Chemistry EFFECT OF TEMPERATURE ON RATE OF REACTIONS-ARRHENIUS THEORY

Arrhenius Concept :

In order to explain the effect of temperature on the reaction rate Arrhenius proposed a theory of reaction rate which states as follows:

i) A chemical reaction takes place by collision between the reactant molecules, and for the collision to be effective the colliding molecules must possess some certain minimum energy called threshold energy of reaction.

ii) Reactant molecules having energy equal or greater than the threshold called active molecules and those having energy less then the threshold are called passive molecules.

iii) At a given temperature there exists a dynamic equilibrium between active molecules. The process of transformation from passive to active molecules being endothermic, increase of temperature increases the number of active molecules and the reaction.

`text(Passive molecules) ⇋ text(Active molecules)`

molecules , `DeltaH = + ve`

iv) Concept of energy of activation `(E_a)`

The extra amount of energy which the reactant molecules (having energy less then the threshold) must acquire so that their mutual collision may lead to the breaking of bond (s) and hence the reaction, is known as energy of the activation of reaction. It is denoted by the symbol `E_a`. Thus,

`E_a = text(Threshold energy - Actual average energy)`

`E_a` is expressed in `kcal` `(mol)^(-1)` or `kJ` `(mol)^(-1)`

Arrhenius Equation :

The variation equilibrium constant of a reaction with temperature is described by Van't Hoff equation of thermodynamics which is as
follows :

`(dlnK_p)/(dT)=(Delta H)/(RT^2)`

If `k_1` and `k_2` be the rate constants of `FR` (forward reaction) and `BR` (backward reaction), respectively then `K_p = k_1//k_2`. Further, `D H` `= E_(a_1)- E_(a_2)`. Putting these in the above equation we get

`(dlnk_1)/(dT)-(dlnk_2)/(dT)=E_(a_1)/(RT^2)-E_(a_2)/(RT^2)`

Splitting into two parts `(d ln k_1)/(dT) =(E_(a_1))/(RT^2)+k` (For FR)

`(d ln k_2)/(dT) =E_(a_2)/(RT^2) +k` (For BR) where `k` is constant

Arrhenius sets `k` equal to zero and without specifying `FR` and `BR`, he gave the following equation called Arrhenius equation.

`(dlnk)/(dT)=E_a/(RT^2)`...............(4)

From this equation it is evident that rate of change of logarithm of rate constant with temperature depends upon the magnitude of energy of activation of the reaction. Higher the `E_(a_1)` smaller the rate of change of logarithm of rate constant with temperature. That is, rate of the reaction with low `E_a` increases slowly with temperature while rate of the reaction with high `E_a` increases rapidly with temperature. It is also evident that rate of increase of logarithm of rate constant will go on decreasing with increase of temperature.

Integrating Equation `4` assuming `E_a` to be constant we get,

`lnk=-(E_a)/(RT)+ln A`.............(5)

or `ln (k/A)=-(E_a)/(RT)` or `k=Ae^((-E_a)// (RT))`.........(6)

Equation (6) is integrated form of Arrhenius equation. The constant `A` called pre-exponential factor or the frequency factor since it is
somewhat related with collision frequency. It is a constant for a given reaction. From Equation (6) it is evident that as `T ->oo` ,`k->A`. Thus, the constant `A` is the rate constant of reaction at infinity temperature. The rate constant goes on increasing with temperature.

So, when `T` approaches infinity, `k` will be maximum. That is to say, `A` is the maximum rate constant of a reaction. It is also to be noted that the exponential term i.e. `e^(-E_a//RT)` measures the fraction of total number of molecules in the activated state or fraction of the total number of effective collisions. If `n_(E_a)` and `n` be the number of molecules of reactant in the activated state and the total number of molecules of the reactant present in the reaction vessel respectively, then

`(n_(Ea))/n=e^(-E_a//RT)`

Equation (5) may also be put as

`log k = (-(E_a)/(2.303 R)) 1/T +log A` ..............(7)

Since `(E_a)/(2.303R)` and `logA` both are constants for a given reaction. So, from equation (7) it is evident that a plot of log `k` vs. `1/T` will be a straight line of the slope equal to `(-E_a)/(2.303 R)` and intercept equal to `logA` as shown in fig.

`(-E_a)/(2.303 R)-tantheta =-tan (180-theta)=-(OA)/(OB)`

`E_a=(OA)/(OB) xx 2.303 R`

`log A=OA`

Thus, from this plot `E_a` and `A` both can be determined accurately

If `k_1`(of first order) be the rate constant of a reaction at two different temperature `T_ 1` and `T_ 2` respectively then from equation (7), we may write

`log k_1 = -E_a/(2.303 R) xx 1/(T_1) +log A`

`log k_2= -E_a/(2.303 R) xx 1/T_2 +log A`

Subtracting former from the latter we get

`log (k_2/k_1) =E_a/(2.303 R) (1/T_1 - 1/T_2)`..............(8)

With the help of this equation it is possible to calculate `E_a` of a reaction provided, rate constants of reaction at two different temperatures are known. Alternatively one can calculate rate constant of a reaction at a given temperature provided that rate constant of the reaction at some other temperature and also `E_a` of the reaction is known.

Temperature Coefficient :

Temperature has very marked effect on the reaction rate. It has been found that the rate of most homogeneous reactions are nearly doubled or tripled by `10^(o)` rise in temperature.

`k_(t+10)/k_1 = 2` to `3` (nearly)

The ratio of the rate constants of a reaction at two different temperatures differing by `10 ^(o)` i.e. `k_(t+10)//k_1` is known as temperature coefficient of reaction rate. This ratio also depends upon temperature and two temperatures generally selected are `25 ^(o) C` and `35^(o)C`. If a reaction has a temperature coefficient of reaction rate equal to `3`, then by raising its temperature from `25 ^(o) C` to `65^(o)C`, the rate will increase by nearly `3xx 3 xx 3xx 3` i.e. `81` times.

Reaction Energy Profile :

The essence of Arrhenius Theory of reaction rate is that there exists an energy barrier in the reaction path between reactant(s) and
product(s) and for reaction to occur the reactant molecules must climb over the top of the barrier which they do by collision. The existence of energy barrier and concept of `E_a` can be understood from the following diagram.

For exothermic reaction : See fig.1.

`H_R=` Average summation of enthalpies of reactants

`H_P=` Average summation of enthalpies of products

`Delta H =` Enthalpy change during the reaction

`E_(a_1) =` Energy of activation of the forward reaction (FR)

`E_(a_2) =` Energy of activation of the backward reaction (BR)

For endothermic reaction : See fig.2.

`text(Notable Points)` :

i) `Delta H =` Energy of activation of FR - Energy of activation of BR

ii) Greater the height of energy barrier, greater will be the energy of activation and more slower will be the reaction at a given temperature.

iii) Rate = Collision frequency `xx` fraction of the total number of collision which is effective.

Collision frequency is the number of collisions per unit volume per unit time. It is denoted by the symbol `Z`. `Z` is directly proportional to `sqrtT`.