If 772 mL of `SO_(2)` gas at STP is adsorbed on `2` g of charcoal at an equilibrium pressure of `16` atmospheres and the value of the constant 'k' in the Freundlich equation is `0.48.` the value of the constant 'n' will be

Explanation

To solve the problem, we will use the Freundlich adsorption isotherm equation, which is given by:

\[ \frac{x}{m} = k \cdot p^{\frac{1}{n}} \]

Where:
- \( x \) = mass of the adsorbate (SO₂) in grams
- \( m \) = mass of the adsorbent (charcoal) in grams
- \( k \) = Freundlich constant
- \( p \) = equilibrium pressure in atmospheres
- \( n \) = constant that we need to find

### Step 1: Calculate the mass of SO₂ adsorbed

Given:
- Volume of SO₂ = 772 mL
- Molar mass of SO₂ = 64 g/mol
- Volume of 1 mole of gas at STP = 2240 mL

Using the formula to find the mass of SO₂:

\[ \text{Mass of SO₂} = \left( \frac{\text{Molar mass of SO₂}}{\text{Volume of 1 mole at STP}} \right) \times \text{Volume of SO₂} \]

\[ \text{Mass of SO₂} = \left( \frac{64 \, \text{g/mol}}{2240 \, \text{mL}} \right) \times 772 \, \text{mL} = \frac{64 \times 772}{2240} \]

Calculating this gives:

\[ \text{Mass of SO₂} = 22.08 \, \text{g/mol} \approx 1.92 \, \text{g} \]

### Step 2: Calculate \( \frac{x}{m} \)

Now, we can calculate \( \frac{x}{m} \):

\[ \frac{x}{m} = \frac{1.92 \, \text{g}}{2 \, \text{g}} = 0.96 \]

### Step 3: Substitute values into the Freundlich equation

Now, we can substitute the values into the Freundlich equation:

\[ 0.96 = 0.48 \cdot (16)^{\frac{1}{n}} \]

### Step 4: Isolate \( (16)^{\frac{1}{n}} \)

To isolate \( (16)^{\frac{1}{n}} \):

\[ (16)^{\frac{1}{n}} = \frac{0.96}{0.48} = 2 \]

### Step 5: Solve for \( n \)

Now, we need to solve for \( n \):

\[ 16^{\frac{1}{n}} = 2 \]

Taking logarithm on both sides:

\[ \frac{1}{n} \cdot \log(16) = \log(2) \]

Since \( \log(16) = 4 \cdot \log(2) \):

\[ \frac{4 \cdot \log(2)}{n} = \log(2) \]

Dividing both sides by \( \log(2) \):

\[ \frac{4}{n} = 1 \]

Thus, solving for \( n \):

\[ n = 4 \]

### Final Answer

The value of the constant \( n \) is \( 4 \).

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