IN the reaction, ` 2NO + CI_(2) to 2NOCI` it has been found that doubling the concentration of both the reactants increases the rate by a factor of eight but doubling the chlorine concentration alone only doubles the rate. Which of the following statements is incorrect ?

Options

A. The reaction is first order in ` CI_(2)`
B. The reaction is second order in NO.
C. The overall order of reaction is 2.
D. The overall order of reaction is 3.

Answer

To solve the problem step by step, we can analyze the given reaction and the provided information about the rate changes when concentrations of the reactants are altered.

### Step 1: Write the Reaction and Rate Law
The reaction given is:
\[ 2NO + Cl_2 \rightarrow 2NOCl \]

The rate law can be expressed as:
\[ \text{Rate} = k [NO]^x [Cl_2]^y \]
where \( x \) and \( y \) are the orders of the reaction with respect to \( NO \) and \( Cl_2 \), respectively.

### Step 2: Analyze the Effect of Doubling Both Reactants
When the concentrations of both reactants are doubled, the new rate can be expressed as:
\[ \text{Rate}_{\text{new}} = k (2[NO])^x (2[Cl_2])^y \]

This can be simplified to:
\[ \text{Rate}_{\text{new}} = k \cdot 2^x [NO]^x \cdot 2^y [Cl_2]^y = k \cdot 2^{x+y} [NO]^x [Cl_2]^y \]

According to the problem, this new rate is 8 times the original rate:
\[ k \cdot 2^{x+y} [NO]^x [Cl_2]^y = 8 \cdot k [NO]^x [Cl_2]^y \]

### Step 3: Set Up the Equation
From the above equation, we can cancel \( k [NO]^x [Cl_2]^y \) from both sides:
\[ 2^{x+y} = 8 \]

Since \( 8 = 2^3 \), we can equate the exponents:
\[ x + y = 3 \] (Equation 1)

### Step 4: Analyze the Effect of Doubling Chlorine Concentration Alone
Now, let's consider the case where only the concentration of \( Cl_2 \) is doubled:
\[ \text{Rate}_{\text{new}} = k [NO]^x (2[Cl_2])^y = k [NO]^x \cdot 2^y [Cl_2]^y = k \cdot 2^y [NO]^x [Cl_2]^y \]

This new rate is stated to be double the original rate:
\[ k \cdot 2^y [NO]^x [Cl_2]^y = 2 \cdot k [NO]^x [Cl_2]^y \]

### Step 5: Set Up the Second Equation
Canceling \( k [NO]^x [Cl_2]^y \) from both sides gives:
\[ 2^y = 2 \]

Thus, we find:
\[ y = 1 \] (Equation 2)

### Step 6: Substitute Back to Find \( x \)
Now, substituting \( y = 1 \) back into Equation 1:
\[ x + 1 = 3 \]
This leads to:
\[ x = 2 \]

### Step 7: Write the Final Rate Law and Determine Overall Order
The rate law can now be expressed as:
\[ \text{Rate} = k [NO]^2 [Cl_2]^1 \]

The overall order of the reaction is:
\[ x + y = 2 + 1 = 3 \]

### Step 8: Identify the Incorrect Statement
Now, we can evaluate the statements given in the question:
1. The reaction is first order in chlorine. (True, since \( y = 1 \))
2. The reaction is second order in \( NO \). (True, since \( x = 2 \))
3. The overall order of the reaction is 2. (False, since the overall order is 3)

Thus, the incorrect statement is:
**The overall order of the reaction is 2.**