The Equilibrium Constant

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Paper No. : 01Fundamentals of Analytical Chemistry Module :13 General concept and chemical equilibrium

Principal Investigator: Dr.NutanKaushik, Senior Fellow

The Energy and Resources Institute (TERI), New Delhi Co-Principal Investigator: Dr. Mohammad Amir, Professor of Pharm. Chemistry,

JamiaHamdard University, New Delhi

Paper Coordinator: Prof. Rajeev Jain, Professor of Chemistry, Jiwaji University, Gwalior

Content Writer: Prof. Rajeev Jain, Professor of Chemistry, Jiwaji University, Gwalior

Content Reviwer: Dr. NimishaJadon, Assistant Professor, Jiwaji University, Gwalior


Description of Module

Subject Name Analytical Chemistry / Instrumentation Paper Name Fundamentals of Analytical Chemistry

Module Name/Title General concept and chemical equilibrium

Module Id 13

Pre-requisites Objectives Keywords


Paper: Fundamentals of Analytical Chemistry Module 13

General Concepts and Chemical Equilibrium

One of the requirements for a reaction used in a titration is that it should proceed to completion at the equivalence point. Likewise, in a gravimetric analysis the reaction used for the separation process must be so complete that an insignificant amount of analyte is left behind.

In this module I shall discus how equilibrium constants are formulated and used for acid- base, precipitation, complex formation, and oxidation-reduction reactions. We shall also apply the principles of equilibrium to determine the feasibility of specific titrations and to decide whether certain separation processes can be made complete.

The Equilibrium Constant

Chemical reactions, such as the formation of hydrogen iodide from hydrogen and iodine in the gaseous phase,

H₂(g) + I₂(g) ↔2HI(g)

are generally reversible, and when the rates of the forward and reverse reactions are equal, the concentrations of the reactants and products remain constant with time. Hence, the reaction has reached a state of equilibrium.

The equilibrium concentrations reflect the intrinsic tendencies of the atoms to exist as

molecules of reactant or product. Although there may be an infinite number of concentrations which satisfy the equilibrium condition, there is only one expression which is found to be


constant at a given temperature for a reaction at equilibrium. For the general reaction in aqueous solution,

A(aq) + B(aq)↔ C(aq) + D(aq) this expression is

K= equilibrium constant.

Here the brackets show concentrations in moles per liter(molarity) at equilibrium. Mostly the fraction is called the mass action expression.

The amount of chemical reactivity is denoted by the active mass of the substance. Analyst today use the term activity as the best meaning of active mass, when activity is not known they use as concentration.

The concentrations employed are usually molality or molarity for reactions in aqueous solution. For gases, molarity or partial pressure is employed. For the completely general reaction.

aA(aq) + bB(aq)↔cC(aq) +dD(aq)

The equilibrium constant is


The exponents in this expression are the coefficients of the reactants and productsin the balanced equation for the reaction.

If a reactant is a solid or liquid, its concentration does not appear in the equilibrium expression. The reason is that the concentration of a solid or liquid is constant. Increasing the amount of a solid or liquid in a reacting system does not change its concentration. The number of moles increases, but the volume also increases, and the number of moles per liter is remaining same.

Aqueous Solutions

Water is one of the most abundant compounds in nature and is essential to life. It dissolves many substances and serves as the medium in which a wide variety of chemical reactions take place.

Weak and Strong Electrolytes

Aqueous solutions of certain compounds are good conductors of electricity, because of the presence of positive and negative ions. Such compounds are called electrolytes, whereas compounds whose aqueous solutions do not conduct movement of ions are called nonelectrolytes. Sodium chloride, NaClis completely dissociated into Na+ and Cl- ions in aqueous solution and is an electrolyte.

Ethylene glycol, CH₂0H-CH₂0H, common antifreeze remainsundissociate in aqueous solution and are an example of a nonelectrolyte. Most ionic compounds are completely dissociated in water and are called strong electrolytes. Many covalent compounds dissociate


to only a slight degree when dissolved in water and are called weak electrolytes. Acetic acid (CH₃COOH), ammonia and water are examples of weak electrolytes.

TABLE 1. Strong and weak Acids and Bases in Aqueous Solution

We write equilibrium constants for the dissociation of weak electrolytes into ions and call them dissociation constants; for example, the dissociation of water is written :

Dissociation of acetic acid and ammonia

Karepresents the dissociation constant of a weak acid; Kb denotes the dissociation constant weak base. Kw used to the dissociation of water and is sometimes called the autoprotolysis constant of water.



