Order of Reaction

Order of reaction

The order of a reaction is the exponent to which the concentration of a reactant is raised in the rate law. It indicates how the rate of the reaction is influenced by the concentration of the reactants. The order of a reaction is determined experimentally and can be an integer or a fraction. It is a crucial concept in chemical kinetics as it helps in understanding the relationship between the concentration of reactants and the rate of reaction.

Definition

The order of reaction is the sum of the exponents of the concentration terms in the rate law. It gives insight into the molecularity of the reaction, which refers to the number of molecules or ions involved in the reaction at the elementary level.

General rate law

The general form of the rate law for a reaction is:

$\text{Rate} = k[A]^m[B]^n$

Where:

• Rate is the rate of the reaction.
• $k$ is the rate constant.
• $[A]$ and $[B]$ are the concentrations of reactants A and B.
• $m$ and $n$ are the orders of reaction with respect to reactants A and B, respectively.

The total order of the reaction is the sum of the exponents: Order = $m + n$

Types of reaction orders

1. Zero order reaction: In a zero-order reaction, the rate of reaction is independent of the concentration of the reactant. The rate law is: $\text{Rate} = k$

2. First order reaction: In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. The rate law is: $\text{Rate} = k[A]$

3. Second order reaction: In a second-order reaction, the rate of reaction is proportional to the square of the concentration of one reactant, or the product of the concentrations of two reactants. The rate law is: $\text{Rate} = k[A]^2 \quad \text{or} \quad \text{Rate} = k[A][B]$

4. Fractional order reaction: A reaction can have a non-integer order (e.g., 1/2 or 3/2), which is determined experimentally. This typically occurs in complex reactions.

Determining the order of a reaction

The order of a reaction can be determined experimentally using the method of initial rates or by using integrated rate laws. In the method of initial rates, the rate of reaction is measured at different concentrations of the reactants, and the order is deduced by analyzing how the rate changes with concentration.

Example: determining the order of reaction

Consider the reaction:

$A \rightarrow B$

If the rate law is given by:

$\text{Rate} = k[A]^m$

To determine the order, the initial rates are measured at different concentrations of A, and the values of $m$ (the order with respect to A) are calculated.

Units of rate constant (k)

The units of the rate constant depend on the order of the reaction. For a reaction of order $n$, the units of the rate constant are:

Zero order: $k$ has units of concentration/time (e.g., M/s).

First order: $k$ has units of 1/time (e.g., 1/s).

Second order: $k$ has units of 1/concentration·time (e.g., 1/M·s).

Calculation examples

Example 1: Zero-order reaction

For a zero-order reaction:

$A \rightarrow B$

The rate law is:

$\text{Rate} = k$

Given that the rate of the reaction is 0.05 M/s, find the rate constant $k$ if the concentration of A is 0.1 M.

Solution: For a zero-order reaction, the rate is independent of the concentration. Thus, we can directly use the given rate:

$\text{Rate} = k = 0.05 \text{ M/s}$

So, the rate constant is 0.05 M/s.

Example 2: First-order reaction

Consider the first-order reaction:

$A \rightarrow B$

The rate law is:

$\text{Rate} = k[A]$

Given that the concentration of A is 0.3 M and the rate of reaction is 0.15 M/s, find the rate constant $k$.

Solution: For a first-order reaction, we use the rate law:

$\text{Rate} = k[A]$

Substituting the known values:

$0.15 \text{ M/s} = k (0.3 \text{ M})$

Solving for $k$:

$k = 0.15 \text{ M/s} \div 0.3 \text{ M} = 0.5 \text{ s}^{-1}$

Thus, the rate constant is 0.5 s⁻¹.

Example 3: Second-order reaction

For a second-order reaction:

$A \rightarrow B$

The rate law is:

$\text{Rate} = k[A]^2$

If the rate of reaction is 0.18 M/s and the concentration of A is 0.6 M, calculate the rate constant $k$.

Solution: For a second-order reaction, we use the rate law:

$\text{Rate} = k[A]^2$

Substituting the known values:

$0.18 \text{ M/s} = k (0.6 \text{ M})^2$

Solving for $k$:

$k = 0.18 \text{ M/s} \div (0.6 \text{ M})^2 = 0.18 \text{ M/s} \div 0.36 \text{ M}^2 = 0.5 \text{ M}^{-1} \text{s}^{-1}$

Thus, the rate constant is 0.5 M⁻¹ s⁻¹.

Example 4: Determining reaction order from experimental data

In an experiment, the following data for the reaction $A \rightarrow B$ was obtained:

Concentration of A (M)	Rate (M/s)
0.2	0.04
0.4	0.16
0.6	0.36

Determine the order of the reaction.

Solution: We will compare the rates at different concentrations of A. Observe the pattern:

1. When the concentration doubles from 0.2 M to 0.4 M, the rate quadruples (0.04 to 0.16).
2. When the concentration doubles from 0.4 M to 0.6 M, the rate quadruples (0.16 to 0.36).

This indicates that the reaction is second-order because the rate increases by the square of the concentration.

Example 5: Integrated rate law for a first-order reaction

For a first-order reaction, the integrated rate law is given by:

$\ln \frac{[A]_0}{[A]} = kt$

Where:

• $[A]_0$ = initial concentration
• $[A]$ = concentration after time $t$
• $k$ = rate constant
• $t$ = time

If the initial concentration of A is 0.8 M, the concentration after 10 seconds is 0.4 M, and the rate constant is 0.1 s⁻¹, find the time taken for the concentration of A to drop to 0.1 M.

Solution: Using the integrated rate law:

$\ln \frac{[A]_0}{[A]} = kt$

Substituting the given values:

$\ln \frac{0.8}{0.1} = 0.1 \times t$

$\ln 8 = 0.1 \times t$

$t = \frac{\ln 8}{0.1} = \frac{2.08}{0.1} = 20.8 \text{ seconds}$

Thus, the time taken for the concentration to drop to 0.1 M is 20.8 seconds.