For the reaction `2AtoB+3C`, if `-(d[A])/(dt)=k_(1)[A]^(2),-(d[B])/(dt)=k_(2)[A]^(2),-(d[C])/(dt)=k_(3)[A]^(2)` the correct reaction between `k_(1),k_(2)` and `k_(3)` is :

Options

A. `k_(1)=k_(2)=k_(3)`
B. `2k_(1)=k_(2)=3k_(2)`
C. `4k_(1)=k_(2)=3k_(2)`
D. `(k_(1))/(2)=k_(2)=(k_(3))/(3)`

Answer

To solve the problem, we need to analyze the given reaction and the rate equations provided. The reaction is:

\[ 2A \rightarrow B + 3C \]

The rate equations given are:

1. \(-\frac{d[A]}{dt} = k_1 [A]^2\)
2. \(-\frac{d[B]}{dt} = k_2 [A]^2\)
3. \(-\frac{d[C]}{dt} = k_3 [A]^2\)

### Step 1: Write the rate of reaction in terms of stoichiometry
From the stoichiometry of the reaction, we know that:

- For every 2 moles of A that react, 1 mole of B is produced and 3 moles of C are produced.

Thus, we can express the rates of change of concentration in terms of the rate of reaction \( r \):

\[ -\frac{1}{2} \frac{d[A]}{dt} = r \]
\[ \frac{d[B]}{dt} = r \]
\[ \frac{1}{3} \frac{d[C]}{dt} = r \]

### Step 2: Relate the rates to the rate constants
From the equations, we can express the rate of reaction \( r \) in terms of the rate constants and concentrations:

1. From \(-\frac{d[A]}{dt}\):
\[ r = -\frac{1}{2} \frac{d[A]}{dt} = \frac{k_1}{2} [A]^2 \]

2. From \(-\frac{d[B]}{dt}\):
\[ r = \frac{d[B]}{dt} = k_2 [A]^2 \]

3. From \(-\frac{d[C]}{dt}\):
\[ r = \frac{1}{3} \frac{d[C]}{dt} = \frac{k_3}{3} [A]^2 \]

### Step 3: Set the equations equal to each other
Since all expressions represent the same rate \( r \), we can set them equal to each other:

\[ \frac{k_1}{2} [A]^2 = k_2 [A]^2 = \frac{k_3}{3} [A]^2 \]

### Step 4: Cancel out \([A]^2\)
Assuming \([A]^2 \neq 0\), we can divide through by \([A]^2\):

\[ \frac{k_1}{2} = k_2 \]
\[ k_2 = \frac{k_3}{3} \]

### Step 5: Express \( k_1 \) in terms of \( k_2 \) and \( k_3 \)
From the equations, we can express \( k_1 \) in terms of \( k_2 \) and \( k_3 \):

1. From \( k_2 = \frac{k_1}{2} \):
\[ k_1 = 2k_2 \]

2. From \( k_3 = 3k_2 \):
\[ k_2 = \frac{k_3}{3} \]

### Step 6: Final relationships
Now we can summarize the relationships:

- \( k_1 = 2k_2 \)
- \( k_2 = k_2 \)
- \( k_3 = 3k_2 \)

Thus, the correct relationship between \( k_1, k_2, \) and \( k_3 \) is:

\[ k_1 = 2k_2 \quad \text{and} \quad k_3 = 3k_2 \]

### Conclusion
The correct answer is that \( k_1 = 2k_2 \) and \( k_3 = 3k_2 \).