For the chemical reaction A + B + C k `to` D, it was found that the rate of the reaction doubled when the concentration of B was doubled, that the rate of reaction doubled when the concentration of both A and B were doubled, and quadrupled when the concentration of both B and C were doubled. The order of the reaction is :

Options

A. 2
B. 3
C. 4
D. 6

Answer

To determine the order of the reaction A + B + C → D based on the given information, we can analyze the effects of changing the concentrations of the reactants on the rate of the reaction.

### Step-by-Step Solution:

1. **Understanding the Rate Law**:
The rate of a reaction can be expressed using a rate law of the form:
\[ \text{Rate} = k[A]^m[B]^n[C]^p \]
where \( k \) is the rate constant, and \( m \), \( n \), and \( p \) are the orders of the reaction with respect to reactants A, B, and C, respectively.

2. **Analyzing the Effect of Doubling [B]**:
It is given that the rate of the reaction doubles when the concentration of B is doubled:
\[ \text{If } [B] \text{ is doubled, } \text{Rate} \propto [B]^n \implies 2 \text{Rate} \propto (2[B])^n \]
This implies:
\[ 2 \text{Rate} = k[A]^m(2[B])^n \implies 2 = 2^n \implies n = 1 \]
Therefore, the reaction is first order with respect to B.

3. **Analyzing the Effect of Doubling [A] and [B]**:
Next, it is stated that the rate of the reaction doubles when both [A] and [B] are doubled:
\[ \text{If } [A] \text{ and } [B] \text{ are both doubled, } \text{Rate} \propto [A]^m[B]^n \implies 2 \text{Rate} \propto (2[A])^m(2[B])^n \]
This gives us:
\[ 2 \text{Rate} = k(2[A])^m(2[B])^n \implies 2 = 2^m \cdot 2^n \implies 2 = 2^{m+n} \]
Since we already found \( n = 1 \), we can substitute:
\[ 2 = 2^{m+1} \implies m + 1 = 1 \implies m = 0 \]
Thus, the reaction is zero order with respect to A.

4. **Analyzing the Effect of Doubling [B] and [C]**:
Finally, it is given that the rate quadruples when both [B] and [C] are doubled:
\[ \text{If } [B] \text{ and } [C] \text{ are both doubled, } \text{Rate} \propto [B]^n[C]^p \implies 4 \text{Rate} \propto (2[B])^n(2[C])^p \]
This leads to:
\[ 4 \text{Rate} = k(2[B])^n(2[C])^p \implies 4 = 2^n \cdot 2^p \implies 4 = 2^{n+p} \]
Since we know \( n = 1 \):
\[ 4 = 2^{1+p} \implies 1 + p = 2 \implies p = 1 \]
Therefore, the reaction is first order with respect to C.

5. **Calculating the Total Order of the Reaction**:
The total order of the reaction is the sum of the individual orders:
\[ \text{Total Order} = m + n + p = 0 + 1 + 1 = 2 \]

### Final Answer:
The order of the reaction is **2**.