For the given reaction ,
`A+Brarr` Products
Following data are given :
`{:("initial conc."(m//L),"initial conc."(m//L),"initial conc."(mL^(-1)S^(-1))),([A]_0,[B]_0,),(0.1,0.1,0.05),(0.2,0.1,0.1),(0.1,0.2,0.05):}`
calculate the Rate constant

Options

A. `0.5"sec"^(-1)`
B. `0.4 "sec"^(-1)`
C. `0.2"sec"^(-1)`
D. `0.7"sec"^(-1)`

Answer

To calculate the rate constant (k) for the reaction \( A + B \rightarrow \text{Products} \) using the provided data, we will follow these steps:

### Step 1: Write the Rate Law Expression
The rate law for the reaction can be expressed as:
\[ \text{Rate} = k [A]^x [B]^y \]
where \( x \) and \( y \) are the orders of the reaction with respect to reactants A and B, respectively.

### Step 2: Set Up the Equations Using Given Data
We have three sets of data for the concentrations of A and B, along with the corresponding rates:

1. For \( [A]_0 = 0.1 \, \text{M}, [B]_0 = 0.1 \, \text{M}, \text{Rate} = 0.05 \, \text{M/s} \)
\[ 0.05 = k (0.1)^x (0.1)^y \quad \text{(Equation 1)} \]

2. For \( [A]_0 = 0.2 \, \text{M}, [B]_0 = 0.1 \, \text{M}, \text{Rate} = 0.1 \, \text{M/s} \)
\[ 0.1 = k (0.2)^x (0.1)^y \quad \text{(Equation 2)} \]

3. For \( [A]_0 = 0.1 \, \text{M}, [B]_0 = 0.2 \, \text{M}, \text{Rate} = 0.05 \, \text{M/s} \)
\[ 0.05 = k (0.1)^x (0.2)^y \quad \text{(Equation 3)} \]

### Step 3: Solve for x and y
To find the values of \( x \) and \( y \), we can manipulate these equations.

#### Dividing Equation 2 by Equation 1:
\[ \frac{0.1}{0.05} = \frac{k (0.2)^x (0.1)^y}{k (0.1)^x (0.1)^y} \]
This simplifies to:
\[ 2 = \frac{(0.2)^x}{(0.1)^x} \]
\[ 2 = 2^x \implies x = 1 \]

#### Dividing Equation 3 by Equation 1:
\[ \frac{0.05}{0.05} = \frac{k (0.1)^x (0.2)^y}{k (0.1)^x (0.1)^y} \]
This simplifies to:
\[ 1 = \frac{(0.2)^y}{(0.1)^y} \]
\[ 1 = 2^y \implies y = 0 \]

### Step 4: Substitute x and y Back into the Rate Law
Now that we have \( x = 1 \) and \( y = 0 \), we can rewrite the rate law:
\[ \text{Rate} = k [A]^1 [B]^0 = k [A] \]

### Step 5: Calculate the Rate Constant k
Using any of the equations, we can now calculate \( k \). Let's use Equation 1:
\[ 0.05 = k (0.1)^1 (0.1)^0 \]
\[ 0.05 = k (0.1) \]
\[ k = \frac{0.05}{0.1} = 0.5 \, \text{M}^{-1}\text{s}^{-1} \]

### Final Result
The rate constant \( k \) for the reaction is:
\[ k = 0.5 \, \text{M}^{-1}\text{s}^{-1} \]