid that donates a proton to water, which acts as a base, to produce Cl^`-`, which is the conjugate base of HCl, and H[3]O^`+`, which is the conjugate acid of water.

     Some chemical substances are neither base nor acid in an absolute sense; a species behaves as an acid or as a base relative to the other species with which it interacts.  Depending on that interaction, a substance might act as an acid or a base; this property is known as amphoterism. Water, for example, acts as a base in the latter equation, whereas in the following reaction its behaviour as an acid is depicted.

NH[3] (aq)  +  H[2]O (l)  <-->  NH[4]^`+` (aq)  +  OH^`-`  (aq)

In this case, ammonia acts as a base according to the definitions both of Bronsted and Lowry and of Arrhenius, but, as NH[3] is a weak base, NH[4]^`+` partially dissociates.  Ammonia accepts a proton from water and gives OH^`-` as a product.  Water donates a proton to ammonia, hence becomes an acid in this reaction.

     Acids and bases are electrolytes:  their solution conducts an electric current to a significant extent:, because they produce ions through their dissociation on dissolution in water, but not all acids and bases are strong electrolytes -- they are classified according to their strength.  As a strong acid, perchloric acid dissociates completely into its corresponding ions when dissolved in water:

HClO[4] (l)  +  H[2]O (l)  -->  ClO[4]^`-`  (aq)  +  H[3]O^`+`  (aq)

Other acids, such as ethanoic acid, dissociate incompletely when dissolved:

HOAc (aq)  +  H[2]O (l)  <--> OAc ^`-` (aq)  +  H[3]O^`+` (aq)

When ethanoic acid is dissolved in water, the concentration of hydronium ions does not correspond to the concentration of ethanoic acid that was added.  The reaction attains an equilibrium that, in this particular case of the dissociation of a weak acid, is called an acid dissociation equilibrium; it is expressed generally as

HA (aq)  +  H[2]O (l)  <-->  H[3]O^`+` (aq)  +  A^`-`  (aq) ;

for this example, we define the equilibrium quotient, which is called the equilibrium quotient for the acid dissociatiion and denoted K[a]:

K[a]  =  [H[3]*O^`+`*`(aq)`]*[A^`-`*`(aq)`]/[HA*`(aq)`]

In succeeding displayed equations we generally omit henceforth the phase designations.  The concentration of water is neglected in this equation because water is the solvent:  we are working with solutions sufficiently dilute that the concentration of water, ~55.4 mol L^(-1), is considered constant.

     According to an analogous case for a weak base, for instance, the dissociation equilibrium of ammonia as described in a chemical equation above, we define an equilibrium quotient K[b] for the basic dissociation:

K[b]  =  [OH^`-`]*[NH[4]^`+`]/[NH[3]]   

For aqueous systems, an intrinsic acid-base equilibrium is the self ionization of water:

2 H[2]O (l)  <-->  H[3]O^`+` (aq)  +  OH^`-` (aq)

for this case, considering again the concentration of water to be constant, the equilibrium quantity is called the ion product:

K[w]  =  [H[3]*O^`+`]*[OH^`-`]

Kw := H*OH = 1*10^(-14);

In this equation, and in others to follow, a symbol such as H or OH denotes both the name of a chemical compound or species and its concentration.  

     This quantity has a value 1.008 10^(-14) for ultrapure water (^`1`H[2]^16O) at 298.15 K.  Defining pH = -log[10]([H^`+`]), we find that the pH of pure water is 6.998; pK[w], defined analogously, is 13.996.  In general, chemical notation pX signifies -log[10](X) of some quantity X, such as an equilibrium quotient, or a concentration of some chemical species.  In contemporary lists of thermodynamic values in tables or chemistry books, these quantities are expressed in terms of only acidity; only values of pK[a], not pK[b], are hence listed.  For the example of ammonia described above, K[b] is defined as

Kb := OH*NH4/NH3;

In an analysis of this case from the perspective of a conjugate acid, the system is described according to this equation,

NH[4]^`+` (aq)  +  H[2]O (l)  <-->  NH[3] (aq)  +  H^`+` (aq)

for which K[a] is

Ka := NH3*H/NH4;

If K[a] be multiplied by K[b],

"Ka Kb" = Ka*Kb;

which is the ion product K[w] of water of approximate value 1.0 10^(-14); any K[b] is hence calculated from K[a] of its conjugate acid and vice versa, because


Kw = Kb*Ka;

isolate(%, Kb);


  example 2   Calculating K[b] using K[w] and K[a]

  example 3   Calculating pH and concentration of an acidic solution

  Neutralization of strong acid with a strong base -- titration

  example 4   Selection of an indicator for neutralization of a strong acid with a strong base

  Solution of a weak acid or base -- titration and neutralization

  Titration of a polyprotic acid

  Solubility equilibria

    When crystalline M[a]*X[b] is in contact with its saturated aqueous solution, an equilibrium exists between the crystal and its ions in solution:

M[a]X[b](s)  <--> a M^`b+`(aq) + b X^`a-`(aq)

The equilibrium quotient for this system is expressed as

K[eq] = [M^`b+`]^a*[X^`a-`]^b/[M[a]*X[b]]

but, as M[a]X[b](s)  has a constant activity because it is a pure crystalline solid, the solubility product K[sp] is defined as

K[sp] = [M^`b+`]^a*[X^`a-`]^b 

  Example of BaSO[4]

  Effect of common ions

  Effect of pH

  Solubility product of a weak electrolyte

  Complex-ion equilibria

    Metal ions in solution form complexes with Lewis bases such as water, ammonia or halides.  For metal ion M and ligand L the following reaction occurs to form complex ML,

M + L <--> ML

for which the reaction quotient K[1] is

K[1] = [ML]/([M]*[L])

Many transition-metal ions can form complexes of more than one ligand; they hence have coordination numbers greater than one,


ML + L <--> ML[2]

for which the reaction quotient is

K[2] = [ML[2]]/([ML]*[L])

For the global formation of ML[2],

M  +  2 L <--> ML[2]

a global formation quantity beta[2] is defined as

beta[2] = [ML[2]]/([M]*[L]^2)

so that

beta[2] = K[1]*K[2]

In a general form for the formation of the complex ion ML[n],

M + n L <--> ML[n]

the corresponding quantity beta[n] is defined as

beta[n] = [ML[n]]/([M]*[L]^n)

that is expressed as a product of the various equilibrium quotients

beta[n] = K[1]*K[2] ... K[n] 


  Copper(II) ammonia complexes and the fraction of the metal ion species alpha

  Solubility and formation of complex ions


     The use of mathematical software for chemical education, such as in an analysis of chemical equilibrium as developed in this worksheet, allows an improved understanding of the chemical phenomena involved.  Simplifying assumptions that are essential for manual calculation are superfluous when computer software is applied; an intensive focus on the real chemical problem is thereby attained and, with a sufficient comprehension of the mathematical language, the algebraic work is performed with the software.  In the cases treated above, we employ the symbolic capacity of Maple program to generate the equations that govern a particular situation; we then insert the particular numerical values of parameters for the specific chemical system to produce the highly meaningful results that are generally plotted for maximal understanding.  This condition enables a profound and productive approach to the learning ot chemical topics and thereby to chemical education.