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 solve the problem step by step, we will follow the process of determining the rate law for the reaction and then calculating the rate constant \( k \).

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

### Step 2: Set Up the Equations Using Given Data
Using the provided data, we can set up equations for the rate based on the initial concentrations of A and B.

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

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

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

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

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

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

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

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

### Final Answer
The rate constant \( k \) is \( 0.5 \, \text{M}^{-1}\text{s}^{-1} \).

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