We can see here that the Fermi energy of sodium is approximately 6.27 x 10⁻²⁰ Joules.
a) To calculate the density of charge carriers, we need to know the molar mass of sodium (Na) and the Avogadro's number (approximately 6.022 x 10²³ mol⁻¹).
Given that the molar mass of sodium is 0.023 kg/mol, we can convert this to grams by multiplying by 1000:
0.023 kg/mol × 1000 g/kg = 23 g/mol
Now, using Avogadro's number, we can determine the number of sodium atoms in one mole:
6.022 x 10²³ mol⁻¹ × 1 mol = 6.022 x 10²³
Since sodium is monovalent, each sodium atom carries one positive charge. Therefore, the number of charge carriers is also 6.022 x 10²³.
To calculate the density of charge carriers, we divide the number of charge carriers by the volume:
Density of charge carriers = Number of charge carriers / Volume
The density of sodium is given as 971 kg/m³. We can convert this to grams per cubic meter by multiplying by 1000:
971 kg/m³ * 1000 g/kg = 971000 g/m³
Now, we can substitute the values into the equation:
Density of charge carriers = (6.022 x 10²³) / (971000 g/m³)
Calculating this expression gives us:
Density of charge carriers ≈ 6.2 x 10¹⁸ charge carriers/m³
Therefore, the density of charge carriers in sodium is approximately 6.2 x 10¹⁸ charge carriers per cubic meter.
b) The Fermi energy is a measure of the highest energy level occupied by electrons in a solid at absolute zero temperature. It depends on the density of charge carriers and can be calculated using the formula:
Fermi energy (Ef) = (h² / (8π²m)) × (3π²n)^(2/3)
Where h is Planck's constant (approximately 6.626 x 10⁻³⁴ J·s), m is the mass of a charge carrier, and n is the density of charge carriers.
In the case of sodium, the charge carriers are sodium ions with a single positive charge (Na⁺).
To calculate the Fermi energy, we need to know the mass of a sodium ion. The molar mass of sodium (Na) is 0.023 kg/mol, so the mass of one sodium ion is:
0.023 kg/mol / Avogadro's number = 0.023 kg / 6.022 x 10²³ = 3.81 x 10⁻²⁵ kg
Now, we can substitute the values into the Fermi energy equation:
Fermi energy (Ef) = (6.626 x 10⁻³⁴ J·s)² / (8π² * 3.81 x 10⁻²⁵ kg) × (3π² × 6.2 x 10¹⁸ charge carriers/m³)^(2/3)
Calculating this expression gives us:
Fermi energy (Ef) ≈ 6.27 x 10⁻²⁰ J
Therefore, the Fermi energy of sodium is approximately 6.27 x 10⁻²⁰ Joules.