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.
840
thanks
CHEAT!
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They deserve to get kicked off the course
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To solve this problem, we need to use the concept of vacancy formation energy and the relationship between temperature and the number of vacancies. Here's how you can approach it:
1. Start by finding the ratio of vacancies to the total number of atoms at the given temperature (777°C):
- The given temperature has 1 vacancy for every 22,200 atoms. So the vacancy-to-atom ratio is 1/22,200.
2. We need to find the temperature at which there will be one vacancy for every 11,100 atoms. Let's denote this temperature as T2.
3. Using the given information, we can set up the following equation:
- At T2, there will be 1 vacancy for every 11,100 atoms, so the vacancy-to-atom ratio is 1/11,100.
4. Now, we can use the Boltzmann relationship, which relates the vacancy formation energy to temperature and vacancy concentration:
- The equation is given as: E = k * T * ln(Cv / Co), where E is the vacancy formation energy, k is the Boltzmann constant, T is the temperature in Kelvin, Cv is the vacancy concentration, and Co is the lattice atom concentration.
5. Rearrange the equation to solve for T:
- T = E / (k * ln(Cv / Co))
6. Plug in the given values into the equation:
- E = 1.11 eV (given in the problem)
- k = 8.617333262145 x 10^-5 eV/K (Boltzmann constant)
- Cv/Co = (1/11,100) / (1/22,200) = 2
7. Now, substitute the values into the equation and solve for T:
- T = (1.11 eV) / (8.617333262145 x 10^-5 eV/K * ln(2))
8. Calculate the natural logarithm of 2:
- ln(2) ≈ 0.6931
9. Plug this value into the equation:
- T ≈ (1.11 eV) / (8.617333262145 x 10^-5 eV/K * 0.6931)
10. Calculate the value of T in Kelvin.
11. To convert the temperature from Kelvin to Celsius, subtract 273.15 Celsius from the Kelvin temperature.
That's how you can find the temperature at which there will be one vacancy for every 11,100 atoms in the given metal.