Answer :
The number of vacancies per cubic meter at 1000°C for the given metal is approximately \(4.47 \times 10^{21}\) vacancies/m³.
To calculate the number of vacancies per cubic meter at 1000°C for the given metal, we can use the formula:
\(N_v = N_A \cdot e^{\frac{-Q_v}{k \cdot T}}\)
where:
- \(N_v\) is the number of vacancies per cubic meter
- \(N_A\) is Avogadro's number (6.022 x 10^23 vacancies/mol)
- \(Q_v\) is the energy for vacancy formation per atom (1.22 eV/atom)
- \(k\) is Boltzmann's constant (8.617 x 10^-5 eV/K)
- \(T\) is the temperature in Kelvin (1000°C = 1273 K)
First, let's convert the given values to the correct units:
- The atomic weight is given as 37.4 g/mol. We can convert this to kg/mol by dividing by 1000, which gives us 0.0374 kg/mol.
- The density is given as 6.25 g/cm³. We can convert this to kg/m³ by multiplying by 1000, which gives us 6250 kg/m³.
Now, we can substitute the values into the formula and calculate \(N_v\):
\(N_v = (6.022 \times 10^{23} \, \text{vacancies/mol}) \times e^{\frac{-1.22 \, \text{eV/atom}}{8.617 \times 10^{-5} \, \text{eV/K} \times 1273 \, \text{K}}} = 4.47 \times 10^{21} \, \text{vacancies/m³}\)
Therefore, the number of vacancies per cubic meter at 1000°C for the given metal is approximately \(4.47 \times 10^{21}\) vacancies/m³.
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