## Determination of Order of Reaction Chemistry Notes

**Determination of Order of Reaction :**

Order of reaction can be measured by following methods:

- Graphical Method
- Ostwald Isolation Method
- Initial Rate Method
- Integrated Rate Law Method

**Graphical Method :**

This method is used only for reaction having one reactant. This reaction can be represented as follows:

nA → Product

Rate of reaction (r) = k[A]^{n}

Here, n = Order of reaction.

There are following steps used in the determination of order of reaction by graphical method :

First of all, concentrations of reactants are measured at different time intervals by suitable method.

Now, plot a graph between concentration and time.

Rate of instantaneous reaction is determined from slope of line after plotting a graph.

Instantaneous rates are represented by plotting a graph with respect to [A], [A]^{2}, [A]^{3} as per requirement.

With the help of these graphs, following conclusions are made :

If straight line is obtained in graph between rate and [A], then the reaction is of first order reaction.

Rate = k[A]

If straight line is obtained in graph between rate and [A]^{2}, then the reaction is of second order reaction.

Rate = k[A]^{2}

Similarly, if straight line is obtained in graph between rate and [A]^{2}, then the reaction is of third order reaction

But, this method is a hit and trial method fundamentally. So, its area of utility is limited. This method is not suitable for more than one reactants.

**Graphs for Different Order of Reaction :**

Graph between Concentration of Reactants and the Rate of Reaction

**Ostwald Isolation Method :**

→ This method is used in that condition when more than one reactants take part in reaction. In this method, the rate of reaction depends on that species taking part in the reaction whose molar concentration is changed in the reaction. If there is either no change in concentration of reactant or concentration is very low, then such type of reactant does not contribute in the determination of rate of reaction.

For example, aA + bB + C → Product

→ Molar concentrations of two reactants Band C of above reaction are taken very high in first step and rate of reaction is studied with A respect to only. Ratek[A]^{x}

→ Here, x is order of reaction with respect to reactant A Similarly, we study rate of reaction with respect to other reactants Band C.

Let, order of reaction is y and z for reactant B and C respectively, then total order of reaction = x + y + z.

**Initial Rate Method :**

→ Initial rate method is used for the determination of order of reaction in the presence of more than one reactant. Initial rate is that rate of reaction which takes place on initiating the reaction. So, at t = 0, rate is called initial rate. It is determined from the slope of concentration-time curve at zero time.

→ Initial rate represents intrinsic tendency of reaction because at this stage t = 0 and rate of reaction does not influenced from products.

→ In this method, initial rates of a reactant at different concentrations are determined by keeping concentrations of other reactants as constant. These data represent order of reaction related to specific reactant. This experiment is also used related to other reactants one by one and determines order of reaction in each condition.

**Integrated Rate Equation :**

→ Integrated rate equation method can be used in the determination of order of various reactions. Different integrated rate equations are used for different order of reactions. Integrated rate equation for various order of reactions is given as below:

Integrated rate expression for zero order reaction

A → Product

Let initial concentration [A]_{0} = a mol L^{-1} and at time ‘t’

[A] =(a – x)molL^{-1}

kt = a – (a – x)

⇒ kt = a – a + x

⇒ kt = \(\frac{x}{t}\) …… (6)

Equation (5) and (6) are integrated rate equations for zero order reaction.

Graphical Representation :

→ If a graph is plotted between concentration and time for zero order reaction then it is obtained as figure 4.14. According to this graph, a straight line (y = mx + C) having slope (-k) is obtained and its intercept is equal to [A]_{0}.

→ If rate equation x = kt in zero order reaction is considered as integrated rate equation then it is an equation of a simple line (y = mx) whose slope (tan) is equivalent to rate constant (k^{0}).

→ Some examples of zero order reactions are given below:

→ The thermal decomposition of gases at catalytic surface, some enzyme catalysed reactions and photochemical reactions are of zero order.

Integrated Rate Equation or Expression of First Order Reaction

A → Product

At t = 0 concentration = [A]_{0}

At t = t concentration = LAI

For first order reaction

Rate = k[A] ………. (1)

Rate = \(\frac{d[A]}{d t}\) ………. (2)

Since both equations i.e., equation (1) and equation (2) represent the rates. So,

Equation (5) and (6) are integrated rate equations of first order reaction.

Interval Equation :

If initial concentration of reactant A is not known then interval equation can be used for the determination of rate constant. In this equation, initial concentration is removed from rate equation. Let, x, and X, quantities of reactant are decomposed at two different times t, and ty, Equation (7) can be written as

**Graphical Representation :**

We can draw various types of graphs for first order reaction. The integrated rate equation of first order reaction is

→ If a graph is plotted between log \(\frac{a}{(a-x)}\) and t then according to figure, a simple line (y = mx) is obtained passing from original point, whose slope is

tan θ = \(\frac{k_{1}}{2.303}\) …..(10)

Rate constant can be calculated simply by above equation. Equation (9) can also be written as follows

log a- log (a – x) = \(\frac{k_{1}}{2.303} t\)

But log a is constant quantity. Therefore,

log (a – x) = – \(\frac{k_{1}}{2.303}\) t + log α ……..(11)

→ It has opposite slope and a straight line (y = mx + c) intersect on y-axis, which is in graph. So, graph is in between log (a – x) and t. The slope of this straight line will be as follows by which the value of k, can be calculated.

All radioactive decompositions are of first order reactions.

**Integrated Rate Equations for Reactions of Different Order :**

Generally, rate equation for nth order of reaction can be expressed as follows:

The above equation is used for all other order of reactions except first order reaction.

**Half-Life Period :**

→ The time which is required to convert half part of reactant into product in reaction, is called half-life period of that reaction. It is represented by the t_{1/2}

Half-Life Period for Zero Order Reaction

For zero order reaction.

Here, half-life period of zero order reaction is directly proportional to the initial concentration of reactant i.e., it depends on initial concentration.

→ Half-Life Period for First Order Reaction :

For first order reaction,

Hence, half-life period of first order reaction does not depend on initial concentration of reactant. Half-life period in first order reaction can be calculated from rate constant and rate constant can be calculated from half-life period.

→ Half-Life Period for n^{th} Order Reaction :

Half-life period (t_{1/2}) for n^{th} order of reaction can be given by following expression

\(t_{1 / 2} \alpha \frac{1}{(a)^{n-1}}\)

Here, a = initial concentration of reactant.

Similarly, for a reaction, the remaining amount of substance after n half-life period can be determined from following formula:

Substance remains after n half-life period = \(\frac{a}{(2)^{n}}\)

The relation between half-life period and initial concentration determined with the help of t_{1/2} α \(\frac{1}{(a)^{n-1}}\)

will be as follows :

**Plots of Half-life v/s Initial Concentration :**