For the reaction : `N_(2)(g)+3H_(2)(g)rarr2NH_(3)(g)` . If rate of appearance of `NH_(3)` is `6.8xx10^(-3)` gm/min, then rate of disappearance of `H_(2)(g)` at the same condition will be :

Options

A. `6.8xx10^(-3)` gm/min
B. `1.02xx10^(-2)` gm/min
C. `16xx10^(-4)` gm/min
D. `1.2xx10^(-3)` gm/min

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Reaction
The reaction given is:
\[ N_2(g) + 3H_2(g) \rightarrow 2NH_3(g) \]

### Step 2: Identify the Rate of Appearance of NH3
We are given the rate of appearance of \( NH_3 \):
\[ \text{Rate of appearance of } NH_3 = 6.8 \times 10^{-3} \text{ gm/min} \]

### Step 3: Convert the Rate of Appearance of NH3 to Moles
To relate the rate of appearance of \( NH_3 \) to the rate of disappearance of \( H_2 \), we need to convert grams to moles using the molar mass of \( NH_3 \).
- Molar mass of \( NH_3 \) = 14 (for N) + 1*3 (for H) = 17 g/mol

Now, convert the rate:
\[ \text{Rate of } NH_3 \text{ in moles} = \frac{6.8 \times 10^{-3} \text{ gm/min}}{17 \text{ g/mol}} = 0.4 \times 10^{-3} \text{ mol/min} \]

### Step 4: Relate the Rates Using Stoichiometry
From the balanced reaction, we can see that:
- For every 2 moles of \( NH_3 \) produced, 3 moles of \( H_2 \) are consumed.

Thus, the relationship between the rate of disappearance of \( H_2 \) and the rate of appearance of \( NH_3 \) can be expressed as:
\[ -\frac{d[H_2]}{dt} = \frac{3}{2} \frac{d[NH_3]}{dt} \]

### Step 5: Substitute the Rate of Appearance of NH3
Now, substituting the rate of appearance of \( NH_3 \):
\[ -\frac{d[H_2]}{dt} = \frac{3}{2} \times 0.4 \times 10^{-3} \text{ mol/min} = 0.6 \times 10^{-3} \text{ mol/min} \]

### Step 6: Convert the Rate of Disappearance of H2 to Grams
Now, we need to convert the rate of disappearance of \( H_2 \) back to grams. The molar mass of \( H_2 \) is 2 g/mol.
\[ \text{Rate of } H_2 \text{ in grams} = 0.6 \times 10^{-3} \text{ mol/min} \times 2 \text{ g/mol} = 1.2 \times 10^{-3} \text{ gm/min} \]

### Conclusion
The rate of disappearance of \( H_2(g) \) at the same conditions is:
\[ \boxed{1.2 \times 10^{-3} \text{ gm/min}} \]

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