An activationenergy of 2.0 eV is required to form a vacancy in a metal. At 800°C there is one vacancy for every 10-4 atoms. At what temperature will there be one vacancy for every 1000 atoms?
Temperature (in degrees Kelvin):
unanswered
1201
To find the temperature at which there will be one vacancy for every 1000 atoms, we can use the equation:
Nv/N = exp(-Qv/kT)
Where:
Nv/N is the ratio of the number of vacancies to the total number of atoms.
exp is the exponential function.
Qv is the activation energy to form a vacancy.
k is the Boltzmann constant.
T is the temperature in Kelvin.
We are given that at 800°C (which is 1073 Kelvin), there is one vacancy for every 10^-4 atoms. So, the current value of Nv/N is 10^-4.
Now, we need to solve for the temperature T when Nv/N is equal to 10^-3 (which corresponds to one vacancy for every 1000 atoms).
Let's rearrange the equation to solve for T:
ln(Nv/N) = -Qv/kT
T = -Qv / (k * ln(Nv/N))
Substituting the given values:
Qv = 2.0 eV
k = 8.617333262145 x 10^-5 eV/K (Boltzmann constant)
T = -2.0 eV / (8.617333262145 x 10^-5 eV/K * ln(10^-3))
Calculating this expression, we find:
T ≈ 4051 Kelvin
Therefore, the temperature at which there will be one vacancy for every 1000 atoms is approximately 4051 degrees Kelvin.