`underset((g))(CO)+underset((g))(2H_(2))to underset((g))(CH_(3)OH)`
In the above reaction Hydrogen disappears at the rate of 0.2 gm/sec. What is the rate fo appearance of Methano,at that moment?

Options

A. 0.1 gm/sec
B. 1.6 gm/sec
C. 3.2 gm/sec
D. 0.05 gm/sec

Explanation

To solve the problem, we need to analyze the given chemical reaction and relate the rates of disappearance of reactants to the rate of appearance of the product.

### Step-by-Step Solution:

1. **Write the balanced chemical equation:**
\[ \text{CO (g)} + 2\text{H}_2 (g) \rightarrow \text{CH}_3\text{OH (g)} \]

2. **Identify the rates of change:**
According to the stoichiometry of the reaction, the rates of disappearance of the reactants and the rate of appearance of the product can be expressed as:
\[ -\frac{d[\text{CO}]}{dt} = \frac{1}{1} \cdot \frac{d[\text{CH}_3\text{OH}]}{dt} \]
\[ -\frac{d[\text{H}_2]}{dt} = \frac{2}{1} \cdot \frac{d[\text{CH}_3\text{OH}]}{dt} \]

3. **Given data:**
The rate of disappearance of hydrogen (\( \text{H}_2 \)) is given as:
\[ -\frac{d[\text{H}_2]}{dt} = 0.2 \text{ gm/sec} \]

4. **Relate the rate of disappearance of hydrogen to the rate of appearance of methanol:**
From the stoichiometric relationship, we can express the rate of appearance of methanol (\( \frac{d[\text{CH}_3\text{OH}]}{dt} \)):
\[ -\frac{d[\text{H}_2]}{dt} = 2 \cdot \frac{d[\text{CH}_3\text{OH}]}{dt} \]
Substituting the given rate of hydrogen:
\[ 0.2 = 2 \cdot \frac{d[\text{CH}_3\text{OH}]}{dt} \]

5. **Solve for the rate of appearance of methanol:**
\[ \frac{d[\text{CH}_3\text{OH}]}{dt} = \frac{0.2}{2} = 0.1 \text{ gm/sec} \]

6. **Convert the rate of methanol to grams per second:**
Since the molecular weight of hydrogen (H₂) is 2 g/mol and that of methanol (CH₃OH) is 32 g/mol, we need to adjust the rate based on the molecular weights:
\[ \text{Rate of appearance of methanol} = 0.1 \text{ gm/sec} \times \frac{32}{2} = 0.1 \times 16 = 1.6 \text{ gm/sec} \]

### Final Answer:
The rate of appearance of methanol at that moment is:
\[ \text{1.6 gm/sec} \]