10 % sites of catalyst bed have adsorbed by `H_(2)` on Heating `H_(2)` gas is evloved from sites and collected at 0.03 atm and 300 K in a small vessel of `2.46 cm^(3)`.
no. of sites available is `5.4 xx 10^(16) ` per `cm^(2)` and surface area is `1000 cm ^(2)` . find out the no. of suface sites occupied per molecule of `H_(2)`.

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To solve the problem step by step, we will calculate the number of surface sites occupied per molecule of \( H_2 \) based on the given data.

### Step 1: Calculate the moles of \( H_2 \) evolved
We can use the ideal gas law to find the number of moles of \( H_2 \) gas collected.

\[ PV = nRT \]

Where:
- \( P = 0.03 \, \text{atm} \)
- \( V = 2.46 \, \text{cm}^3 = 2.46 \times 10^{-3} \, \text{L} \)
- \( R = 0.0821 \, \text{L atm K}^{-1} \text{mol}^{-1} \)
- \( T = 300 \, \text{K} \)

Rearranging the equation to solve for \( n \):

\[ n = \frac{PV}{RT} \]

Substituting the values:

\[ n = \frac{0.03 \times 2.46 \times 10^{-3}}{0.0821 \times 300} \]

Calculating this gives:

\[ n \approx 3 \times 10^{-6} \, \text{moles of } H_2 \]

### Step 2: Calculate the number of molecules of \( H_2 \)
To find the number of molecules, we use Avogadro's number (\( 6.022 \times 10^{23} \, \text{molecules/mol} \)):

\[ \text{Number of molecules} = n \times N_A = 3 \times 10^{-6} \times 6.022 \times 10^{23} \]

Calculating this gives:

\[ \text{Number of molecules} \approx 1.81 \times 10^{18} \, \text{molecules} \]

### Step 3: Calculate the total number of surface sites available
The total number of sites available can be calculated using the given number of sites per cm² and the total surface area:

\[ \text{Total sites} = \text{Number of sites per cm}^2 \times \text{Surface area} \]

Given:
- Number of sites per cm² = \( 5.4 \times 10^{16} \)
- Surface area = \( 1000 \, \text{cm}^2 \)

Calculating this gives:

\[ \text{Total sites} = 5.4 \times 10^{16} \times 1000 = 5.4 \times 10^{19} \, \text{sites} \]

### Step 4: Calculate the number of occupied sites
Since 10% of the sites are occupied by \( H_2 \):

\[ \text{Occupied sites} = 0.10 \times \text{Total sites} = 0.10 \times 5.4 \times 10^{19} = 5.4 \times 10^{18} \, \text{sites} \]

### Step 5: Calculate the number of surface sites occupied per molecule of \( H_2 \)
To find the number of surface sites occupied per molecule of \( H_2 \):

\[ \text{Sites occupied per molecule} = \frac{\text{Occupied sites}}{\text{Number of molecules}} = \frac{5.4 \times 10^{18}}{1.81 \times 10^{18}} \]

Calculating this gives:

\[ \text{Sites occupied per molecule} \approx 3 \]

### Final Answer:
The number of surface sites occupied per molecule of \( H_2 \) is **3**.

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