the energy of vacancy formation in palladium is 1.5 eV.at 888 C there is one vacancy formation for every million (10^6) atoms sites. what temperature would be achieved a vacancy fraction of one for every thousand (10^3)atoms sites?
To answer this question, we need to understand the relationship between temperature and vacancy formation in a crystal lattice. The probability of vacancy formation increases with temperature, so we can use the concept of a temperature-dependent equilibrium constant to solve the problem.
The equilibrium constant, K, for vacancy formation can be written as follows:
K = exp(-ΔE_vacancy / kT)
Where:
- ΔE_vacancy is the energy of vacancy formation in electron volts (eV), which is given as 1.5 eV in the problem.
- k is the Boltzmann constant (8.617333262145 × 10^-5 eV/K).
- T is the temperature in Kelvin.
Since we are given the vacancy formation for a million (10^6) atom sites at 888 °C, we have:
K = 1 / 10^6
Now, let's calculate the temperature at which the vacancy fraction is one for every thousand (10^3) atom sites.
First, substitute the given values into the equation for K:
1 / 10^6 = exp(-1.5 / (8.617333262145 × 10^-5 × T))
Next, solve for T by taking the natural logarithm of both sides:
ln(1 / 10^6) = -1.5 / (8.617333262145 × 10^-5 × T)
Now, rearrange the equation to solve for T:
T = -1.5 / (8.617333262145 × 10^-5 × ln(1 / 10^6))
Finally, calculate the value of T:
T ≈ 1304.75 K
Therefore, a vacancy fraction of one for every thousand (10^3) atom sites would be achieved at a temperature of approximately 1304.75 Kelvin (K).