The correct equation for calculating the free-electron density in sodium (Na) involves understanding its body-centered cubic (BCC) structure and the number of free electrons per unit volume. The calculation begins with the formula n = 2 / (4.23 x 10^-10 m)^3, resulting in a free-electron density of approximately 2.6 x 10^28 m^-3. Each sodium atom contributes one free electron due to its monovalent nature, and the total number of atoms in the unit cell is two. This approach clarifies the distinction between atomic density and free-electron density.
PREREQUISITESStudents in materials science, physicists, and chemists who are studying the properties of metals, particularly those focusing on electron behavior in crystalline structures.
chaoseverlasting said:Its given that the structure is a BCC type unit cell. The total number of atoms present in a BCC are 2. You are given the dimensions of this unit cell. Since its a cubic cell, you can find the volume enclosed by it.
Each cell contains two atoms, how many cells do you need to get Na (Avagardo Number) atoms?
Once you have that, you know the atomic weight of Na sodium atoms. This weight is equal to the weight of the protons+neutrons+electrons. You know the weight of one electron. You know how many free electrons are present in one sodium atom (valency). Hence you know how many electrons are present in Na sodium atoms.
You also know the total volume enclosed by Na atoms (which you calculated in the beginning). So now you have the total number of free electrons present in a given volume, and can now find the free electron density.