Twenty precent of the surface sites of a catalyst is occupied by nitrogen molecules. The density of surface sites is `6.023xx10^(14)cm^(-2)` . The total sarface area is `1000cm^(2)` . The catalyst is is henced to `300K` and nitrogen is completely desorbed a pressure of `0.001` atm and volume of `2.46cm^(3)` . Calculate the number of sites occupied by niitrogen molecules.

Explanation

To solve the problem step-by-step, we will follow the outlined approach in the transcript and break it down into clear steps.

### Step 1: Calculate the Total Number of Surface Sites
The total number of surface sites can be calculated using the formula:
\[ \text{Total Surface Sites} = \text{Density of Surface Sites} \times \text{Total Surface Area} \]
Given:
- Density of surface sites = \(6.023 \times 10^{14} \, \text{cm}^{-2}\)
- Total surface area = \(1000 \, \text{cm}^2\)

Calculating:
\[ \text{Total Surface Sites} = 6.023 \times 10^{14} \, \text{cm}^{-2} \times 1000 \, \text{cm}^2 = 6.023 \times 10^{17} \, \text{sites} \]

### Step 2: Calculate the Number of Sites Occupied by Nitrogen Molecules
We know that 20% of the total surface sites are occupied by nitrogen molecules. Therefore, we can calculate the number of occupied sites as follows:
\[ \text{Occupied Sites} = 20\% \times \text{Total Surface Sites} = \frac{20}{100} \times 6.023 \times 10^{17} \]
Calculating:
\[ \text{Occupied Sites} = 0.20 \times 6.023 \times 10^{17} = 1.2046 \times 10^{17} \approx 1.204 \times 10^{16} \, \text{sites} \]

### Step 3: Calculate the Number of Moles of Nitrogen
We will use the Ideal Gas Law to find the number of moles of nitrogen desorbed:
\[ PV = nRT \implies n = \frac{PV}{RT} \]
Given:
- Pressure \(P = 0.001 \, \text{atm}\)
- Volume \(V = 2.46 \, \text{cm}^3 = 2.46 \times 10^{-3} \, \text{L}\) (since \(1 \, \text{cm}^3 = 1 \, \text{mL} = 0.001 \, \text{L}\))
- Gas constant \(R = 0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}\)
- Temperature \(T = 300 \, \text{K}\)

Calculating:
\[ n = \frac{(0.001 \, \text{atm}) \times (2.46 \times 10^{-3} \, \text{L})}{(0.0821 \, \text{L} \cdot \text{atm} \cdot \text{K}^{-1} \cdot \text{mol}^{-1}) \times (300 \, \text{K})} \]
Calculating:
\[ n \approx \frac{2.46 \times 10^{-6}}{24.63} \approx 9.98 \times 10^{-8} \, \text{mol} \]

### Step 4: Calculate the Total Number of Nitrogen Molecules
Using Avogadro's number (\(6.023 \times 10^{23} \, \text{molecules/mol}\)):
\[ \text{Total Molecules} = n \times N_A = 9.98 \times 10^{-8} \, \text{mol} \times 6.023 \times 10^{23} \, \text{molecules/mol} \]
Calculating:
\[ \text{Total Molecules} \approx 6.02 \times 10^{16} \, \text{molecules} \]

### Step 5: Calculate the Number of Sites Occupied by One Nitrogen Molecule
To find the number of sites occupied by one nitrogen molecule, we divide the total occupied sites by the total number of nitrogen molecules:
\[ \text{Sites per Molecule} = \frac{\text{Occupied Sites}}{\text{Total Molecules}} = \frac{1.204 \times 10^{16}}{6.02 \times 10^{16}} \]
Calculating:
\[ \text{Sites per Molecule} \approx 2 \]

### Final Answer
Each nitrogen molecule occupies approximately **2 sites** on the catalyst.

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