Question

Compute the concentration (number density, count per volume) of vacancies in copper at room temperature if the lattice parameter of FCC copper is 0.3615 nm at room temperature (25oC, 298 K),

and the activation energy to form a single vacancy is 0.9 eV. Use 8.617x10-5 eV/(atom-K), exactly, as Boltzmann's Constant.

Note: You could look up the density and atomic weight of copper to compute the intermediate value of atom concentration you need for this problem.

But here I give you information that allows you to compute it even faster than that approach (and the matching algorithm is based on this calculation).

Be sure to use values exactly as given and do not round until the final answer.

Answer Format: XEX where EX stands for *10x Example: So for an answer of 158123.126 you would type: 2E5 since NO decimal is prescribed in the prefix.

Units: vacancies/cm3

Answer 1

The atomic weight and density (at 25 ° C) for copper are 63.5 g/mol or 0.0635 kg/mol and 8,960 kg/m³

First lets find the value of N, number of atomic sites per cubic meter for copper, from its atomic weight A_cu, its density, and Avogadro’s number N_A

N = N_A x ρ / A_cu = 6.023 x 1023 x 8,960 kg/m³ / 0.0635 kg/mol = 8.4985953 x 10²⁸ atoms / m³

Temperature dependency can be given as :

Arrhenius equation

N_v = N e^-EA/KT

N_v = 8.4985953 x 10²⁸ atoms / m³ exp-(0.9ev/atom /8.617x10-5 eV/(atom-K) x 298 K

N_v = 5.1044844 x 10¹³ /m³

vacancies per cm³ = 5.1044844 x 10⁷/cm³ = 5E7