If the atom`(_100)Fm^(257)` follows the Bohr model the radius of `_(100)Fm^(257)` is `n` time the Bohr radius , then find `n` .

Options

A. 4
B. `1//4`
C. 100
D. 200

Explanation

To find the value of \( n \) for the atom \( _{100}Fm^{257} \) following the Bohr model, we can use the formula for the radius of an electron in a hydrogen-like atom:

\[ R_m = \frac{R_0 m^2}{Z} \]

where:
- \( R_m \) is the radius of the orbit,
- \( R_0 \) is the Bohr radius (approximately \( 5.29 \times 10^{-11} \) m),
- \( m \) is the principal quantum number (orbit number),
- \( Z \) is the atomic number.

### Step 1: Identify the values
For the atom \( _{100}Fm^{257} \):
- The atomic number \( Z = 100 \).
- The outermost electron is in the fifth orbit, so \( m = 5 \).

### Step 2: Substitute the values into the formula
Now we can substitute these values into the formula:

\[ R_m = \frac{R_0 (5^2)}{100} \]

### Step 3: Calculate \( R_m \)
Calculating \( 5^2 \):

\[ 5^2 = 25 \]

Now substituting this back into the equation:

\[ R_m = \frac{R_0 \cdot 25}{100} \]

### Step 4: Simplify the expression
Now simplify the expression:

\[ R_m = \frac{25}{100} R_0 = \frac{1}{4} R_0 \]

### Step 5: Relate \( R_m \) to \( n \)
From the equation \( R_m = n R_0 \), we can equate:

\[ n R_0 = \frac{1}{4} R_0 \]

### Step 6: Solve for \( n \)
Dividing both sides by \( R_0 \):

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

Thus, the value of \( n \) is:

\[ \boxed{\frac{1}{4}} \]