Rate Of A Chemical Reaction

INTRODUCTION

Chemical Kinetics is the branch of science that deals with rate of reaction, factors affecting the rate of reaction and reaction – mechanism.

Different reactions occur at different rate. In fact a chemical reaction involves redistribution of bonds – breaking of bond(s) in the reactant molecule(s) and making of bonds in the product molecule(s). The rate of a chemical reaction actually depends upon the strength of the bond(s) and number of bonds to be broken during the reaction. It takes longer time for the reactant molecules to acquire higher amount of energy which they do by collision. Hence reactions involving strong bond – breaking occur at relatively slower rate while those involving weak

bond – breaking occur at relatively faster rate. On the basis of rate, reactions are classified as.

$\bullet$ Instantaneous or extremely fast reactions i.e. reactions with half-life of the order of fraction of second.

$\bullet$Extremely slow reactions i.e. reactions with half-life of the order of years.

$\bullet$Reactions of moderate or measurable rate.

Ionic reactions are instantaneous. If a drop of silver nitrate solution is added to a solution of the chloride salt of any metal or solution of HCl, a white precipitate of silver chloride appears within twinkling of eye. This is because of the fact that in aqueous solution an ionic compound exists as its constituent ions. No bond needs to be broken during the reaction. Hence reaction takes no time to complete. The half life period of an ionic reaction is of the order of 10^–10 s.

$N{a}^{+}+C{l}^{-}+A{g}^{+}+N{O}_3^{-}\to AgCl\downarrow +N{a}^{+}+N{O}_3^{-}$

Free radicals being very unstable (reactive) due to the presence of unpaired electron, reactions involving free radicals also occur instantaneously. Thus, the reactions, are instantaneous.

${}^{\bullet }C{H}_3+C{l}_2\to C{H}_{3}Cl+{}^{\bullet }Cl$

${}^{\bullet }C{H}_3{+}^{\bullet }C{H}_3\to H_{3}C-C{H}_3$

Some molecular reactions involving reactant(s) containing odd electron completes within a fraction of second. The speed of such reactions is attributable to the tendency of the odd electron molecule (paramagnetic in nature) to transform into stable spin-paired molecule (diamagnetic) by dimerisation. An example of such reaction is the dimerisation of nitrogen dioxide into nitrogen tetraoxide as mentioned below.

${}^{\bullet }N{O}_2+{}^{\bullet }N{O}_2\to N_2{O}_4$

There are some molecular reactions which are known to be extremely slow. Their half-lives are of the order of several years. Some examples of the type of reactions are as given below:

$4Fe+x{H}_{2}O+3O_2\to 2F{e}_2{O}_3.x{H}_{2}O$

${\left[Cr{\left(H_{2}O\right)}_6\right]}^{3+}+I^{-}\to {\left[Cr{\left(H_{2}O\right)}_5{I}^2\right]}^{+}+H_{2}O$

Note that the first reaction given above is called "rusting of iron". The second one is not ionic reaction as it appears at the first sight. Here in this reaction it is the co-ordinate bond between central metal ion i.e. Cr³⁺ (acceptor) and water molecule (donor) that is broken and covalent bond between Cr³⁺ and I^– that is formed. The half-life of this reaction is in years.

Most molecular reactions especially organic reactions occur at measurable rate. The half-life of such reactions are of the order of minutes, hours, days. Examples of such reactions are numerous. Some of these are given below.

$C{H}_{3}COOC{}_2{H}_5+H_{2}O\stackrel{H^{+}}{\to }C{H}_{3}COOH+C_2{H}_{5}OH$

$\underset{\left(SUCROUSE\right)}{C_{12}{H}_{22}{O}_{11}}.+H_{2}O\to \underset{\left(Glucoos\right)}{C_6{H}_{12}{O}_6}+\underset{\left(Fructos\right)}{C_6{H}_{12}{O}_6}$

$H_2{O}_2\left(aq\right)\to H_{2}O+\frac{1}{2}{O}_2\uparrow$

$2N_2{O}_5\to 4N{O}_2+O_2\uparrow$

$N{H}_{4}N{O}_2\left(aq\right)\to 2H_{2}O+N_2\uparrow$

In Chemical Kinetics we deal with the rates of only those reactions which occur with measurable rate i.e. which are neither too fast nor too slow. The rates of fast reactions are also determined using lasers.

RATE OF REACTION

The rate of a reaction means the speed with which the reaction takes place. This is expressed either in terms of decrease in the concentration of a reactant per unit time or increase in the concentration of a product per unit time.

