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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This is a question from the second midterm of 3.091x by edx. Stop cheating, it's silly and pointless.
stop spamming. hes not the only one who wants answers
To find the temperature at which there will be one vacancy for every 11,100 atoms, we need to use the concept of the equilibrium concentration of vacancies.
The equilibrium concentration of vacancies, denoted as "n_v", can be calculated using the equation:
n_v = exp(-E_v / (k * T))
Where:
- n_v is the equilibrium concentration of vacancies.
- E_v is the formation energy of a vacancy (1.11 eV in this case).
- k is the Boltzmann constant (8.617333262145 x 10^-5 eV/K).
- T is the temperature in Kelvin.
Given that there is one vacancy for every 22,200 atoms at 777°C, we can calculate the equilibrium concentration of vacancies at this temperature using the formula above.
Let's calculate the equilibrium concentration of vacancies at 777°C:
T1 = 777 + 273.15 (Converting temperature from Celsius to Kelvin)
= 1050.15 K
n_v1 = exp(-1.11 eV / (8.617333262145 x 10^-5 eV/K * 1050.15 K))
≈ exp(-11621.02)
Now, we need to find the temperature at which there will be one vacancy for every 11,100 atoms. Let's call this temperature T2.
n_v2 = 1 vacancy / 11100 atoms
Using the equation, we can solve for T2:
1 / 11100 = exp(-1.11 eV / (8.617333262145 x 10^-5 eV/K * T2))
Taking the natural logarithm (ln) on both sides:
ln(1 / 11100) = -1.11 eV / (8.617333262145 x 10^-5 eV/K * T2)
Rearranging the equation:
-1.11 eV / (8.617333262145 x 10^-5 eV/K * T2) = ln(1 / 11100)
Calculating the value on the right side:
ln(1 / 11100) ≈ -9.31297469
Substituting the values:
-1.11 / (8.617333262145 x 10^-5 * T2) = -9.31297469
Now, we can solve for T2:
T2 = -1.11 / (8.617333262145 x 10^-5 * -9.31297469)
T2 ≈ 1551.24 K
Therefore, the temperature at which there will be one vacancy for every 11,100 atoms is approximately 1551.24 Kelvin.