A salt is the product other than water formed when an acid reacts with a base. For example, when hydrochloric acid and sodium hydroxide react, the products are a salt (sodium chloride) and water.

Shown through reaction

Given in ionic form,

The netionic reaction is (H⁺ is hydrated in water)

NaCl, is not a molecule, but is a strong electrolyte, completely dissociated into Na⁺andCl⁻

ions. Most salts are strong electrolytes, their molecular formula areNa₂SO₄, KNO₃, CaCl₂ ,etc. A few salts are weak electrolytes; one we will encounter later is mercury(II) chloride, HgCl2. The reaction

is sufficiently complete at the equivalence point.


Reactions of Salts of Weak Electrolytes

Salts of weak electrolytes react with water to produce either hydrogen or hydroxideions. For example, an aqueous solution of the salt sodium acetate (NaOAc), is basic because the acetate ion reacts with water to produce hydroxideions (hydrolysis)-

OAc⁻ + H₂O ↔ HOAc + OH⁻

The reason is this reaction occurs to an appreciable extent is that HOAc is aweak acid and prefers to remain as molecules. Na⁺ ion does not react with water to produce NaOH

molecules and H₃O+ ions, since NaOH is a strong base and prefers to remain as Na⁺ and OH⁻

ions. Salt ammonium chloride(NH₄Cl) is acidic, since ammonium ions react with water to produce H₃O⁺ ions-

NH₄⁺ + H₂O↔ NH₃+ H₃O⁺

The Cl⁻ ion does not react with water to produce HCl and OH⁻ions, since HCl is a strong acid.

The equilibrium constant for the reaction of acetate ion with water is

The symbol Kbdenotes the acetate ion is a base; it is said that acetate ion is the conjugatebase of acetic acid. There is a simple relation between the Ka of an acid and the Kb of its

conjugate base.This can be seen by multiplying the two expressions:

[H₃O⁺][OAc⁻] [HOAc][OH⁻]

[HOAc] X [OAc⁻] = [H₃O⁺][OH⁻] = Kw


KaX Kb = Kw

The same relation holds for the weak base NH3 and its conjugate acid, NH₄⁺, orfor any such conjugate pair.In discussing solutions containing salts such as sodium acetate or ammonium chloride, chemists sometimes refer to the ion OAc⁻ or NH₄⁺as the salt. It is understood that the salt is NaOAc or NH₄Cl. Since the acetate or ammonium ions undergo reaction with water molecules, they are the species of interest in equilibrium calculations of the hydrogen ion concentration.

Activity and Activity Coefficients

Strong electrolytes, such as NaClare completely dissociated into Na⁺ and Cl⁻ ions in the solid state,

In order to get an agreement between experimental and theoretical equilibrium calculations, the analyst multiplies actual concentrations (molarities,example) by certain numbers, called activity coefficients, to obtain effectiveconcentrations,called activities.

The activity of a species A is defined as follows:

where is the activity, f Athe activity coefficient, and [A] the molarity of species A.

For example, the activity of the hydronium ion is aH₃0⁺= f H₃O⁺[H₃0⁺]

the activity of hydroxide ion is aOH⁻= f OH⁻[OH⁻]


true constant for the dissociation of water, Kw, is either the activity or the activity coefficient could be made dimensionless generally it is the activity coefficient which is so treated. Hence activity is expressed in the same units as concentration.

Activity coefficients depend upon the size of the hydrated ions. Generally, smaller ions show greater movement from ideal behaviour than do larger ones at the same ionic strength.In general the presence of ions will have a less effect upon the activity of aneutral molecule than upon that of another electrolyte.

Ions do influence molecules to some degree by interacting with existing dipoles or even inducing them. But it is reasonable to take the activity coefficient of a neutral molecule as unity under normal solution conditions.

The Debye-Huckel Limiting Law

In 1923, Peter Debye and Erich Huckel presented a theory of interionic-attraction effects which shows how to calculate activity coefficients theoretically.They assumed that ions were point charges at relatively greater distances from each other, that the dielectric constant of the electrolyte solution was independent of the concentration of the solute, and that the dielectric constant of water could be used in all calculations. The equation is

Where fi is the activity coefficient of a single ionic species, such as Na⁺orCl⁻, Zis the charge of the ion, μ the ionic strength of the solution, and Aa constant.

