What fraction (n/N) of the lattice sites are vacant at 298 K for a crystal in which the energy required to make a defect is 1 eV.`(1 eV = 1.602 xx 10^(-19) J)`

Options

A. `1.24 xx 10^(-17)`
B. `2.6 xx 10^(-6)`
C. `4.00 xx 10^(-12)`
D. `10.24 xx 10^(-13)`

Explanation

To find the fraction of vacant lattice sites (n/N) at 298 K for a crystal with a defect formation energy of 1 eV, we can use the following formula:

\[ \frac{n}{N} = e^{-\frac{E_v}{kT}} \]

Where:
- \( E_v \) is the energy required to create a vacancy (1 eV),
- \( k \) is the Boltzmann constant (\( 1.38 \times 10^{-23} \, \text{J/K} \)),
- \( T \) is the absolute temperature in Kelvin (298 K).

### Step 1: Convert the energy from eV to Joules
Given that \( 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \), we can convert the energy required to create a defect:

\[ E_v = 1 \, \text{eV} = 1.602 \times 10^{-19} \, \text{J} \]

### Step 2: Substitute values into the formula
Now, we substitute the values into the formula for the fraction of vacant sites:

\[ \frac{n}{N} = e^{-\frac{1.602 \times 10^{-19}}{(1.38 \times 10^{-23})(298)}} \]

### Step 3: Calculate the denominator
Calculate \( kT \):

\[ kT = (1.38 \times 10^{-23} \, \text{J/K}) \times (298 \, \text{K}) = 4.11 \times 10^{-21} \, \text{J} \]

### Step 4: Calculate the exponent
Now, calculate the exponent:

\[ -\frac{E_v}{kT} = -\frac{1.602 \times 10^{-19}}{4.11 \times 10^{-21}} \approx -39.0 \]

### Step 5: Calculate the fraction
Now, we can calculate \( \frac{n}{N} \):

\[ \frac{n}{N} = e^{-39.0} \]

Using a calculator or an exponential function, we find:

\[ \frac{n}{N} \approx 1.9 \times 10^{-17} \]

### Final Answer
Thus, the fraction of the lattice sites that are vacant at 298 K is approximately:

\[ \frac{n}{N} \approx 1.9 \times 10^{-17} \]