Copper has a Fermi energy of 7.02 eV. What is its electron number
density?
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Using Ef=h^2/8m(3n/pi)^2/3
I get an answer of 2.246*10^7 but this is incorrect. I know the answer is 1.8*10^29 but am struggling to see where I went wrong. Can anyone help?
To find the electron number density (n) using the Fermi energy (Ef), we can use the formula you mentioned, Ef = (ħ^2 / 8m)(3π^2n)^(2/3), where ħ is the reduced Planck's constant (h/2π), and m is the mass of an electron.
Given that the Fermi energy (Ef) for copper is 7.02 eV, we need to convert it to joules to maintain consistency in units. 1 eV is equal to 1.6022 x 10^-19 joules, so the Ef for copper is 7.02 x 1.6022 x 10^-19 = 1.1227944 x 10^-18 J.
Now, rearranging the equation, we have:
3π^2n = (8mEf / ħ^2)^(3/2)
n = (8mEf / ħ^2)^(3/2) / (3π^2)
The mass of an electron (m) is approximately 9.109 x 10^-31 kg, and the reduced Planck's constant (ħ) is approximately 1.055 x 10^-34 J s.
Substituting these values into the equation, we have:
n = (8 * 9.109 x 10^-31 kg * 1.1227944 x 10^-18 J)^(3/2) / (3 * (3.14159)^2)
Evaluating the expression, we get:
n ≈ 8.49 x 10^28 m^(-3)
Therefore, the correct electron number density for copper is approximately 8.49 x 10^28 electrons per cubic meter.