For water at 25°C, A, a collection of constants including temperature, the dielectric constant, and Avogadro's number, is 0.512. The equation has been of great value in providing a law for


extrapolation to zero concentration and is commonly referred to as the Debye-Huckel limiting law.

It is impossible to test the above equation experimentally, since it is not possible to prepare a solution which contains only a single ionic species. The mean activity coefficient for a 1: 1 type electrolyte is defined by the equation

This quantity can be measured by various physicochemical techniques and has been found to agree satisfactorily with values calculated by the Debye-Huckel equation.

For example, the activity coefficient calculated for a 0.01 M solution of HCl is 0.89, and the mean value measured experimentally is 0.90.For a binary electrolyte AmBnthe Debye- Huckel equation is-

where ZA and ZB are the charges on the cation and anion taken without regard to sign.

Thermodynamic Equilibrium Constant

In thermodynamic terms, the true equilibrium constant (designated K˚) for the general reaction is expressed in terms of activities

aA +bB ↔cC +dD,



Here K representing the common equilibrium constant in terms of concentrations and it is often referred to as the "concentration quotient." Since K° is a true constant and since the activity coefficients of reactants and products change with the ionic strength of a solution, Kis not strictly constant. However, at infinite dilution the activity coefficients approach unity, and Kdoes become equal to K˚.

Equilibrium calculations should be made using the thermodynamic value of the equilibrium constantwhere activity coefficients are known. However, it is rare that activity coefficients are known in the complex, concentrated solutions frequently encountered in analytical chemistry.

Therefore in most calculations in the text we shall use molarities as an approximation of activities and shall assume that Kis constant. The feasibility of a titration, involve relative changes in equilibrium concentrations and are not greatly affected by neglecting activities. In equilibrium calculations we should use Activities.

Standard States

Reactants those are in aqueous solution areuse of molarity as an approximation of the activity of such substances. Many of the reactions we see in analytical chemistry involve reactants which are gases, liquids, or solids, and relation between activity and free energy is given by the equation


where ΔG is the free energy change for the transfer of 1 mol of a substance froma state of activity a1 to one of activity a2. The free energy of a chemical speciesdepends upon the nature, quantity of the substance, temperature and pressure.

Hence it is customary to get an arbitrary reference or standard state and to assign to it an activity of unity; then values for the absolute free energy of an element or the free energy of formation of a compound can be calculated and tabulated.

The customary choices for standard states, all at 25°C, are as follows:

1. In very dilute solution, the activity of a solute is the same as its molality, where ideal behavior may be assumed. That is,

a/m=1 when m→0

where mis the molality . Since molality and molarity are very nearly the same in dilute aqueous solutions, use molarity rather than molality as an approximation of activity. The standard state is a hypothetical one in which the solute is at 1M concentration, but the environment about the solute is the same as that in an ideal solution.

2. For a pure gas the standard state is 1 atm, and the activity is then the same as the pressure of the gas. For a real gas the standard state is that in which the so-called fugacity is unity.

Since a real gas approaches ideal behaviour at low pressures, making fugacity and pressure approximately equal, we will take the pressure of a gas as its activity. Thus

a/p =1 when p→0

3. The activity of a pure liquid or solid, acting as a solvent for other substances is unity. That is, the standard state is a molefraction of unity, where X is the mole fraction of the solvent.

a/x =1 when P → 0


If the activity of the liquid or solid is changed by dissolving in it a solute, still we can find the activity by mole fraction. In most examples thatwe shall encounter; it will still be acceptable to take a value of unity as theactivity of the solvent.

For example, a liter of a 0.1 M aqueous solution of a solute contains 0.1 mol of that solute and about 55.3 mol of water. The mole fraction of water is thus about 55.3/55.4 = 1.

Equilibrium Calculations

There are four types of chemical reactions used in titrimetric analyses.

1. Acid-Base Equilibria

Water is a weak electrolyte, it dissociates into H₃O⁺and OH⁻ions :- 2H₂O↔ H₃O⁺ + OH⁻

The extent of dissociation of water has been measured experimentally, and at 25°C the hydronium and hydroxide ion concentrations have been found to be1.0 X 10-7 M. This means that the value of Kw, the autoprotolysis constant ofwater, is 1.0 x10-14 at 25°C :-

Kw = [H₃O⁺][OH⁻]

Kw = (1.0 x 10-7)(1.0 x 10-7) Kw = 1.0 X 10-14


The value of Kw at several other temperatures is shown in following table

The term pH is convenient for expressing hydrogen ion concentrations, since the latter values are very small and may vary during titrations.

