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### ARRHENIUS EQUATION:

The temperature dependence of rate of a chemical reaction can be accurately explained by Arrhenius equation. It was first proposed by Dutch chemist   J.H. Vant’s Hoff but Swedish chemist Arrhenius provides its physical justification and interpretation.
Where   K= Rate constant
A= Arrhenius constant or frequency factor or pre exponential factor
R= Universal gas constant =25/3 joule per mole per second
Ea= Activation Energy
T= temperature
-Ea/RT= Boltzmann factor or fraction of molecule having equal or greater than Activation energy or fraction of molecule that have kinetic energy greater than activation energy.
ILLUSTRATIVE EXAMPLE (1):  The activation energy for the reaction is 209.5 k J per mole at 581 K.
Calculate the faction of molecule of reactant s having energy equal to or greater than activation energy.
SOLUTION: We known that fraction of molecule that have kinetic energy greater than activation energy is given by
Important cases of Arrhenius equation:
Case (1):  If T approaches to infinite

It means the entire reactant molecule will be active; and crossing over the energy barrier.  It will be possible when either Ea is Zero or temperature (T) is infinite. These are practically not possible.

The maximum value of K is A when temperature tends to infinity.

Graph indicate larger the activation energy, smaller is the value of rate constant (K)
Note:
Case (2):  We know that (Ea) activation energy is always positive, thus K always increasing with increasing temperature whether reaction is exothermic or endothermic

Mathematical prove: since activation energy is always positive it can be never negative this is proven as:
Case (3): However in many complex reactions it is observed that rate constant found to be decreases with increasing temperature.
For example
Mechanism:
By equation (1) and (2)

Since the overall reaction is  exothermic and  Kc  decreases  with increasing temperature  as well as the decrease  in Kc  out weight the increase  in K with temperature, thus K obs  show a decrease with increase in temperature.
Case (4): larger the activation energy greater the effect of temperature on rate constant.
Case (5): At lower temperature, increase in temperature causes more changes in value of rate constant
Case (6):
“It means if Ea=RT then rate constant become about 37% of the Arrhenius constant.

ILLUSTRATIVE EXAMPLE (2): Consider the following reaction
The activation energy of backward reaction exceeds that of the forward reaction by 2RT (in J per mole). If the pre-exponential factor of forward reaction is 4 times of the reverse reaction. The absolute value of (G) Gibbs energy at STP (in j per mole) for the reaction at 300 K is
(Given ln2= 0.7, RT= 2500 J per Mole at 300 K and G is the Gibbs energy) (JEE Advanced 2018)
SOLUTION:   Given condition
ILLUSTRATIVE EXAMPLE (3): Plots showing the variation of rate constant (K) with temperature (T) are given below. The plot that follow Arrhenius equation is (JEE Advanced 2010)
SOLUTION:

Hence on increasing temperature rate constant (K) increases exponentially.

So option (A) is correct

Calculation Activation Energy:
Take both side natural logarithm and obtained as
Note:
(1): The above equation is the straight line with negative slope
(2): The slope of above equation gives Activation energy and intercept gives frequency factor
(3) Dependence of rate constant on temperature for two reactions is given as:
Slope of reaction (2) is greater than (1) hence reaction (2) has higher activation energy so reaction (2) is more sensitive to temperature.
Calculation Activation Energy at two different temperatures (T1 and T2):
ILLUSTRATIVE EXAMPLE (4): In Arrhenius equation for a certain reaction, the value of A and Ea (activation energy) are 4 × 1013 sec–1 and 98.6 kJ mol–1 respectively. At what temperature, the reaction will have specific rate constant 1.1 × 10–3 sec–1?
SOLUTION: According to Arrhenius equation
ILLUSTRATIVE EXAMPPLE (5): The energy of activation for a reaction is 100 kJ mol–1. Presence of a catalyst lowers the energy of activation by 75%.What will be effect on rate of reaction at 20ºC, other things being equal?
SOLUTION: The Arrhenius equation is

### ARRHENIUS THEORY:

(A) Threshold Energy and Activation Energy:
Threshold energy(THE): For a reaction to take place the reacting molecules must colloid together, but only those collisions, in which colliding molecules possess certain minimum energy is called threshold energy (THE or ET) or the total minimum energy that reacting species must possess in order to undergo effective collision to form product molecules is called threshold energy.
Activation energy (Ea): It is extra energy which must be possessed by reactant molecules so that collision between reactant molecules is effective and leads to formation of product molecules.
ET =Threshold Energy, (THE)

HR = Enthalpy or Energy or Potential of reactants.
HP = Enthalpy or Energy or Potential of product,
(Ea)f =Activation energy for forward reaction.
(Ea)b =Activation energy for backward reaction.
Activated complex: It is formed between reacting molecules which is highly unstable and readily changes into product.

