Method of Integration (Integrated Rate Equation Method)

The Method of Integration, also known as the Integrated Rate Equation Method, is one of the most widely used methods for determining the order of a chemical reaction. In this method, the experimentally measured concentrations of the reactant at different time intervals are substituted into the integrated rate equations corresponding to zero-order, first-order, and second-order reactions. The equation that gives a constant value of the rate constant \(k\) throughout the reaction indicates the correct order of the reaction.

Unlike the differential method, this method does not require the measurement of instantaneous reaction rates. Instead, it uses concentration-time data obtained experimentally, making it more accurate and convenient for many reactions.

Principle

Suppose the concentration of a reactant is measured at different times during a reaction. The experimental data are tested using the integrated rate equations for different reaction orders.

Case I: Zero-Order Reaction

The integrated rate equation is

$$
[A]=[A]_0-kt
$$

or

$$
k=\frac{[A]_0-[A]}{t}
$$

If the calculated value of \(k\) remains constant for all observations, the reaction is a zero-order reaction.

Case II: First-Order Reaction

The integrated rate equation is

$$
k=\frac{2.303}{t}
\log\left(\frac{[A]_0}{[A]}\right)
$$

If the calculated value of \(k\) is constant at different time intervals, the reaction follows first-order kinetics.

Case III: Second-Order Reaction

The integrated rate equation is

$$
k=\frac{1}{t}
\left(
\frac{1}{[A]}
–
\frac{1}{[A]_0}
\right)
$$

If the calculated value of \(k\) remains constant, the reaction is a second-order reaction.

Graphical Method

The order of a reaction can also be determined graphically by plotting the appropriate functions of concentration against time.

Procedure

1. Measure the concentration of the reactant at different time intervals.
2. Calculate the rate constant using the integrated rate equation for zero-order, first-order, and second-order reactions separately.
3. Compare the calculated values of the rate constant.
4. The equation that produces a constant value of \(k\) represents the correct order of the reaction.

Advantages

• Simple and accurate method for determining reaction order.
• Does not require measurement of instantaneous reaction rates.
• Applicable to most homogeneous reactions.
• Provides both the reaction order and the rate constant simultaneously.

Limitations

• Requires accurate concentration-time data.
• Errors in concentration measurements affect the calculated rate constant.
• Less suitable for highly complex reactions involving multiple mechanisms.

Important Points

• The method is based on integrated rate equations.
• The correct reaction order is identified by obtaining a constant value of the rate constant.
• It is one of the most commonly used experimental methods in chemical kinetics.
• The graphical approach provides an additional confirmation of the reaction order.