Around `20%` surface sites have adsorbed `N_(2)`. On heating `N_(2)` gas evolved form sites and were collected at 0.001 atm and 298 K in a container of volume `2.46cm^(3)` the density of surface sites is `6.023xx10^(14)cm^(-2)` and surface area is `1000cm^(2)` find out the number of surface sites occupied per molecule of `N_(2)`.

Answer

To solve the problem step by step, we will follow the instructions given in the video transcript and include calculations along the way.

### Step 1: Calculate the number of moles of \(N_2\) evolved

We can use the ideal gas law, which is given by the equation:

\[ PV = nRT \]

Where:
- \(P\) = pressure (in atm)
- \(V\) = volume (in liters)
- \(n\) = number of moles
- \(R\) = universal gas constant (0.0821 L·atm/(K·mol))
- \(T\) = temperature (in Kelvin)

Given:
- \(P = 0.001 \, \text{atm}\)
- \(V = 2.46 \, \text{cm}^3 = 2.46 \times 10^{-3} \, \text{L}\)
- \(T = 298 \, \text{K}\)

Substituting the values into the equation:

\[ n = \frac{PV}{RT} = \frac{0.001 \times 2.46 \times 10^{-3}}{0.0821 \times 298} \]

Calculating this gives:

\[ n = \frac{2.46 \times 10^{-6}}{24.4758} \approx 1.0 \times 10^{-7} \, \text{moles of } N_2 \]

### Step 2: Calculate the number of molecules of \(N_2\)

To find the number of molecules, we use Avogadro's number (\(6.022 \times 10^{23} \, \text{molecules/mol}\)):

\[ \text{Number of molecules} = n \times N_A = 1.0 \times 10^{-7} \times 6.022 \times 10^{23} \]

Calculating this gives:

\[ \text{Number of molecules} \approx 6.022 \times 10^{16} \, \text{molecules of } N_2 \]

### Step 3: Calculate the total number of surface sites

The total number of surface sites can be calculated using the density of surface sites and the surface area:

Given:
- Density of surface sites = \(6.023 \times 10^{14} \, \text{cm}^{-2}\)
- Surface area = \(1000 \, \text{cm}^2\)

Total surface sites:

\[ \text{Total surface sites} = \text{Density} \times \text{Area} = 6.023 \times 10^{14} \times 1000 \]

Calculating this gives:

\[ \text{Total surface sites} \approx 6.023 \times 10^{17} \, \text{sites} \]

### Step 4: Calculate the number of occupied surface sites

Since around 20% of the surface sites have adsorbed \(N_2\):

\[ \text{Occupied surface sites} = 0.20 \times \text{Total surface sites} = 0.20 \times 6.023 \times 10^{17} \]

Calculating this gives:

\[ \text{Occupied surface sites} \approx 1.2046 \times 10^{17} \, \text{sites} \]

### Step 5: Calculate the number of surface sites occupied per molecule of \(N_2\)

Finally, we can find the number of surface sites occupied per molecule of \(N_2\):

\[ \text{Sites occupied per molecule} = \frac{\text{Occupied surface sites}}{\text{Number of molecules}} = \frac{1.2046 \times 10^{17}}{6.022 \times 10^{16}} \]

Calculating this gives:

\[ \text{Sites occupied per molecule} \approx 2 \]

### Final Answer

The number of surface sites occupied per molecule of \(N_2\) is approximately **2**.

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