Finely divided catalyst has greater surface area and has greater catalytic activity than the compact solid. If a total surface area of 6291456 `cm^(2)` is required for adsorption in a catalytic gaseous reaction, then how many splits should be made in a cube of exactly 1 cm in length to achieve required surface area?
[Given : One split of a cube gives eight cubes of same size]

Options

A. `60`
B. `80`
C. `20`
D. `22`

Explanation

To solve the problem, we need to determine how many splits are required to achieve a total surface area of 6291456 cm² from a cube with a length of 1 cm.

### Step-by-Step Solution:

1. **Understanding the Surface Area of a Cube**:
The surface area \( A \) of a cube with side length \( a \) is given by the formula:
\[ A = 6a^2 \]
For a cube with a length of 1 cm, the surface area is:
\[ A = 6 \times (1)^2 = 6 \, \text{cm}^2 \]

2. **Effect of Splitting the Cube**:
When we split a cube, each split divides the cube into 8 smaller cubes. After one split, the new length of each smaller cube becomes:
\[ \text{New length} = \frac{1}{2} \, \text{cm} \]
After \( n \) splits, the length of each smaller cube will be:
\[ \text{Length after } n \text{ splits} = \frac{1}{2^n} \, \text{cm} \]

3. **Calculating the Surface Area after n Splits**:
The surface area of one of the smaller cubes after \( n \) splits is:
\[ A_n = 6 \left(\frac{1}{2^n}\right)^2 = 6 \times \frac{1}{4^n} = \frac{6}{4^n} \]
Since there are \( 8^n \) smaller cubes after \( n \) splits, the total surface area \( A_{total} \) after \( n \) splits is:
\[ A_{total} = 8^n \times \frac{6}{4^n} = 6 \times \frac{8^n}{4^n} = 6 \times 2^{2n} = 6 \times 2^{n} \]

4. **Setting Up the Equation**:
We know that the total surface area required is 6291456 cm². Therefore, we set up the equation:
\[ 6 \times 2^n = 6291456 \]

5. **Solving for \( n \)**:
Dividing both sides by 6:
\[ 2^n = \frac{6291456}{6} = 1048576 \]
Now, we need to express 1048576 as a power of 2. We can calculate:
\[ 1048576 = 2^{20} \]
Therefore, we have:
\[ n = 20 \]

### Final Answer:
The number of splits required is **20**.