Sorensen, in 1909, defined pH as the negative logarithm of the hydrogen ion concentration. It was later realized that the electromotive force (emf) of a galvanic cell used to measure pH was dependent more upon the activity of hydrogen ion than upon the concentration. Hence the correct definition of pH is:-


pH is defined as, to convert a negative power of ten into a small positive number. Thus a hydrogen ion concentration of l.0 x 10-1 corresponds to a pH value of 1.00 and a value of 1.0 x 10-13 becomes pH= 13.00.

Such numbers, ranging from, say, 0 or 1 up to perhaps 13 or 14, are conveniently plotted on titration curves. We shall consistently express pH values to two decimal places.


The following examples illustrate the conversion of hydrogen ion concentration to pH, and vice versa.


(A) The hydrogen ion concentration of a solution is 5.0 x 10-7 Calculate the pH.

(B) The pHof a solution is 10.70. Calculate the hydrogen ion concentration.

It is often easy to use other p-functions analogous to pH. For example,the following functions are frequently used:-

Since [H₃0⁺][OH⁻] = Kw = 1.0x10-14 at 25°C, then pH + pOH = pKw = 14.00


In neutral water [H₃O⁺] = [OH⁻] and

At 25°C the values of the pH and pH are 7.00 in neutral solutions.

In acidic solutions[ H₃0⁺] > 10.0 x 10-7 and pH <7.00.

Inbasic solutions [H₃O+] <1.0 x 10-7 and pH> 7.00.

Solubility Equilibria

The equilibrium constant expressing the solubility of a precipitate in water is the familiar solubility product constant. For a precipitate of silver chloride,the equilibrium constant of the reaction

AgCl(s) ↔ Ag⁺(aq) + Cl⁻(aq)

For solid AgCl,activity is constant, and by convention we take it to be unity. The AgCl is only slightly soluble; hence the concentrations of Ag⁺ and Cl⁻ ions are small and, unless large concentrations of other ions are present. Activities can be approximated by molarities, giving:-

The constant Ksp is called the solubility product constant.

A general expression for the salt AxBy, dissociating as follows:-


The numerical value of a solubility product constant can be easily calculated from the solubility of the compound.

Complex-Formation Equilibria

Reactions involving complex formation are utilized by the analyst, in both titrimetric or volumetric and gravimetric procedures. Solid AgClwill dissolve in a solution of ammonia.

The equation can be written molecularly as AgCl(s) + 2NH3(aq) ↔ Ag(NH3)₂Cl

The compound Ag(NH₃)₂Cl is called a complex. The compound is ionic, dissociating into Ag(NH₃)₂⁺andCl⁻ ions, and the species Ag(NH₃)₂⁺ is called a complex ion.The silver- ammonia complex ion is formed in steps, by the addition of molecules of ammonia, called the ligand, to silver ion, called the central metal ion.

The equilibrium constants for the two reactions are


They are called the stability or formation constants of the complexes.


The equilibrium constant of an oxidation-reduction reaction is obtained from the potential of an appropriate galvanic cell. The equilibrium expressions for such reactions are formulated in the usual way.

For example, when iron (II) is titrated with cerium(IV), the reaction is

The equilibrium constant for this reaction is

Systematic Equilibrium Calculations

In analytical chemistry, we need to calculate the concentration of some analyte in an aqueous solution in which there are several interacting equilibria. The mathematical calculations maybe quite complex. However, through systematic treatment of equilibrium, which can be used in such cases, and often reasonable assumptions can be made which greatly simplify the mathematical operations.

The general approach is to identify all the chemical species in the solution and to find a set of equations sufficient in principle to permit calculation of the concentration of each species.

When there are more equations than can be solved simultaneously with convenience, hence invoke our chemical knowledge to simplify the mathematical problem.The procedure is as follows:

1. Identify all the species in the solution. Water molecules are not counted since their


2. Write all the equilibrium constants involving the unknown species. These constants must be independent equations. For example, if one uses the dissociation constant of a weak acid, Ka, and the autoprotolysis constant of water, Kw one could not also use the dissociation constant of the anion of the weak acid,Kb since Kb, = Kw/Ka. No additional information on the relations betweenconcentrations is given by the constant Kb.

