**Order of the Reaction**

The order of a reaction refers to the mathematical relationship between the concentration of a reactant and the rate of the reaction. It is determined by analyzing the data obtained from experimental observations.

To determine the order of the reaction from the given information, we can use the concept of half-life. The half-life of a reaction is the time taken for the concentration of a reactant to decrease to half of its initial value.

Given that the half-life period for the reaction at initial concentrations of 0.2 mol/L and 0.4 mol/L are 100 s and 200 s respectively, we can proceed with the following steps:

1. **Write the Integrated Rate Equation for a First-Order Reaction**:
The integrated rate equation for a first-order reaction is given by the equation:

ln([A]t/[A]0) = -kt

Where [A]t is the concentration of reactant A at time t, [A]0 is the initial concentration of reactant A, k is the rate constant, and ln is the natural logarithm.

2. **Derive the Half-Life Equation for a First-Order Reaction**:
By substituting [A]t = [A]0/2 and rearranging the equation, we can derive the half-life equation for a first-order reaction:

t1/2 = ln(2)/k

Where t1/2 is the half-life of the reaction.

3. **Compare the Half-Life Equations**:
By comparing the half-life equations for the two different initial concentrations, we can determine their relationship:

t1/2(0.2 mol/L) = ln(2)/k
t1/2(0.4 mol/L) = ln(2)/k

Dividing the second equation by the first equation, we get:

t1/2(0.4 mol/L) / t1/2(0.2 mol/L) = (ln(2)/k) / (ln(2)/k)

Simplifying the equation, we obtain:

t1/2(0.4 mol/L) / t1/2(0.2 mol/L) = 1

Since the ratio of the half-lives is equal to 1, we can conclude that the reaction is first order.

Therefore, the order of the reaction is first order.