Knowing that the energy associated with the formation of 1 mole of vacancies in copper (Cu) is 104 kJ, determine the vacancy density (nv/cm3) in Cu at 1284 K. Assume the pre-exponential factor is one.
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To determine the vacancy density in copper (Cu) at 1284 K, we can use the concept of thermal equilibrium and the equation for vacancy concentration. The equation is given by:
nv = N * exp(-Qv / (k*T))
Where:
nv is the vacancy density (in cm^3)
N is the number of lattice sites per unit volume (in cm^3)
Qv is the energy associated with the formation of 1 mole of vacancies (in J)
k is the Boltzmann constant (1.38 x 10^-23 J/K)
T is the temperature in Kelvin (K)
First, we need to convert the energy given from kJ to J:
Qv = 104 kJ * (1000 J / 1 kJ) = 104000 J
Next, we need to find the number of lattice sites per unit volume (N). For a simple cubic lattice (as in copper), the number of lattice sites per unit volume is given by:
N = (atomic weight / density) * (6.022 x 10^23 / Avogadro's number)
The atomic weight of copper (Cu) is approximately 63.55 g/mol, and the density of copper is approximately 8.96 g/cm^3.
N = (63.55 g/mol / 8.96 g/cm^3) * (6.022 x 10^23 / 6.022 x 10^23) = 7.08 x 10^22 cm^-3
Now, we can substitute the values into the vacancy concentration equation:
nv = (7.08 x 10^22 cm^-3) * exp(-104000 J / (1.38 x 10^-23 J/K * 1284 K))
Simplifying the equation gives us the vacancy density (nv) in cm^3.
Please note that the pre-exponential factor is given as one, so we don't need to consider it in this calculation.