3. Write the mass-balance(or material balance) equation. This equation is simplya statement of the conservation of matter. For example, in a 0.10 F solution ofacetic acid, all of the acid ends up as HOAc molecules or OAc- ions. Themass-balance equation on acetate is

[HOAc] + [OAc⁻] = 0.10

4. Write the charge-balance equation, which is based on what is called the electroneutrality condition: The total concentration of positive charge must equalthe total concentration of negative charge.

5. Using chemical principles, make approximations to simplify the mathematicalproblem.

Frequently, for example, we are able to drop certain terms becauseour knowledge of chemistry tells us that they are negligible.

6. Solve the equations for the unknown concentrations. Then substitute them intothe original equations to determine the validity of the approximations which were made.

End Point Detection in Titrations

If the titration reaction goes well to completion, there will be a large, abrupt change in the concentration of the analyte (and of the titrant) near the end point titration. The analyst can convert this abrupt change into a signal to stop the addition of titrant-the EPt has been reached.


If the titration is followed graphically by plotting [A] Vs. milliliters of B this rapid rate of change is obscured because the linear concentration scale is not sufficiently large to reveal the enormous change that occurs.

Since the concentration extends over several orders of magnitude, it is best displayed graphically by plotting a logarithmic function of the concentration, pA = -log A, against the volume of titrant. In such a plot, called a titration curve, the value of pA rises slowly at the start of the titration, rapidly increases as the EPt is approached, and increases slowly after the Ept. Values of pA are also shown in Table 6, where the change in paper change in volume, ΔpA / ΔV, can be seen to increase rapidly at the EPt.

Table 6.Contains data for the titration of 50 ml, of 0.10 M A with 50Ml of 0.10 M B for the precipitation reaction


Figure 1 Plots of [A] and pA in titration of 50 mL of 0.10MA with 0.10 MB for three values of K.

Use of linear plots in titrations

Sometimes the analyst does employ linear plots to depict a titration curve. Examples are photometric and amperometric titrations.In such titrations a physical property which is proportional to the concentration of the analyte (or titrant) is measured.

If the titration reaction is essentially complete, the end point is easily determined by the intersection of two straight lines on either side of the EPt. If the reaction is appreciably incomplete, the curve will be rounded near the Ept.

In this case the end point is located by the intersection of the extrapolated straight linesdrawn through points taken well before and after the EPt.

If these measurements are made well away' from equivalence, the excess analyte or titrant will be sufficient to force the reaction to completion by the common-ion effect. Then the


points should fall on a straight line, and extrapolation should yield an accurate determination of the EPt volume,


I. Do you know?

1. When the rates of the forward and reverse reactions are equal, the concentrations of the reactants and products remain constant with time.

Hence that the reaction has reached a state of equilibrium.

2. The equilibrium concentrations reflect the intrinsic tendencies of the atoms to exist as molecules of reactant or product.

3. Water is one of the most plentiful compounds in nature and is essential to life processes. It dissolves many substances and serves as the medium in which a wide variety of chemical reactions take place.

4. The simple Debye-Huckel equation, or limiting law, has been found to give useful results only in very dilute solutions. As the ionic strength increases, mean activity coefficients calculated from the equation are significantly smaller than the experimental values.

5. The free energy of a chemical speciesdepends not only upon the nature and quantity of the substance but also upon temperature and pressure.


Interesting Facts


The equilibrium condition exists in relation to thermal, mechanical, and chemical changes. For example, within a closed flask, liquid water evaporates to form vapour, and at the same time the vapour condenses to form liquid.


Le Chatelier's principle states that if a stress is brought to bear upon a system at equilibrium, the equilibrium reaction shifts in a direction that relieves the stress.


Pure solids do not appear in the equilibrium expression and pure liquids do not



Dynamic equilibrium occurs when the chemical reaction continues to proceed, but a number of products and reactants remain constant. This is one type of chemical equilibrium.


For any reaction, the equilibrium constant Keq equals the ratio of the forward rate constant kf to the reverse rate constant kr. Because a catalyst accelerates the forward and reverse reactions to the same degree, it does not change the Keq of a reaction.




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