Chemical kinetics

 1. Chemical Kinetics: It is the branch of physical chemistry which deals with the study of the rate of chemical reaction and the mechanism by which the reaction occurs.

2. Rate of Reaction: It may be defined as the change in concentration of a reactant or product in unit time.

For a general reaction, interval of R → P, the rate of reaction may be expressed as

Rate of reaction = DecreaseinconcentrationofRTimetaken

= IncreaseinconcentrationofPTimetaken

Rate of reaction = –Δ[R]Δt=Δ[P]Δt

The negative sign in the rate expression indicates the decrease in the concentration of the reactant and gives a positive value of the rate.

Units of rate are mol L–1 s–1 or atm s–1 (in gaseous reactions).

The above expression of rate gives us the average rate of reaction.

3. Instantaneous Rate of Reaction: It is the rate of reaction at a particular moment of time and measured as a very small concentration change over a very small interval of time.

Mathematically, Instantaneous rate = (Average rate) Dt → 0

For a general reaction, R → P

Instantaneous rate = –d[R]dt=d[P]dt

Instantaneous rate can be determined graphically by drawing a tangent at time t on either side of the curve for concentration of A or B vs time and calculating its slope.

Thus, rinst = –d[R]dt = – slope (for R)

rinst = +d[P]dt = slope (for P)

4. General Expression for Rate of Reaction: For a general reaction,

aA + bB → cC + dD

rav = –1aΔ[A]Δt=–1bΔ[B]Δt=1cΔ[C]Δt=1dΔ[D]Δt

rinst = –1ad[A]dt=–1bd[B]dt=1cd[C]dt=1dd[D]dt

5. Factors Affecting the Rate of a Chemical Reaction:

Rate of a reaction is influenced by following factors:

(a) Nature of reactants: It has been observed that ionic substances react more rapidly than the substances with covalent bond. This is because ions are immediately available in aqueous solution on dissociation hence, react rapidly but covalent molecules consume part of energy in breaking of bonds.

(b) Concentration of reactants: Rate of a reaction is directly proportional to the concentration of reactants.

(c) Temperature: Rate of a reaction increases with the increase in temperature.

(d) Presence of catalyst: In presence of catalyst, the rate of reaction generally increases and the equilibrium state is attained quickly in reversible reactions.

(e) Surface area of the reactants: The smaller the particle size, greater the surface area and faster is the reaction.

(f) Radiations: There are many reactions which either do not take place at all or are quite slow in the dark but take place at a considerable speed when exposed to sunlight or ultraviolet radiations, such reactions are called photochemical reactions. Examples are photosynthesis of carbohydrates, photography, etc.

6. Rate Law: It is an experimentally determined expression which relates the rate of reaction with concentration of reactants.

For a hypothetical reaction,

A + B → Products

Rate [A]m [B]n

or Rate = k[A]m [B]n

where k is a constant called specific rate of reaction or rate constant.

If [A] = [B] = 1 mol L–1 then

Rate = k

Thus, rate constant may be defined as the rate of reaction when the concentration of each reactant in the reaction is unity.

7. Order of Reaction: It may be defined as the sum of powers of the concentration of the reactants in the rate law expression.

Order of a reaction can be 0, 1, 2, 3 and even a fraction.

For a hypothetical reaction,

aA + bB + cC → Products

Let rate = k[A]m [B]n [C]p

where, m = order of reaction with respect to A

n = order of reaction with respect to B

p = order of reaction with respect to C

Overall order of reaction = m + n + p

Units of rate constant:

For an nth order reaction, A → Product

Rate = k[A]n

or k = Rate[A]n=concentrationtime×1(concentration)n

= (concentration)1 – n time–1

On considering S.I. unit of concentration as mol L–1 and time as seconds, the unit of k = (mol L–1) 1–n s–1

(a) Examples of zero order reactions

(i) Some enzyme catalysed reactions and reactions which occur on metal surfaces.

(ii) Decomposition of gaseous ammonia on a hot platinum surface.

2NH3(g)−→−PtCatalyst1130KN2(g)+3H2(g)

(iii) H2 (g) + Cl2 (g) −→hv 2HCl (g)

(iv) 2HI (g) −→gold H2 (g) + I2 (g)

Unit of k = mol L–1s–1

(b) Examples of 1st order reactions

(i) All radioactive disintegrations are of the first order.

