The volume of nitrogen gas Um (measured at STP) required to cover a sample of silica get with a mono-molecular layer is `129 cm^(3) g^(-1)` of gel. Calculate the surface area per gram of the gel if each nitrogen molecule occupies `16.2 xx 10^(-20) m^(2).`

Explanation

To solve the problem, we need to calculate the surface area per gram of silica gel based on the given volume of nitrogen gas required to form a monolayer and the area occupied by each nitrogen molecule. Here’s a step-by-step solution:

### Step 1: Calculate the number of nitrogen molecules in 129 cm³ of nitrogen gas at STP

At Standard Temperature and Pressure (STP), 1 mole of any gas occupies 22,400 cm³. Therefore, we can calculate the number of moles of nitrogen gas in 129 cm³:

\[ \text{Number of moles of } N_2 = \frac{129 \, \text{cm}^3}{22400 \, \text{cm}^3/mol} \]

### Step 2: Convert moles to molecules

Using Avogadro's number (approximately \(6.022 \times 10^{23}\) molecules/mol), we can find the number of nitrogen molecules:

\[ \text{Number of molecules} = \text{Number of moles} \times 6.022 \times 10^{23} \, \text{molecules/mol} \]

### Step 3: Calculate the total surface area covered by the nitrogen molecules

Each nitrogen molecule occupies an area of \(16.2 \times 10^{-20} \, \text{m}^2\). Therefore, the total surface area covered by the nitrogen molecules can be calculated as:

\[ \text{Total Surface Area} = \text{Number of molecules} \times \text{Area occupied by each molecule} \]

### Step 4: Calculate the surface area per gram of silica gel

Since the volume of nitrogen gas is given per gram of silica gel, the total surface area calculated in Step 3 will be the surface area per gram of silica gel.

### Putting it all together:

1. **Calculate the number of moles of \(N_2\)**:
\[ \text{Number of moles of } N_2 = \frac{129}{22400} \approx 0.00576 \, \text{moles} \]

2. **Calculate the number of nitrogen molecules**:
\[ \text{Number of molecules} = 0.00576 \times 6.022 \times 10^{23} \approx 3.47 \times 10^{21} \, \text{molecules} \]

3. **Calculate the total surface area**:
\[ \text{Total Surface Area} = 3.47 \times 10^{21} \times 16.2 \times 10^{-20} \approx 562.74 \, \text{m}^2 \]

Thus, the surface area per gram of silica gel is approximately \(560 \, \text{m}^2\).

### Final Answer:
The surface area per gram of the silica gel is approximately **560 m²**.

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