Number Density Of Free Electrons


Free electrons in a metal are assumed to collide occasionally with the lattice ions, but otherwise the details of the interactions between electrons and ions are neglected, as is the Coulomb repulsion between electrons. Left undisturbed, the electron gas settles down to a state of thermal equilibrium, characterised by a particular temperature and number density. The temperature of the electron gas is the same as the temperature of the metal,. The number density of free electrons (the number of free electrons per unit volume, n) depends on the choice of metal.

In Drude’s model, the valency of a metal is simply the number of free electrons released per atom. For example, the valency of aluminium is three, so each aluminium atom releases three of its 27 electrons into the electron gas.


Equations:


To calculate the number density of free electrons (n):

$n = \frac{z N_{A}}{V_{A}}$

,where
z is the valency,
NA is the Avogadro’s constant,
VA is the molar volume.


To calculate the molar volume (VA):

$V_{A} = \frac{M_{r} \times 10^{-3}}{\rho}$

, where
Mr is the relative atomic mass (the 10-3 is to convert Mr from grams to kg)
ρ is the density.


Substituting equation for VA into the first equation, we get:

$n = \frac{z \rho N_{A}}{M_{r} \times 10^{-3}}$

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