For a reaction `A+2Brarr2C`, the following data were obtained . Initial concentration
`{:(,[A],[B],"Rate"(moll^(-1)min^(-1))),(i.,1.0,1.0,0.15),(ii,2.0,1.0,0.30),(iii.,3.0,1.0,0.45),(iv.,1.0,2.0,0.15),(v.,1.0,3.0,0.15):}`
The rate law for this reaction

Options

A. `R=k[A][B]^0`
B. `R=k[A][B]^2`
C. `R=k[A][B]`
D. `R=k[A]^2[B]`

Answer

To determine the rate law for the reaction \( A + 2B \rightarrow 2C \) based on the provided data, we will follow these steps:

### Step 1: Write the general rate law expression
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k[A]^{\alpha}[B]^{\beta} \]
where \( k \) is the rate constant, \( \alpha \) is the order with respect to \( A \), and \( \beta \) is the order with respect to \( B \).

### Step 2: Analyze the data
We have the following experimental data:

| Experiment | [A] (mol/L) | [B] (mol/L) | Rate (mol/L/min) |
|------------|-------------|-------------|-------------------|
| (i) | 1.0 | 1.0 | 0.15 |
| (ii) | 2.0 | 1.0 | 0.30 |
| (iii) | 3.0 | 1.0 | 0.45 |
| (iv) | 1.0 | 2.0 | 0.15 |
| (v) | 1.0 | 3.0 | 0.15 |

### Step 3: Determine the order with respect to \( A \) (α)
Using experiments (i) and (ii):
- For experiment (i):
\[ 0.15 = k(1.0)^{\alpha}(1.0)^{\beta} \]
- For experiment (ii):
\[ 0.30 = k(2.0)^{\alpha}(1.0)^{\beta} \]

Dividing the second equation by the first:
\[ \frac{0.30}{0.15} = \frac{k(2.0)^{\alpha}(1.0)^{\beta}}{k(1.0)^{\alpha}(1.0)^{\beta}} \]
\[ 2 = \frac{(2.0)^{\alpha}}{(1.0)^{\alpha}} \implies 2 = 2^{\alpha} \]
Thus, \( \alpha = 1 \).

### Step 4: Determine the order with respect to \( B \) (β)
Using experiments (iv) and (v):
- For experiment (iv):
\[ 0.15 = k(1.0)^{\alpha}(2.0)^{\beta} \]
- For experiment (v):
\[ 0.15 = k(1.0)^{\alpha}(3.0)^{\beta} \]

Dividing the fifth equation by the fourth:
\[ \frac{0.15}{0.15} = \frac{k(1.0)^{\alpha}(3.0)^{\beta}}{k(1.0)^{\alpha}(2.0)^{\beta}} \]
\[ 1 = \frac{(3.0)^{\beta}}{(2.0)^{\beta}} \implies 1 = \left(\frac{3.0}{2.0}\right)^{\beta} \]
Since the left side is 1, this implies \( \beta = 0 \).

### Step 5: Write the final rate law
Now we can write the rate law as:
\[ \text{Rate} = k[A]^{1}[B]^{0} = k[A] \]

### Conclusion
The rate law for the reaction \( A + 2B \rightarrow 2C \) is:
\[ \text{Rate} = k[A] \]