Rate of reaction $=\frac{Decreaseintheconcentrationofareac\tan t}{Time\mathrm{int}erval}$

Or, $=\frac{increaseintheconcentrationofaproduct}{Time\mathrm{int}erval}=\frac{\Delta x}{\Delta t}$

The term $\Delta x$ means $x_{final}-x_{initial}and\Delta t$ is the amount of time elapsed. For example, a car driver starts his journey at 9.00 AM with odometer reading x miles. At 11.00 AM, he reaches his destination. The odometer reading at destination is y miles. The rate of his travel can be calculated as

$Rate=\frac{\Delta (dis\tan ce)}{\Delta (time)}=\frac{dis\tan c{e}_{(f)}-dis\tan c{e}_{(in)}}{tim{e}_{(f)}-tim{e}_{(in)}}$

$=\frac{y-x}{11.0-9.0}=\frac{y-x}{2}miles{h}^{-1}$

The above example indicates that the car has been driven with uniform rate but actually it has been driven sometimes faster and sometimes slower depending upon the condition of road. Thus, the overall rate is an average rate and the rate at which the car was moving at any instan

t, called instantaneous rate. The rate measured over a long time interval is called average rate and the rate measured for an infinitesimally small time interval is called instantaneous rate.

In general, for any reaction of the type

$A+B\to C+D$

Average rate of reaction $=-\frac{\Delta [A]}{\Delta t}=-\frac{\Delta [B]}{\Delta t}=+\frac{\Delta [C]}{\Delta t}=+\frac{\Delta [D]}{\Delta t}$

Where [A] signifies the molar conc. of (A) and $\Delta [A]$ stands for the change in molar concentration of A. The negative sign placed before a reaction rate symbol signifies a decrease in concentration of the reactant with increase of time and a positive sign before the rate symbol signifies that the concentration of product increases with increase of time.

Average rate of reaction

The rate measured over a long time interval is called average rate. The rate of reaction (average rate) is defined as the change of concentration of any one of the reactants (or products) per unit time. Average rate of reaction $r_{average}=\frac{Changeofconcentrationofoneofreac\tan tsorproducts}{Timetakenforthechange}$

Consider the reaction between CO and $N{O}_2$

$CO+N{O}_2\to C{O}_2+NO$

This equation shows that one mole of CO reacts with one mole of $N{O}_2,$ one mole each of $C{O}_{2}andNO$ are formed. The average rate of reaction can be expressed either by decrease in conc. of reactant $(CO,N{O}_2)$ or by the increase in conc. of any one of products $(C{O}_{2}orNO).$

Thus, $-\frac{\Delta (CO)}{\Delta t}=-\frac{\Delta (N{O}_2)}{\Delta t}=\frac{\Delta (C{O}_2)}{\Delta t}=\frac{\Delta (NO)}{\Delta t}$

For the reaction, $2H_2{O}_2\to 2H_{2}O+O_2$

When 2 moles of $H_2{O}_2$ decomposed, one mole of O₂ and 2 moles of $H_{2}O$ is formed. The rate of increase in the conc. of $O_2,$ therefore is half that of the disappearance of the conc. of $H_2{O}_2$ and increase in conc. of $H_{2}O$ is the same of the disappearance of the conc. of $H_2{O}_2$in the same time interval.

So $\frac{\Delta [O_2]}{\Delta t}=\frac{-1}{2}\frac{\Delta [H_2{O}_2]}{\Delta t}=\frac{1}{2}\frac{\Delta [H_{2}O]}{\Delta t}$

In general, for a reaction, $n_{1}A+n_{2}B\to m_{1}C+m_{2}D$

The rate is expressed as:

$-\frac{1}{n_1}\frac{\Delta [A]}{\Delta t}=\frac{-1}{n_2}\frac{\Delta [B]}{\Delta t}=\frac{1}{m_1}\frac{\Delta [C]}{\Delta t}=\frac{1}{m_2}\frac{\Delta [D]}{\Delta t}$

Instantaneous rate

With the progress of reaction the conc. of reactants decreases while that of product increases. According to law of mass action the rate of reaction decreases moment to moment as shown by graph of rate vs. time.

Rate varies from moment to moment so rate of reaction has to be specified at a given instant of time called instantaneous rate $r_{ins}or{r}_t=\pm \frac{dC}{dt}$

Where dC is the infinitesimal change in conc. during infinitesimal time interval dt after time t i.e. between t and t + dt.

