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.