A formation energy of 1.11 eV is required to create a vacancy in a particular metal. At 777oC there is one vacancy for every 22,200 atoms. At what temperature will there be one vacancy for every 11,100 atoms?
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To solve this problem, we will use the equation for the concentration of vacancies in a metal:
n = N * exp(-Ef / (k * T))
where:
- n is the concentration of vacancies (number of vacancies per unit volume)
- N is the total number of atoms in the metal
- Ef is the formation energy of a vacancy (in electronvolts)
- k is the Boltzmann constant (8.617333262145 x 10^-5 eV/K)
- T is the temperature in Kelvin
First, let's calculate the concentration of vacancies at 777°C. We need to convert the temperature to Kelvin by adding 273 to the Celsius value:
T1 = 777 + 273 = 1050 K
Given that there is one vacancy for every 22,200 atoms, we can calculate the concentration of vacancies (n1):
n1 = 1 / 22,200
Now, let's solve for the formation energy (Ef) at T1:
1 / 22,200 = N * exp(-Ef / (k * 1050))
Next, let's calculate the concentration of vacancies at the desired condition where we want one vacancy for every 11,100 atoms:
n2 = 1 / 11,100
We can now solve for the temperature (T2) using the same equation with the formation energy (Ef) as the unknown:
1 / 11,100 = N * exp(-Ef / (k * T2))
Now, we have two equations:
1 / 22,200 = N * exp(-Ef / (k * 1050)) ---(1)
1 / 11,100 = N * exp(-Ef / (k * T2)) ---(2)
Dividing equation (1) by equation (2):
(1 / 22,200) / (1 / 11,100) = (N * exp(-Ef / (k * 1050))) / (N * exp(-Ef / (k * T2)))
2 = exp(-Ef / (k * 1050)) / exp(-Ef / (k * T2))
Since the bases of the exponents are the same, the exponents must be equal:
-Ef / (k * 1050) = -Ef / (k * T2)
Dividing both sides by -Ef and simplifying:
1 / (k * 1050) = 1 / (k * T2)
k * T2 = k * 1050
T2 = 1050 K
Therefore, the temperature at which there will be one vacancy for every 11,100 atoms is 1050 K.