(ii) Decomposition of sulphuryl chloride.

SO2Cl2 → SO2 + Cl2

Unit of k = s–1 . Therefore, change in unit concentration does not alter the value of k.

(c) Examples of 2nd order reactions

(i) CH3COOC2H5 + NaOH → CH3COONa + C2H5OH

(ii) NO(g) + O3(g) → NO2(g) + O2(g)

Unit of k = litre mol–1 second–1

(d) Examples of 3rd order reactions

(i) 2NO(g) + O2(g) → 2NO2(g)

(ii) 2NO(g) + Br2(g) → 2NOBr(g)

Unit of k = litre2 mol–2 second–1

8. (a) Elementary reaction: A reaction which take place in one step is called an elementary reaction. When a sequence of elementary reactions gives the products, the reaction is called complex reaction.

(b) Molecularity: The number of reacting species (molecules, atoms, ions) taking part in an elementary reaction which must collide simultaneously in order to bring about a chemical reaction.

Reactions are classified as unimolecular, bimolecular and trimolecular for molecularity 1, 2 and 3 respectively.

Examples:

NH4NO2 → N2 + 2H2O (Unimolecular reaction)

2HI(g) → H2(g) + I2(g) (Bimolecular reaction)

2NO(g) + O2(g) → 2NO2(g) (Trimolecular reaction)

The probability of more than three molecules colliding simultaneously is rare. Therefore, molecularity of a reaction does not extend beyond three. Molecularity can be defined only for an elementary reaction and has no meaning for a complex reaction.

(c) Intermediates: The species which are produced in one step and consumed in another are called intermediates.

(d) Mechanism of reaction: A series of elementary reactions proposed to account for the overall reaction is called mechanism of reaction. The overall rate of the reaction is controlled by the slowest step in a reaction and is called the rate determining step.

Consider the reaction, 2H2O2 −→−−−AlkalinemediumI– 2H2O + O2

The rate equation for this reaction is found to be

Rate = –12ddt[H2O2] = k[H2O2] [I –]

Evidences suggest that this reaction takes place in two steps as follows:

Step I. H2O2 + I– −→Slow H2O + IO– (Intermediate)

Step II. H2O2 + IO– −→Fast H2O + I – + O2

The first step, being slow, is the rate determining step. Thus, the rate of formation of intermediate, IO– will determine the rate of reaction.

9. Pseudo First Order Reaction: A reaction which is not truly of first order but under certain conditions becomes reaction of the first order is called a pseudo first order reaction. For example, the inversion of cane sugar is a bimolecular reaction but it is a first order reaction as concentration of H2O is quite large and does not change appreciably.

C12H22O11 + H2O −→H+ C6H12O6 + C6H12O6

Rate = k [C12H22O11]

10. Zero Order Reactions: In such reactions, the rate remains constant throughout the course of reaction, i.e., the rate does not change with the change in concentration of reactants.

Rate = k [Reactant]0 or Rate = k

Zero order reactions generally occur in a heterogeneous system, wherein the reactant is absorbed on the surface of a solid catalyst (here it is converted into product). The fraction of the surface of the catalyst covered by the reactant is proportional to its concentration at low values and the rate of reaction is of the first order. However, after certain concentration limit of the reactant, the surface of the catalyst is fully covered. As the concentration of the reactant further increases, no change in it takes place. Thus, rate becomes independent of concentration and the order of reaction becomes zero.

Integrated rate law for zero order reaction:

Consider a general zero order reaction

R → P

As it is a reaction of zero order

\ –d[R]dt = k[R]0 = k ⇒ – d[R] = kdt

– ∫ dt[R] = k ∫ dt

– [R] = kt + C …(i)

where C is constant of integration.

When t = 0, [R] = [R]0

C = – [R]0

Substituting the value of C in equation (i), we get

– [R] = kt – [R]0

kt = [R]0 – [R]

t = 1k{[R]0–[R]} …(ii)

or k = 1t{[R]0–[R]}

Half-life of a reaction: It is the time in which the concentration of a reactant is reduced to half of its original value.

Half-life period of a zero order reaction:

When [R] = [R]02,t=t1/2

Substituting these values in equation (ii), we get

\ t1/2 = 1k{[R]0–[R]02}

t1/2 = [R]02k

t1/2 [R]0

11. First Order Reactions: In this class of reactions, the rate of reaction is directly proportional to the first power of the concentration of reacting substance.

