For the reaction ` A + 2B to ` C, the reaction rate is doubled if the concentration of A is doubled , the rate is increased by four times when concentrations of both A and B are increased by four times . The order of the reaction is _______.

Options

A. 3
B. 0
C. 1
D. 2

Answer

To determine the order of the reaction \( A + 2B \to C \), we need to analyze how the reaction rate changes with varying concentrations of the reactants \( A \) and \( B \).

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

### Step 2: Analyze the first condition
According to the problem, when the concentration of \( A \) is doubled, the reaction rate is doubled. This can be expressed mathematically as:
\[ R_2 = 2R_1 \]
Substituting into the rate law:
\[ k [2A]^x [B]^y = 2(k [A]^x [B]^y) \]
This simplifies to:
\[ k (2^x [A]^x [B]^y) = 2k [A]^x [B]^y \]
Dividing both sides by \( k [A]^x [B]^y \):
\[ 2^x = 2 \]
From this, we can conclude:
\[ x = 1 \]

### Step 3: Analyze the second condition
Next, we consider the second condition where the concentrations of both \( A \) and \( B \) are increased by four times, resulting in a fourfold increase in the reaction rate:
\[ R_3 = 4R_1 \]
Substituting into the rate law:
\[ k [4A]^x [4B]^y = 4(k [A]^x [B]^y) \]
This simplifies to:
\[ k (4^x [A]^x)(4^y [B]^y) = 4k [A]^x [B]^y \]
This can be rewritten as:
\[ k (4^{x+y} [A]^x [B]^y) = 4k [A]^x [B]^y \]
Dividing both sides by \( k [A]^x [B]^y \):
\[ 4^{x+y} = 4 \]
From this, we can conclude:
\[ x + y = 1 \]

### Step 4: Solve for \( y \)
We already found that \( x = 1 \). Substituting this into the equation \( x + y = 1 \):
\[ 1 + y = 1 \]
Thus, we find:
\[ y = 0 \]

### Step 5: Determine the overall order of the reaction
The overall order of the reaction is given by the sum of the orders with respect to each reactant:
\[ \text{Order} = x + y = 1 + 0 = 1 \]

### Conclusion
The order of the reaction is \( \boxed{1} \).

---