At 1 atm and 273 K the volume of nitrogen gas required to cover a sample of silica gel, assuming Langmuir monolayer adsorption , is found to be `1.30 cm^(3)g^(-1)` of the gel. The area occupied by a nitrogen molecule is `0.16nm^(2)`. Find out the no. of surface sites occupied per molecule of `N_(2)`.

Options

A. `5.568 m^(2)g^(-1)`
B. `3.34 m^(2) g^(-1)`
C. `1.6 m^(2) g^(-1)`
D. none of these

Answer

To solve the problem step by step, we need to find the number of surface sites occupied per molecule of nitrogen gas (N₂) on the silica gel surface.

### Step 1: Calculate the number of nitrogen molecules per gram in the monolayer.

Given:
- Volume of nitrogen gas required = 1.30 cm³/g
- Volume of 1 mole of gas at STP (Standard Temperature and Pressure) = 22400 cm³
- Avogadro's number (N_A) = 6.022 x 10²³ molecules/mol

Using the formula:
\[ \text{Number of molecules per gram} = \frac{\text{Volume of gas}}{\text{Volume of 1 mole of gas}} \times N_A \]

Substituting the values:
\[ \text{Number of molecules per gram} = \frac{1.30 \, \text{cm}^3/g}{22400 \, \text{cm}^3/mol} \times 6.022 \times 10^{23} \, \text{molecules/mol} \]

Calculating:
\[ \text{Number of molecules per gram} = \frac{1.30}{22400} \times 6.022 \times 10^{23} \approx 3.48 \times 10^{19} \, \text{molecules/g} \]

### Step 2: Calculate the total area occupied by these nitrogen molecules.

Given:
- Area occupied by one nitrogen molecule = 0.16 nm² = \(0.16 \times 10^{-18} \, \text{m}^2\)

The total area occupied by the nitrogen molecules per gram can be calculated as:
\[ \text{Total area} = \text{Number of molecules per gram} \times \text{Area per molecule} \]

Substituting the values:
\[ \text{Total area} = 3.48 \times 10^{19} \, \text{molecules/g} \times 0.16 \times 10^{-18} \, \text{m}^2 \]

Calculating:
\[ \text{Total area} = 3.48 \times 10^{19} \times 0.16 \times 10^{-18} \approx 5.568 \, \text{m}^2/g \]

### Step 3: Determine the number of surface sites occupied per molecule of N₂.

The number of surface sites occupied per molecule can be determined by considering the total area available for adsorption and the area occupied by a single nitrogen molecule.

Since we have already calculated the total area occupied by nitrogen molecules per gram, we can conclude that each molecule occupies one site.

Thus, the number of surface sites occupied per molecule of N₂ is 1.

### Final Answer:
The number of surface sites occupied per molecule of nitrogen (N₂) is **1**.