Rate = k[Reactant]1

Integrated rate law for 1st order reaction:

Consider the general first order reaction

R → P

As the reaction follows first order kinetics,

\ –d[R]dt [R]

–d[R]dt = k[R] ⇒ –d[R][R]= k[dt]

Integrating both sides, we get

– ln [R] = kt + C ...(i)

where C is constant of integration

When t = 0, [R] = [R]0

– ln [R]0 = 0 + C

Substituting the value of C in (i), we get

– ln [R] = kt – ln [R]0

ln [R] = – kt + ln [R]0

kt = ln[R]0[R]=2.303log[R]0[R]

log[R]0[R] = kt2.303

t = 2.303klog[R]0[R]

where [R]0 is initial concentration and [R] is the final concentration.

Half-life period for a first order reaction

When t = t1/2, [R] = [R]02

t1/2 = 2.303klog[R]0[R]0/2=2.303klog2

t1/2 = 2.303k×0.3010ort1/2=0.693k

Since no concentration term is involved, therefore, t1/2 for a first order reaction is independent of initial concentration.

12. Integrated Rate Equation for a Gaseous System: Consider a typical first order gas phase reaction.

A(g) → B(g) + C(g)

Let Pi be the initial pressure of A and Pt the total pressure at time ‘t’ and PA, PB and PC be the partial pressures of A, B and C respectively at time t.

Total pressure, Pt = PA + PB + PC (pressure units)

If x atm be the decrease in pressure of A at time t and one mole each of B and C is being formed, the increase in pressure of B and C will also be x atm each.

	A(g)	→	B(g)	+	C(g)
At t = 0	Pi atm		0 atm		0 atm
At time = t	(Pi – x) atm		x atm		x atm

Pt = (Pi – x) + x + x = Pi + x or x = Pt – Pi

PA = Pi – x = Pi – (Pt – Pi) = 2Pi – Pt

k = 2.303tlogPiPA=2.303tlogPi(2Pi–Pt)

13. Determination of Order of Reaction:

There are many methods available for the determination of order of reaction.

(a) Graphical method

(b) Initial rate method

(c) Integrated rate law method

(a) Graphical method: This method is applicable to those reactions wherein only one reactant is involved.

(b) Initial rate method: This method is used to determine the order of reaction in such cases where more than one reactant is involved. It involves determination of order of reaction with respect to each reactant separately. For this, order of a particular reactant is determined. A series of experiment are carried out in which the concentration of that particular reactant is changed whereas the concentration of other reactants are kept constant. In each experiment, the rate is determined from the plot of concentration vs time. Similarly, concentration of another reactant is varied keeping the concentration of rest of the reactant constant and initial rate is determined. The data obtained are then compared to see how the initial rate depends on the initial concentration of each reactant. Thus, on the basis of the results the form of rate law is determined.

(c) Integrated rate law method: There are integrated rate law equations which are very convenient to understand the variation in concentration with time, for different order of reactions. After studying the concentrations at various intervals of time, the data are put in all the integrated rate law equations one by one. The expression which gives a constant value of the rate constant decides the order of the reaction.

Zero order equation; k = [R]0–[R]t,

First order equation; k = 2.303tlog[R]0[R]

14. Temperature Dependence of Rate of a Reaction:

(a) Temperature coefficient: It is defined as the ratio of rate constants of the reaction at two temperatures differing by 10°.

Temperature coefficient = Rateconstantat(T+10)°RateconstantatT°

For most of the reactions, temperature coefficient lies between 2 and 3.

(b) Collision frequency (z): It is defined as total number of collisions per unit volume per unit time.

(c) Effective collisions: Collisions which lead to the formation of product molecules are called effective collisions.

Rate of reaction = f × z, where z is the collision frequency and f is the fraction of collisions, which are effective.

(d) Threshold energy: The minimum energy that the reacting molecules must possess in order to undergo effective collisions to form the product is called threshold energy.

(e) Activated complex: The arrangement of atoms corresponding the energy maxima (threshold energy) during the course of a reaction is called activated complex or transition state. The activated complex has partial reactant character and partial product character.

Characteristics of an activated complex

(i) The potential energy of the activated complex is maximum.

(ii) The activated complex has a transient existence and breaks up at a definite rate to form the products.

(f) Activation energy: The energy required to form activated complex is called activation energy. It is the difference between the threshold energy and the average energy possessed by the reacting molecules.

