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 answer this question, we need to use the concept of the Arrhenius equation, which relates the formation energy of vacancies to the temperature.
The Arrhenius equation is given by:
k = A * exp(-Ea / (RT))
Where:
- k is the rate constant,
- A is the pre-exponential factor,
- Ea is the activation energy,
- R is the gas constant,
- T is the temperature.
In this case, we can assume that the rate of formation of vacancies (k) is equal to the rate of annihilation of vacancies, which means that the rate of creation and rate of annihilation are equal.
We can set up the following equation:
k1 * N1 = k2 * N2
Where:
- k1 and k2 are the rate constants at temperatures T1 and T2, respectively,
- N1 and N2 are the number of vacancies per atom at temperatures T1 and T2, respectively.
Given that at T1 = 777oC, N1 = 1 vacancy / 22,200 atoms, and at T2 (to be determined) we want N2 = 1 vacancy / 11,100 atoms.
Now, let's express the rate constants in terms of the formation energies using the Arrhenius equation:
k1 = A * exp(-Ea1 / (R * T1))
k2 = A * exp(-Ea2 / (R * T2))
We can cancel out the pre-exponential factor (A) since it's the same for both rate constants.
Now, we can rewrite the equation with the equations for k1 and k2:
exp(-Ea1 / (R * T1)) / T1 = exp(-Ea2 / (R * T2)) / T2
Next, let's substitute the given values for Ea1 (1.11 eV) and T1 (777oC) in terms of Kelvin (K):
Ea1 = 1.11 eV * (1.6 x 10^-19 J/eV) = 1.776 x 10^-19 J
T1 = 777 + 273 = 1050 K
Now we can rewrite the equation:
exp(-(1.776 x 10^-19 J) / (R * 1050 K)) / 1050 = exp(-Ea2 / (R * T2)) / T2
Finally, we need to find Ea2 and T2. Rearranging the equation, we have:
exp(-Ea2 / (R * T2)) = (exp(-(1.776 x 10^-19 J) / (R * 1050 K)) / 1050) * T2
Since both sides of the equation are equal to exp(-Ea2 / (R * T2)), we can equate them:
exp(-Ea2 / (R * T2)) = (exp(-(1.776 x 10^-19 J) / (R * 1050 K)) / 1050) * T2
Taking the natural logarithm (ln) of both sides:
(-Ea2 / (R * T2)) = ln((exp(-(1.776 x 10^-19 J) / (R * 1050 K)) / 1050) * T2)
Now, we can solve for T2 by rearranging the equation and isolating T2:
T2 = (-Ea2 / (R * ln((exp(-(1.776 x 10^-19 J) / (R * 1050 K)) / 1050))))
Now, substitute the given values for Ea2 (1.11 eV), R (8.314 J/(mol·K)), and solving for T2:
T2 = (-1.11 eV * (1.6 x 10^-19 J/eV)) / (8.314 J/(mol·K) * ln((exp(-(1.776 x 10^-19 J) / (8.314 J/(mol·K) * 1050 K)) / 1050))
Calculating the value of T2 will give you the temperature at which there will be one vacancy for every 11,100 atoms.