10% sites of catalyst bed have adsorbed `H_2`. On heating, `H_2` gas is evolved from sites and collected at 0.03 atm and 300K in a small vessel of `2.46 cm^3`. Number of sites available is `7.2×(10)^(16)` per cm^2 and surface area is 1000`cm^2`. The number of surface sites are occupied by one molecule of `H_2` is [Given: `N_A=6×(10)^(23)]`

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the number of moles of hydrogen gas evolved.

We can use the ideal gas law to find the number of moles of hydrogen gas (`n`):

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

Where:
- \( P = 0.03 \, \text{atm} \)
- \( V = 2.46 \, \text{cm}^3 = 2.46 \times 10^{-3} \, \text{L} \) (since \( 1 \, \text{cm}^3 = 1 \, \text{mL} = 0.001 \, \text{L} \))
- \( R = 0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \)
- \( T = 300 \, \text{K} \)

Substituting the values:

\[ n = \frac{0.03 \, \text{atm} \times 2.46 \times 10^{-3} \, \text{L}}{0.082 \, \text{L atm K}^{-1} \text{mol}^{-1} \times 300 \, \text{K}} \]

Calculating this gives:

\[ n = \frac{0.0000738}{24.6} \approx 3 \times 10^{-6} \, \text{mol} \]

### Step 2: Calculate the number of hydrogen molecules.

To find the number of molecules of hydrogen gas, we use Avogadro's number (\( N_A = 6 \times 10^{23} \, \text{molecules/mol} \)):

\[ \text{Number of molecules} = n \times N_A \]

Substituting the values:

\[ \text{Number of molecules} = 3 \times 10^{-6} \, \text{mol} \times 6 \times 10^{23} \, \text{molecules/mol} = 18 \times 10^{17} \, \text{molecules} \]

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

The total number of surface sites can be calculated using the given number of sites per cm² and the total surface area:

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

Substituting the values:

\[ \text{Total sites} = 7.2 \times 10^{16} \, \text{sites/cm}^2 \times 1000 \, \text{cm}^2 = 7.2 \times 10^{19} \, \text{sites} \]

### Step 4: Calculate the number of sites occupied by hydrogen.

Since 10% of the total sites are occupied by hydrogen:

\[ \text{Occupied sites} = 0.10 \times \text{Total sites} \]

Substituting the values:

\[ \text{Occupied sites} = 0.10 \times 7.2 \times 10^{19} = 7.2 \times 10^{18} \, \text{sites} \]

### Step 5: Calculate the number of surface sites occupied by one molecule of \( H_2 \).

To find the number of surface sites occupied by one molecule of hydrogen, we divide the total occupied sites by the number of hydrogen molecules:

\[ \text{Sites per molecule} = \frac{\text{Occupied sites}}{\text{Number of molecules}} \]

Substituting the values:

\[ \text{Sites per molecule} = \frac{7.2 \times 10^{18}}{18 \times 10^{17}} = 4 \]

### Final Answer:

The number of surface sites occupied by one molecule of \( H_2 \) is **4**.

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