Activation energy (Ea) = Threshold energy – Average energy possessed by reacting molecules

Reactants−→−−−ActivationenergyActivatedcomplex−→EnergyProducts

For fast reactions, activation energies are low whereas for slow reactions activation energies are high.

(g) Arrhenius equation: It relates rate constant with temperature in the following way:

k = Ae–Ea/RT

where A is constant called frequency factor, Ea is the energy of activation.

ln k = lnA–EaRT

log k = logA–Ea2.303RT

A plot of log k vs. 1/T is a straight line whose slope is –Ea2.303R and intercept is log A.

If k1 and k2 are the rate constants at two temperatures T1 and T2 ,then

log k1 = logA–Ea2.303RT1 ...(i)

log k2 = logA–Ea2.303RT2 ...(ii)

Subtracting (i) from (ii), we get,

log k2 – log k1 = Ea2.303R[1T1–1T2]

or logk2k1 = Ea2.303R[T2–T1T1T2]

(h) Effect of temperature on rate of reaction:

Increasing the temperature of a reaction mixture increases the fraction of molecules, which collide with energies greater than Ea. It is clear from the diagram alongside that with 10° rise in temperature, the area showing the fraction of molecules having energy equal to or greater than activation energy gets almost double leading to almost doubling of the rate of reaction.

15. Catalyst: A catalyst is a substance which alters the rate of reaction without itself undergoing any chemical change at the end of the reaction.

For example, catalyst MnO2 increases the rate of decomposition of potassium chlorate to a great extent.

2KClO3 −→−HeatMnO2 2KCl + 3O2

According to intermediate complex theory, a catalyst participates in a chemical reaction by forming temporary bonds with the reactants resulting in an intermediate complex. This has a transitory existence and decomposes to yield products and the catalyst.

It is believed that the catalyst provides an alternate pathway by reducing the activation energy between reactants and products hence lowering the potential energy barrier as shown in Fig. 4.11.

It is clear from the Arrhenius equation (k=Ae–Ea/RT)that lower the value of activation energy faster will be the rate of reaction.

For example, SO2 is oxidised to SO3 in the presence of nitric oxide as catalyst.

2SO2(g) + O2(g) −→−NO(g) 2SO3

O2(g)Reactant+2NO(g)Catalyst→2NO2(g)Intermediate

NO2(g)Intermediate+SO2(g)Reactant→SO3(g)Product+NO(g)Catalyst

Characteristics of a catalyst

(i) It can only catalyse the spontaneous reaction but not the non-spontaneous reaction.

(ii) It does not change the equilibrium constant, but only helps in attaining equilibrium faster.

(iii) It can catalyse both forward and backward reactions to the same extent to maintain the equilibrium state in case of reversible reaction.

(iv) It does not alter the free energy change (DG) of a reaction.

(v) A small amount of the catalyst can catalyse a large amount of reactions.

16. Collision Theory of Chemical Reactions:

(i) Only effective collisions bring about a chemical reaction. The collisions in which molecules collide with sufficient kinetic energy (threshold energy) and proper orientation, so as to facilitate breaking of bonds between reacting species and formation of new bonds to form products are called as effective collisions.

(ii) In collision theory, activation energy and proper orientation of the molecules together determine the criteria of an effective collision and hence the rate of chemical reaction.

Rate = PZABe–Ea/RT

Where, ZAB = The collision frequency of reactants A and B

P = Probability factor or steric factor

(It take into accounts the fact that in a collision, molecules must be properly oriented)

e–Ea/RT = Fraction of molecules with energies equal to or greater than Ea.

Important Formulae

1. Integrated Rate Equations

(i) For a zero order reaction:

t=[R]0–[R]kandt1/2=[R]02k

(ii) For a first order reaction:

t=2.303klog[R]0[R]andt1/2=0.693k

Amount of the substance left after n half lives of Ist order reaction = [R]02n.

2. Arrhenius Equation

(i) k = A e–Ea/RT

where k = Rate constant, A = Arrhenius factor or frequency factor, Ea = Activation energy, R = Gas constant, T = Temperature in Kelvin

(ii) logk2k1= Ea2.303R[T2–T1T1T2]

where k1 = Rate constant at T1 and k2 = Rate constant at T2

(iii) Ea = – 2.303 × R × slope (inaplotoflogkvs1T)