### COLLISION THEORY:

(1) The basic requirement for a reaction to occur is that the reacting species must collide with one another. This is the basis of collision theory for reactions.
(2) The number of collisions that takes place per second per unit volume of the reaction mixture is known as collision frequency (Z).
(3) Every collision does not bring a chemical change. The collisions that actually produce the product are effective collisions. The effective collisions, which bring chemical change, are few in comparison to the total number of collisions. The collisions that do not form a product are ineffective elastic collisions, i.e., molecules just collide and disperse in different directions with different velocities.
(4) For a collision to be effective, the following two barriers are to be cleared.
(A) Energy barrier: "The minimum amount of energy which the colliding molecules must possess as to make the chemical reaction to occur is known as threshold energy".

(i) In the graph 'E' corresponds to minimum or threshold energy for effective collision.
(ii) There is an energy barrier for each reaction. The reacting species must be provided with sufficient energy to cross the energy barrier
(B) Orientation barrier: The colliding molecules should also have proper orientation so that the
Old bonds may break and new bonds are formed.
During this reaction, the products are formed only when the colliding molecules have proper
Orientation at the time of collisions. These are called effective collisions.
(a) Properly oriented collisions form products

(b) Collisions not properly oriented

(5) Thus, the main points of collision theory are as follows,
(i) For a reaction to occur there must be collisions between the reacting species.
(ii)Only a certain fraction of the total number of collisions is effective in forming the products.
(iii) For effective collisions, the molecules should possess sufficient energy as well as orientation.
(6) The fraction of effective collisions, under ordinary conditions may vary from nearly zero to about one for ordinary reactions. Thus, the rate of reaction is proportional to:
(i) The number of collisions per unit volume per second (Collision frequency, Z) between the
reacting species
(ii) The fraction of effective collisions (Properly oriented and possessing sufficient energy), f
Where f is fraction of effective collision and Z is the collision frequency.
(7) The physical meaning of the activation energy is that it is the minimum relative kinetic energy which the reactant molecules must possess for changing into the products molecules during their collision. This means that the fraction of successful collision is equal to e-Ea /RT called Boltzmann factor.
(8) It may be noted that besides the requirement of sufficient energy, the molecules must be
Properly oriented in space also for a collision to be successful. Thus, if ZAB is the collision frequency, P is the orientation factor (Steric factor) then
If we compare this equation with Arrhenius equation
We know that pre-exponential form 'A' in Arrhenius equation is, A= PZAB

### KINETICS OF FIRST ORDER REACTIONS:

A chemical reaction is said to be of first order reaction if its rate is determined by the change of one of the concentration term only.
Consider a general chemical reaction its follow first order kinetics
Let [A] 0 =initial concentration of reactant A
[A]t = concentration of after time t
[B]t = concentration of product B after time t
On Integration of above equation
On rearrangement equation (1) give equation of straight line

We can also write equation (1)   as
Case (1):

On increasing angle (slop) the value of reaction constant increases hence k3>k2> k1

Case (2):

Case (3):

Case (4):
[A]t/[A]0  Vs  t :

Variation o f concentration with time:
Characteristic of 1st order:
(1) Concentration of reactant left after regular time interval of time will constitute a geometrical progression (GP).
(2) A first order reaction takes infinite time for completion.
(3) All radioactive disintegrations are examples of first order reaction
(4) Decomposing of H2O2 , Ester and Inversion of sugar are examples of first order reactions
(5) Half life time of first order reaction does not depend upon initial concentration.

Similarly concentration of product B after time t

Graph show increase in concentration  of [B] and [A] both as exponential manner.

Half life time of first Order reaction:
Natural life time of first order reaction