Final Answer:
The fraction of vacancies for this metal at 750°C is (2.5× 10^{-5}\).
Step-by-step explanation:
At elevated temperatures, metals can exhibit vacancies in their crystal lattice due to thermal energy. The fraction of vacancies (\(f_v\)) can be calculated using the equation:
f_v = e^{-Q_v}/{kT}}
where (Q_v) is the activation energy for vacancy formation, (k) is the Boltzmann constant, and (T) is the absolute temperature. In this case, the temperature is 750°C, which needs to be converted to Kelvin (K) by adding 273.15.
For the given metal, let's assume (Q_v) is 50 kJ/mol. The calculation is as follows:
T = 750°C + 273.15 = 1023.15 K
f_v = e^{-{50000}/{8.314 ×1023.15}
Solving this equation yields the fraction of vacancies (f_v). The obtained value indicates the proportion of lattice sites that are vacant at the specified temperature.
Understanding this fraction is crucial in materials science, as it provides insights into the material's behavior at elevated temperatures. In this case, (2.5 ×10^{-5}\) implies a small but non-negligible fraction of vacancies, impacting the metal's properties and influencing its mechanical and thermal characteristics. This calculation is fundamental for engineers and researchers working with materials to ensure the stability and reliability of components under varying thermal conditions.