Consider a reaction $AB$,To know the rate of reaction at any time t, a tangent is drawn to curve at the point corresponding to that time and it is extended on either side so as to cut the axes, say at the point A and B. Then

$Rateofreaction=\frac{Changeintheconc.}{Time}$

$=\frac{\Delta x}{\Delta t}=\frac{OA}{OB}=slopeofthe\tan gent$

Thus the rate of reaction at time 10 minutes

$\Delta t=20\times 60=1200\sec .$

$\Delta x=0.04moles$

$r_{ins}=\frac{0.04}{1200}=3.35\times {10}^{-5}molli{t}^{-1}{\sec }^{-1}$

Units of the rate of reaction

Since concentration is usually expressed in moles / time and time is taken in seconds or minutes, the unit of the rate of reaction is moles $li{t}^{-1}{\sec }^{-1}(M{L}^{-1}{\sec }^{-1})ormolesli{t}^{-1}{\min }^{-1}(M{L}^{-1}{\min }^{-1})$or $molesli{t}^{-1}hou{r}^{-1}(M{L}^{-1}{h}^{-1})$ or moles $li{t}^{-1}da{y}^{-1}(M{L}^{-1}{d}^{-1})ormolesli{t}^{-1}yea{r}^{-1}(M{L}^{-1}{y}^{-1}).$

Illustration 1. Why we prefer instantaneous rate of reaction over average rate of reaction?

Solution: The rate of reaction decreases continuously with time except for a zero order reaction. Therefore, average rate of reaction has no significance for the reaction. But instantaneous rate of reaction for a given instant of time does not change with time.

Illustration 2. Define rate of a reaction.

Solution: Rate of a reaction may be defined as the change in any one of the reactants or products per unit time.

Illustration 3. Define specific rate constant.

Solution: It is defined as the rate of a chemical reaction when the concentration of each reactant appearing in the rate equation is taken as unity.

LAW OF MASS ACTION

"At a given temperature, the rate of a reaction at a particular instant is proportional to the product of the active masses of the reactants at that instant raised to powers which are numerically equal to the numbers of their respective molecules in the stoichiometric equation describing the reaction".

Active mass = molar concentration of the substance

$=\frac{numberofgmmolesofthesubs\tan ce}{Volumeinlitre}$

$=\frac{\frac{W}{M}}{V}=\frac{n}{V}$

where W = mass of substance, M = molecular mass in grams.

V = volume in litres

Consider a simple reaction $A\to B$

If C_A is the molar concentration or active mass of A at a particular instant, then

$\frac{dx}{dt}\propto C_{A}or\frac{dx}{dt}=K{C}_A$

Where K is a proportionality constant or rate constant.

If C_A = 1 then

Rate $=\frac{dx}{dt}=K$

Let us consider a general reaction

$aA+bB\to \mathrm{Pr}oduct$

$Rate=\left(\frac{dx}{dt}\right)\propto {(A)}^a{(B)}^b$

$Rate=\frac{dx}{dt}=K{(A)}^a{(B)}^b$

If [A] = [B] = 1 mole / lit, then

Rate = K

Rate of reaction at unit concentration of reactant is called rate constant.

The value of rate constant depends on :

(i) Nature of reactant

(ii) Temperature

(iii) Catalyst

MOLECULARITY

A chemical reaction that take place in one and only one step i.e., all that occurs in a single step is called elementary reaction while a chemical reaction occurring in the sequence of two or more steps is called complicated reaction. The sequence of steps through which a complicated reaction takes place is called reaction mechanism. Each step in a mechanism is an elementary step reaction.

The molecularity of an elementary reaction is defined as the minimum number of molecules, atoms or ions of the reactants required for the reaction to occur and is equal to the sum of stoichiometric coefficient of the reactants in the chemical equation of the reaction. Thus, the molecularity of some elementary reactions are as mentioned below :

Elementary reactions Molecularity

$PC{l}_5\rightleftharpoons PC{l}_3+C{l}_2$ 1

$H_2+I_2\rightleftharpoons 2HI$ 2

Reaction with molecularity equal to one, two, three etc; are called unimolecular, biomolecular, trimolecular etc. respectively.

A complicated reaction has no molecularity of its own but molecularity of each of the steps (elementary reactions) involved in the mechanism.

For example; consider the reaction; $2NO+2H_2\to N_2+2H_{2}O;$which is complicated reaction and takes place in the sequence of following three steps:

(i) $NO+NO\rightleftharpoons N_2{O}_2$ (fast and reversible)

(ii) $N_2{O}_2+H_2\stackrel{r.d.s.}{\to }{N}_{2}O+H_{2}O$ (slow)

(iii) $N_{2}O+H_2\to N_2+H_{2}O$ (fast)

The molecularity of each step in the mechanism is two, so that we say that the reaction takes in the sequence of three steps each of which is bimolecular. There is another way also. According to which molecularity of a complicated reaction is taken to be equal to the molecularity of the slowest step i.e. rate determining step (r.d.s) in the mechanism.

