For the chemical reaction
`N_(2)(g)+3H_(2)(g)hArr2NH_(3)(g)`
The correct option is:
(a) `3(d[H_(2)])/(dt)=2(d[NH_(3)])/(dt)`
(b) `-(1)/(3)(d[H_(2)])/(dt)=-(1)/(2)(d[NH_(3)])/(dt)`
(c) `-(d[N_(2)])/(dt)=2(d[NH_(3)])/(dt)`
(d) `-(d[N_(2)])/(dt)=(1)/(2)(d[NH_(3)])/(dt)`

Options

A. `3(d[H_(2)])/(dt)=2(d[NH_(3)])/(dt)`
B. `-(1)/(3)(d[H_(2)])/(dt)=-(1)/(2)(d[NH_(3)])/(dt)`
C. `-(d[N_(2)])/(dt)=2(d[NH_(3)])/(dt)`
D. `-(d[N_(2)])/(dt)=(1)/(2)(d[NH_(3)])/(dt)`

Explanation

To solve the problem, we need to analyze the given chemical reaction:

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

From this reaction, we can derive the rate expressions for the reactants and products.

### Step 1: Write the rate expressions

For a general reaction of the form:

\[ aA + bB \rightleftharpoons cC + dD \]

The rate of reaction can be expressed in terms of the change in concentration of the reactants and products:

- For reactant A:
\[ \text{Rate} = -\frac{1}{a} \frac{d[A]}{dt} \]
- For reactant B:
\[ \text{Rate} = -\frac{1}{b} \frac{d[B]}{dt} \]
- For product C:
\[ \text{Rate} = \frac{1}{c} \frac{d[C]}{dt} \]
- For product D:
\[ \text{Rate} = \frac{1}{d} \frac{d[D]}{dt} \]

### Step 2: Apply it to the given reaction

For the reaction \( N_2 + 3H_2 \rightleftharpoons 2NH_3 \):

- The rate in terms of \( N_2 \) is:
\[ \text{Rate} = -\frac{d[N_2]}{dt} \]
- The rate in terms of \( H_2 \) is:
\[ \text{Rate} = -\frac{1}{3} \frac{d[H_2]}{dt} \]
- The rate in terms of \( NH_3 \) is:
\[ \text{Rate} = \frac{1}{2} \frac{d[NH_3]}{dt} \]

### Step 3: Set the rates equal to each other

From the above expressions, we can equate the rates:

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

### Step 4: Rearranging the equations

1. From \( -\frac{d[N_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \):
\[ -\frac{d[N_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \implies -d[N_2] = \frac{1}{2} d[NH_3] \]

2. From \( -\frac{1}{3} \frac{d[H_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \):
\[ -\frac{d[H_2]}{dt} = \frac{3}{2} \frac{d[NH_3]}{dt} \]

### Step 5: Evaluate the options

Now we can evaluate the options provided:

(a) \( 3 \frac{d[H_2]}{dt} = 2 \frac{d[NH_3]}{dt} \) - **Incorrect**
(b) \( -\frac{1}{3} \frac{d[H_2]}{dt} = -\frac{1}{2} \frac{d[NH_3]}{dt} \) - **Incorrect**
(c) \( -\frac{d[N_2]}{dt} = 2 \frac{d[NH_3]}{dt} \) - **Incorrect**
(d) \( -\frac{d[N_2]}{dt} = \frac{1}{2} \frac{d[NH_3]}{dt} \) - **Correct**

Thus, the correct option is (d).