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? Update: Express your answer in Celsius.
To solve this problem, we can use the equation for vacancy concentration:
C_v = N_v / N_total
where C_v is the concentration of vacancies, N_v is the number of vacancies, and N_total is the total number of atoms.
We have two sets of data:
At 777°C, there is one vacancy for every 22,200 atoms.
At an unknown temperature T, we want to find when there will be one vacancy for every 11,100 atoms.
Let's assume that the total number of atoms N_total remains constant.
From the equation for vacancy concentration, we can set up the following ratio:
C_v1 / C_v2 = N_v1 / N_v2
where C_v1 is the vacancy concentration at 777°C, C_v2 is the vacancy concentration at temperature T, N_v1 is the number of vacancies at 777°C, and N_v2 is the number of vacancies at temperature T.
At 777°C, C_v1 = 1 / 22,200
At temperature T, C_v2 = 1 / 11,100
We can now set up the ratio:
(1 / 22,200) / (1 / 11,100) = N_v1 / N_v2
Simplifying the ratio:
1 / 2 = N_v1 / N_v2
Cross-multiplying:
N_v1 = 2 * N_v2
We know that N_v1 is the number of vacancies at 777°C and N_v2 is the number of vacancies at temperature T. By multiplying N_v2 by 2, we can find N_v1.
Next, we are given that the formation energy to create a vacancy in this metal is 1.11 eV. Let's convert this energy to temperature units.
Using the equation:
E_f = k * T
where E_f is the formation energy, k is the Boltzmann constant (8.617333262145 x 10^-5 eV/K), and T is the temperature in Kelvin.
Substituting the given values:
1.11 eV = (8.617333262145 x 10^-5 eV/K) * T
Simplifying:
T = (1.11 eV) / (8.617333262145 x 10^-5 eV/K)
Computing this value gives us the temperature T in Kelvin.
Finally, we can convert the temperature from Kelvin to Celsius to get the answer in the desired unit.
That's how you can approach solving this problem step by step.