For `N` atoms in a crystal with `N_(i)` interstitial position in its structure. If there are `n` Frenekel defects in the crystal, then `n =` ……. .

Answer

To find the number of Frenkel defects \( n \) in a crystal with \( N \) atoms and \( N_i \) interstitial positions, we can use the relationship derived from the principles of solid-state chemistry. Here’s a step-by-step solution:

### Step 1: Understand the Frenkel Defect
A Frenkel defect occurs when an atom leaves its normal lattice position and occupies an interstitial site. This means that for every Frenkel defect, one atom is missing from its original position and one atom occupies an interstitial position.

### Step 2: Set Up the Relationship
Given:
- \( N \): Total number of atoms in the crystal
- \( N_i \): Number of interstitial positions available
- \( n \): Number of Frenkel defects

The relationship for the number of Frenkel defects can be expressed as:
\[ n = \frac{N}{N_i^{1/2}} \cdot e^{-\frac{E_1}{2kT}} \]
where:
- \( E_1 \) is the energy required to create a Frenkel defect,
- \( k \) is the Boltzmann constant,
- \( T \) is the absolute temperature.

### Step 3: Substitute Values (if any)
If specific values for \( N \), \( N_i \), \( E_1 \), \( k \), and \( T \) are provided, substitute them into the equation to calculate \( n \).

For example, if \( N = 10 \) and \( N_i = 4 \), and assuming \( E_1 = 0 \) (for simplicity):
\[ n = \frac{10}{4^{1/2}} \cdot e^{-\frac{0}{2kT}} = \frac{10}{2} \cdot 1 = 5 \]

### Step 4: Conclusion
Thus, the number of Frenkel defects \( n \) can be calculated using the formula provided, and it depends on the total number of atoms, the number of interstitial positions, and the temperature.