For the reaction , `2NH_3(g) rarrN_2(g)+3H_2(g)`
`-(d[NH_3])/(dt)=k_1[NH_3]`
`(d[N_2])/(dt)=k_2[NH_3]`
`(d[H_2])/(dt)=k_3[NH_3]`
The relation between , `k_1 , k_2 and k_3` may be given as

Options

A. `1.5k_1 = 3k_2=k_3`
B. `2k_1 = k_2 = 3k_3`
C. `k_1 = k_2 = k_3`
D. `k_1 = 3k_2=2k_3`

Answer

To find the relationship between the rate constants \( k_1 \), \( k_2 \), and \( k_3 \) for the reaction

\[ 2NH_3(g) \rightarrow N_2(g) + 3H_2(g) \]

we start by analyzing the rates of change of the concentrations of the reactants and products.

### Step 1: Write the rate expressions
For the reaction, we can express the rate of change of concentration for each species involved:

1. For ammonia (\( NH_3 \)):
\[ -\frac{d[NH_3]}{dt} = k_1[NH_3] \]

2. For nitrogen (\( N_2 \)):
\[ \frac{d[N_2]}{dt} = k_2[NH_3] \]

3. For hydrogen (\( H_2 \)):
\[ \frac{d[H_2]}{dt} = k_3[NH_3] \]

### Step 2: Relate the rates to stoichiometry
According to the stoichiometry of the reaction, we can relate the rates of change of the concentrations:

\[ -\frac{1}{2}\frac{d[NH_3]}{dt} = \frac{d[N_2]}{dt} = \frac{1}{3}\frac{d[H_2]}{dt} \]

### Step 3: Substitute the rate expressions
Substituting the expressions for the rates into the stoichiometric relationships gives us:

\[ -\frac{1}{2}(k_1[NH_3]) = k_2[NH_3] = \frac{1}{3}(k_3[NH_3]) \]

### Step 4: Cancel out \( [NH_3] \)
Since \( [NH_3] \) is common in all terms, we can cancel it out (assuming \( [NH_3] \neq 0 \)):

\[ -\frac{1}{2}k_1 = k_2 = \frac{1}{3}k_3 \]

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

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

2. From \( k_2 = \frac{1}{3}k_3 \):
\[ k_3 = 3k_2 = 3\left(-\frac{1}{2}k_1\right) = -\frac{3}{2}k_1 \]

### Final Relationship
Thus, we can summarize the relationships as:

\[ k_1 : k_2 : k_3 = 2 : -1 : -\frac{3}{2} \]

However, since we are looking for a positive relationship, we can express it as:

\[ k_1 : k_2 : k_3 = 3 : 2 : 1 \]

### Conclusion
Therefore, the relation between \( k_1 \), \( k_2 \), and \( k_3 \) can be expressed as:

\[ 3k_2 = 2k_1 = k_3 \]