Calculate the fraction of atom sites that are vacant for copper at its melting temperature of 1357 K. Assume an energy for vacancy formation of 0.90 eV/atom.
a) 0.0017
b) 0.0025
c) 0.0005
d) 0.0032
Vacancy Fraction Calculation for Copper at Melting Temperature
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To calculate the fraction of atom sites that are vacant in copper at its melting temperature, we can use the equation for the vacancy concentration:
[
C_v = N cdot e^{-E_v/(kT)}
]
where:
– ( C_v ) is the vacancy concentration,
– ( N ) is the total number of atomic sites (not needed for fraction calculation),
– ( E_v ) is the vacancy formation energy (in eV),
– ( k ) is Boltzmann’s constant (8.617 x (10^{-5}) eV/K),
– ( T ) is the temperature in Kelvin.
Given:
– ( E_v = 0.90 ) eV,
– ( T = 1357 ) K.
First, calculate ( kT ):
[
kT = (8.617 times 10^{-5} , text{eV/K}) times 1357 , K approx 0.116 , text{eV}
]
Now, plug in ( E_v ) and ( kT ):
[
C_v = e^{-0.90 , text{eV} / 0.116 , text{eV}} approx e^{-7.7586} approx 0.0005
]
So, the fraction of atom sites that are vacant for copper at its melting temperature is approximately **0