For example, the reaction

$ROH+HCl\stackrel{ZnC{l}_2}{\to }RCl+H_{2}O$

is said to be unimolecular nucleophilic substitution$(S_{N}1)$. Since the reaction occurs in the sequence of the following three steps and the slowest step i.e. r.d.s. is unimolecular.

(i) $R\underset{\bullet \bullet }{\overset{\bullet \bullet }{O}}H+HCL\to R\stackrel{+}{O}{H}_2+C{l}^{-}\left(Fast\right)$

(ii) $R\stackrel{\oplus }{O}{H}_2\rightleftharpoons R^{\oplus }+H_{2}O(slow)$

(iii) $R^{+}+C{l}^{-}\to RCl(fast)$

Reactions of higher molecularity (molecularity > 3) are rare. This is because a reaction takes place by collision between reactant molecules and as number of a reactant molecules i.e. molecularity increases the chance of their coming together and colliding simultaneously decreases.

ORDER OF REACTION

The mathematical expression showing the dependence of rate on the concentration of reactant is known as rate law or rate expression of the reaction and sum of the indices (powers) of the concentration terms appearing in the rate law as observed experimentally is called order of reaction. To understand what is order of reaction, consider the reaction :

$2NO(g)+2H_2(g)\to N_2(g)+2H_{2}O(g)$

Kinetic experiment carried out at 1100ºK upon this reaction has shown following rate data.

From the experiment number 1 and 2, it is evident that rate increases 4 fold when conc. of NO is doubled keeping the conc. of $H_2$ constant i.e. $Rate\propto [NO{[}^{2}when[H_2]$is constant again from experiment number 2 and 3, it is evident that when concentration of $H_2$ is doubled keeping the conc. of NO constant, the rate is just doubled i.e.

$Rate\propto [H_2]when[NO]$is constant

$Overallrate\propto {[NO]}^2[H_2]$

Order of reaction with respect to NO is 2 and with respect to $H_{2}is1.$ The overall order of reaction is 2 + 1 = 3. This order of reaction suggest that the reaction is complicated and it does not occur in single step. In order to explain this reaction following mechanism has been proposed.

(i) $NO+NO\rightleftharpoons N_2{O}_2(fastandreversible)$

(ii) $N_2{O}_2+H_2\to N_{2}O+H_{2}O(slow)$

(iii) $N_{2}O+H_2\to N_2+H_{2}O(fast)$

Rate of overall reaction = Rate of step II $=K\left[N_2{O}_2\right][H_2]$

where K = Rate constant of step II

$N_2{O}_2$being intermediate for the overall reaction, its concentration has to be evaluated in terms of the concentration of reactant and this can be done by applying law of mass action upon the equilibrium of step I. Thus,

$K_c=\frac{[N_2{O}_2]}{{[NO]}^2}$

or $[N_2{O}_2]=K_c{[NO]}^2$

where $K_c=$equilibrium constant of step I, putting this value of concentration of $N_2{O}_2$ in the above rate expression, we get

$Rateofreaction=K.K_c{[NO]}^2[H_2]$

or Rate of reaction $=K^{\prime }{[NO]}^2[H_2]$

Rate of reaction $\propto {[NO]}^2[H_2]$

Where $K^{\prime }=K.K_c$ is another constant, rate constant of overall reaction.

In general, if rate law of a reaction represented by the equation.

$aA+bB\to \mathrm{Pr}oducts$is experimentally found to be as follows :

$Rate\propto {[A]}^m{[B]}^n$

Then order w.r.t. A = m, order w.r.t. B = n

Overall order = m + n

It may be noted that 'm' may or may not be equal to a and similarly 'n' may or may not be equal to b, m and n are experimental values, which really depends upon reaction mechanism and experimental condition, may not be predicted by just seeing the chemical equation of the reaction. An example of this is as follows:

(i) $2N_2{O}_5\to 4N{O}_2+O_2$

$Rate\propto [N_2{O}_5]$

Order of reaction is 1.

(ii) $2N{O}_2\to N_2{O}_2+O_2$

$Rate\propto {[N{O}_2]}^2$

Order of reaction is 2.

Illustration 4. For the reaction $A+2B\to 2A{B}_2$, the rate constant is $1.26\times 10^{-3}Lmo{l}^{-1}{s}^{-1}.$ What is the order of the reaction?

Solution: The units of rate constant is $Lmo{l}^{-1}{s}^{-1}or(mol{L}^{-1}{)}^{-1}{s}^{-1+.}$ Equate this with general expression of (mol L^-1)^1-ns^-1.

${\left(mol{L}^{-1}\right)}^{-1}{s}^{-1}={\left(mol{L}^{-1}\right)}^{1-n}{s}^{-1}$

$\therefore -1=1-norn=2$

The order of reaction = 2