Chemistry Rate Expression, Rate Constant and Order and Molecularity of Reaction

### Topics Covered :

● Factors Influencing Rate of a Reaction
● Rate Expression
● Rate Constant
● Units of Rate Constant
● Order of a Reaction
● Molecularity of a Reaction

### Factors Influencing Rate of a Reaction :

Rate of reaction depends upon the experimental conditions such as concentration of reactants (pressure in case of gases), temperature and catalyst.

### Dependence of Rate on Concentration :

The rate of a chemical reaction at a given temperature may depend on the concentration of one or more reactants and products. The representation of rate of reaction in terms of concentration of the reactants is known as rate law. It is also called as rate equation or rate expression.

### Rate Expression and Rate Constant :

=> The results in Table clearly show that rate of a reaction decreases with the passage of time as the concentration of reactants decrease.

=> Conversely, rates generally increase when reactant concentrations increase. So, rate of a reaction depends upon the concentration of reactants.

=> Consider a general reaction aA + bB → c C + d D

where a, b, c and d are the stoichiometric coefficients of reactants and products. The rate expression for this reaction is

Rate prop [A]^x [B]^y ..................(1).

where exponents x and y may or may not be equal to the stoichiometric coefficients (a and b) of the reactants. Above equation
can also be written as

Rate = k [A]^x [B]^y ...............(2).

 - (d [R])/(dt) = k [A]^x [B]^y ...............(3).

● This form of equation (3) is known as differential rate equation, where k is a proportionality constant called rate constant.

color{purple}(✓✓)color{purple} " DEFINITION ALERT"
Rate constant may be defined as the rate of the reaction when the molar concentration of each reactant is taken as unity. That is why the rate constant is also called specific reaction rate.

color{green} ✍️ color{green} mathbf("Characteristics of rate constant"):
•Greater is the value of the rate constant, faster is the reaction.
•Each reaction has a definite value of the rate constant at a particular temperature.
•The value of the rate constant for the same reaction changes with temperature.
•The value of the rate constant of a reaction does not depend upon the concentrations of the reactants.
•The units of the rate constant depends upon the order of the reaction.

text(Rate Law of Rate Expression) : The equation which relates the rate of a reaction to concentration of reactants is called rate law or rate expression.

● Thus, rate law is the expression in which reaction rate is given in terms of molar concentration of reactants with each term raised to some power, which may or may not be same as the stoichiometric coefficient of the reacting species in a balanced chemical equation.

● For example :

2NO(g) + O_2 (g) → 2NO_2 (g)

We can measure the rate of this reaction as a function of initial concentrations either by keeping the concentration of one of the reactants constant and changing the concentration of the other reactant or by changing the concentration of both the reactants.

The following results are obtained:

-> It is obvious, after looking at the results, that when the concentration of NO is doubled and that of O_2 is kept constant then the initial rate increases by a factor of four from 0.096 to 0.384 mol L^(–1)s^(–1).

-> This indicates that the rate depends upon the square of the concentration of NO.

-> When concentration of NO is kept constant and concentration of O_2 is doubled the rate also gets doubled indicating that rate depends on concentration of O_2 to the first power.

-> Hence, the rate equation for this reaction will be

Rate = k[NO]_2[O_2]

-> The differential form of this rate expression is given as :

- (d [R])/(dt) = k [NO]^2 [O_2]

-> Here, we observe that for this reaction in the rate equation derived from the experimental data, the exponents of the concentration terms are the same as their stoichiometric coefficients in the balanced chemical equation.

● Some other examples are given below :

Reaction Experimental rate expression
1. CHCl_3+Cl_2 → C Cl_4+HCl Rate = k [CHCl_3] [Cl_2]^(1/2)
2. CH_3COOC_2H_5+H_2O → CH_3COOH +C_2H_5OH Rate = k [CH_3COOC_2H_5]^1 [H_2O]^0

-> In these reactions, the exponents of the concentration terms are not the same as their stoichiometric coefficients.

=> Thus, we can say that Rate law for any reaction cannot be predicted by merely looking at the balanced chemical equation, i.e., theoretically but must be determined experimentally.

### Order of a Reaction :

=> In the rate equation (1)

Rate = k [A]^x [B]^y

x and y indicate how sensitive the rate is to the change in concentration of A and B.

=> Sum of these exponents, i.e., x + y in (1) gives the overall order of a reaction whereas x and y represent the order with respect to the reactants A and B respectively.

=> Hence, the sum of powers of the concentration of the reactants in the rate law expression is called the order of that chemical reaction.

color{purple}(✓✓)color{purple} " DEFINITION ALERT"
The sum of powers to which the molar concentrations in the rate law equation are raised to express the observed rate of the reaction is called the order of reaction.

