### Integrated rate law for a first order reaction

For reactions where many reactants are involved, the order of the reaction is determined by the method of isolation. This was developed by Ostwald. In this method, the concentrations of all the reactants except one are taken in excess.

For e.g., in the reaction

A + B + C + D Products

If the rate law is r = k [A]m [B]n [C]o [D]p and if B, C and D are taken in large excess, such that [B] » [B]o, [C] » [C]o and [D] » [D]o then the rate law changes to a first order form as given by

Rate = k'[A]m where k' = k [B]o[C]o[D]o

The concentrations of A are measured at different intervals of time and the order of the reaction is determined graphically or the initial rate method is used where different initial concentrations of A are taken and the order is determined with respect to the reactant A. In succession, the above procedure can be used for the rest of the reactants, and then a probable rate law is determined. In this method, the integrated rate equation can be used by trial and error method, to determine the order and the rate constant of the reaction. What is the integrated rate expression? This will be discussed in the following section.

## Integrated Rate Expressions

Concentration dependence of rate is a differential equation (equation 8). It is tedious to determine the instantaneous rates from the slopes of the tangents for each value of t and this in turn makes the rate law determination difficult. To avoid this difficulty, the integrated form of the rate law expression is used.

From the integrated rate equation, the extent of a reaction in terms of the concentration of the reactant can be measured, if the rate constant is known. If the extent of the reaction is known, then the rate constant can be easily calculated. The integrated rate expressions are different for reactions of different orders. The integrated rate expressions for zero order and first order are derived in the subsequent sections.

## Integrated Rate Equation For Zero-Order Reaction

For a zero order reaction, the rate is independent of concentration of the reactant.

For e.g., A Products

Integration of the equation gives

At t = 0, [A] = [A]o = constant

Therefore equation becomes

A plot of [A], that is concentration, versus time t, will be linear with slope equal to (-k) and the intercept equal to [A]o.

The rate constant k of the reaction is given by,

The time taken for the initial concentration to reduce by half is given by,

From the t1/2 relation, the rate constant k can be calculated.

Reactions in heterogenous system usually follow zero order kinetics. In such a system the reactants are adsorbed on the surface of a solid catalyst. At low concentrations of the reactant, the rate of the reaction depends on the fraction of the surface covered by the reactant. However, at higher concentrations the surface of the catalyst becomes fully covered and any further increase in the concentration of the reactants does not affect the rate of the reaction. One very common example, is the decomposition of ammonia (NH3) on finely divided platinum (e.g., 4). At very low pressures of ammonia, the decomposition reaction has rate r given by

Where k1 and k2 are constants. At low pressures, k2 [NH3] term does not count as it is very less than one. Under such condition, the reaction is first order in ammonia.

At higher pressures, the value of k2[NH]3 is much larger than unity and therefore, the rate follows a zero order kinetics. The rate at high pressures of ammonia is given by

## Integrated Rate Equation For First Order Reaction

For the reaction A Products, which follows first order kinetics, the rate of the reaction is proportional to the concentration of A raised to power 1. i.e.,

Integration of the differential equation gives

In [A] = - kt + constant

At t = 0, [A] = [A]o

ln [A]o = constant

On substitution, the integrated equation is transformed to

The rate constant k1, of a first order reaction can be determined from the expression,

The integrated rate equation can also be written in the form,

This equation shows that the concentration of the reactant decreases exponentially. From the equation it can be seen that if ln[A] is plotted against t, a straight line is obtained with the slope equal to (-k).

fig 6.9 - Linear form of a first order reaction

A special feature of a first order reaction is the half life (t1/2 of the reaction. When the reaction has proceeded half way then [A]=[A]o/2. On substitution of this value in the integrated equation the expression of t1/2 becomes

### Half-Life of a First Order Reaction is Independent of the Initial Concentration of the Reactant

Example 6:

A first order reaction is 40% complete in 50 minutes. What is the rate constant? In what time will the reaction be 80% complete?