To determine the order of the reaction with respect to \( A_2 \) and \( B_2 \) from the given data, we will analyze the rate of reaction using the method of initial rates. The rate of reaction is given by the equation:
\[
\text{R.O.R.} = k[A_2]^a[B_2]^b
\]
We will use the provided experimental data to find the values of \( a \) and \( b \).
### Step 1: Analyze the given data
We have three sets of initial concentrations and their corresponding rates of reaction:
1. Experiment 1: \([A_2] = 0.2 \, \text{M}, [B_2] = 0.2 \, \text{M}, \text{R.O.R.} = 0.04 \, \text{Ms}^{-1}\)
2. Experiment 2: \([A_2] = 0.1 \, \text{M}, [B_2] = 0.4 \, \text{M}, \text{R.O.R.} = 0.04 \, \text{Ms}^{-1}\)
3. Experiment 3: \([A_2] = 0.2 \, \text{M}, [B_2] = 0.4 \, \text{M}, \text{R.O.R.} = 0.08 \, \text{Ms}^{-1}\)
### Step 2: Compare Experiments 1 and 3 to find \( b \)
In Experiments 1 and 3, the concentration of \( A_2 \) is the same (0.2 M), while \( B_2 \) changes:
- From Experiment 1:
\[
\text{R.O.R.} = k[0.2]^a[0.2]^b = 0.04
\]
- From Experiment 3:
\[
\text{R.O.R.} = k[0.2]^a[0.4]^b = 0.08
\]
Now, we can set up the ratio of the rates:
\[
\frac{0.08}{0.04} = \frac{k[0.2]^a[0.4]^b}{k[0.2]^a[0.2]^b}
\]
This simplifies to:
\[
2 = \frac{[0.4]^b}{[0.2]^b}
\]
Which can be written as:
\[
2 = \left(\frac{0.4}{0.2}\right)^b = 2^b
\]
From this, we can equate the powers:
\[
b = 1
\]
### Step 3: Compare Experiments 2 and 3 to find \( a \)
Next, we will compare Experiments 2 and 3, where the concentration of \( B_2 \) is the same (0.4 M):
- From Experiment 2:
\[
\text{R.O.R.} = k[0.1]^a[0.4]^b = 0.04
\]
- From Experiment 3:
\[
\text{R.O.R.} = k[0.2]^a[0.4]^b = 0.08
\]
Setting up the ratio of the rates gives us:
\[
\frac{0.08}{0.04} = \frac{k[0.2]^a[0.4]^b}{k[0.1]^a[0.4]^b}
\]
This simplifies to:
\[
2 = \frac{[0.2]^a}{[0.1]^a}
\]
Which can be written as:
\[
2 = \left(\frac{0.2}{0.1}\right)^a = 2^a
\]
From this, we can equate the powers:
\[
a = 1
\]
### Final Result
The order of the reaction with respect to \( A_2 \) is \( a = 1 \) and with respect to \( B_2 \) is \( b = 1 \). Therefore, the overall reaction order is:
- Order with respect to \( A_2 \): 1
- Order with respect to \( B_2 \): 1