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

=> A zero order reaction means that the rate of reaction is independent of the concentration of reactants.

=> A balanced chemical equation never gives us a true picture of how a reaction takes place since rarely a reaction gets completed in one step.

=> color{red} ("The reactions taking place in one step are called elementary reactions ").

=> color{red} ("When a sequence of elementary reactions (called mechanism) gives us the products, the reactions
are called complex reactions").

=> These may be consecutive reactions (e.g., oxidation of ethane to CO_2 and H_2O passes through a series of intermediate steps in which alcohol, aldehyde and acid are formed), reverse reactions and side reactions (e.g., nitration of phenol yields o-nitrophenol and p-nitrophenol).
Q 2917480389

Calculate the overall order of a reaction which has the rate expression
(a) text(Rate) = k [A]^(1/2) [B]^(3/2)
(b) text(Rate) = k [A]^(3/2) [B]^(-1)

Solution:

(a) text(Rate) = k [A]^x [B]^y

order  = x+y

So order = 1/2 + 3/2 = 2, i.e., second order

(b) text(order) = 3/2+(-1) = 1/2 i.e., half order.
Q 2967580485

Identify the reaction order from each of the following rate constants .
(i). k = 2.3 xx 10^(-5) L mol^(-1) s^(-1)

(ii). k = 3 xx 10^(-4) s^(-1)

Solution:

(i) The unit of second order rate constant is L mol^(-1) s^(-1) , therefore k = 2.3 × 10^(–5) L mol^(–1) s^(–1) represents a second order reaction.

(ii). The unit of a first order rate constant is s^(–1) therefore k = 3xx10^(-4) s^(-1) represents a first order reaction.

### Units of Rate Constant :

For a general reaction

aA + bB → c C + d D

text(Rate) = k [A]^x [B]^y

Where x + y = n = order of the reaction

k = text(Rate)/([A]^x [B]^y)

 = text(concentration)/text(time) xx 1/(text{concerntration})^n

Taking SI units of concentration, mol L^(–1) and time, s, the units of k for different reaction order are listed in Table.

### Molecularity of a Reaction :

=> Molecularity of a reaction helps in understanding its mechanism.

text(Definition) : The number of reacting species (atoms, ions or molecules) taking part in an elementary reaction, which must collide simultaneously in order to bring about a chemical reaction is called molecularity of a reaction.

=> The reaction can be unimolecular when one reacting species is involved, for example, decomposition of ammonium nitrite.

NH_4NO_2 → N_2+2H_2O

=> Bimolecular reactions involve simultaneous collision between two species, for example, dissociation of hydrogen iodide.

2HI → H_2 +I_2

=> Trimolecular or termolecular reactions involve simultaneous collision between three reacting species, for example,

2NO +O_2 → 2NO_2

=> The probability that more than three molecules can collide and react simultaneously is very small. Hence, the molecularity greater than three is not observed.

=> It is, therefore, evident that complex reactions involving more than three molecules in the stoichiometric equation must take place in more than one step.

KClO_3 + 6FeSO_4 + 3H_2SO_4 → KCl + 3Fe_2 (SO_4)_3 + 3H_2O

● This reaction which apparently seems to be of tenth order is actually a second order reaction. This shows that this reaction takes place in several steps.

● Which step controls the rate of the overall reaction can be answered if we go through the mechanism of reaction, for example, chances to win the relay race competition by a team depend upon the slowest person in the team.

● Similarly, the overall rate of the reaction is controlled by the slowest step in a reaction called the rate determining step.

=> Consider the decomposition of hydrogen peroxide which is catalysed by iodide ion in an alkaline medium.

2H_2O_2 undersettext(Alkaline medium) overset(-I) → 2H_2O +O_2

● The rate equation for this reaction is found to be

text(Rate) = (- d [H_2O])/(dt) = kappa [H_2O_2] [I^(-)]

● This reaction is first order with respect to both H_2O_2 and I^(-).

● Evidences suggest that this reaction takes place in two steps

(a) H_2O + I^(-) → H_2O +I O^-

(b) H_2O_2+I O^(-) → H_2O + I^- + O_2

● Both the steps are bimolecular elementary reactions. Species IO^- is called as an intermediate since it is formed during the course of the reaction but not in the overall balanced equation.

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

=> From the above discussion, we conclude the following :

(i) Order of a reaction is an experimental quantity. It can be zero and even a fraction but molecularity cannot be zero or a non integer.

(ii) Order is applicable to elementary as well as complex reactions whereas molecularity is applicable only for elementary reactions. For complex reaction molecularity has no meaning.

(iii) For complex reaction, order is given by the slowest step and generally, molecularity of the slowest step is same as the order of the overall reaction.