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Na is a monovalent metal (BCC) with a density of 0.9712 g cm-3. Its atomic mass is 22.99 g mol-1. The drift mobility of electrons in Na is 53 cm2(V-s)-1.
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Solution
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22.99 g = 6.023 x 1023 atoms
so, in 0.9712 g of Na we have = (6.023 x 1023 x 0.9712)/22.99 = 2.544 x 1022 atoms.
It means 1 cm3 has 2.544 x 1022 atoms.
Since, each Na atom contributes one electron, the total number of electrons in 1 cm3 (n) = 2.544 x 1022 electrons.
Given that the particles's mean separation d = 1/n1/3 (it should be noted that it is an approximation)
Plugging the value of n gives d = 3.4 x 10-8 cm or 0.34 nm.
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The crystal structure of sodium is BCC (body centered cubic), which has 2 sodium atoms per unit cell (one at the center and one shared by eight corners). The figure below shows the (110) plane of a sodium unit cell.
If a is the lattice parameter (size of the unit cell) and R is the radius of the atom, then the relation between them is:
The density of sodium (0.9712 g cm-3) can be used to determine the lattice parameter (a) since we know that it has 2 atoms per unit cell. Density of sodium = (number of atoms per unit cell x weight of each atom)/volume of the unit cell (a3). Weight of each sodium atom = atomic weight of sodium divided by the Avogadro's number = 22.99/6.023 x 1023 = 3.817 x 10-23 g. Plugging the values gives the lattice parameter (a) = 4.2836 x 10-8 cm or 0.4284 nm.
Plugging the value of the lattice parameter (a) in the equation shown above gives R = 1.8548 x 10-8 or 0.1855 nm.
If the electron spends most of its time between two neighbouring Na+ ions, the electron is most likely to be at a distance R between the sodium ions. Thus, the mean separation between an electron (e-) and an Na+ ion is 0.1855 nm.
The approximate Coulombic energy (CE) is given by the equation:
(equation 1)
Solving the above equation gives CE = - 1.242 x 10-18 J or - 7.7628 eV.
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According to the kinetic molecular theory of matter, the average kinetic energy (KEthermal) for gas atoms is
. At 298 K, KEthermal = 6.1686 x 10-21 or 0.03855 eV, which is very low compared to the Coulombic interaction energy (-7.7628 eV). It simply means that the Coulombic interaction is very strong and that the electrons in sodium are very tighly bound and not free. Hence, the kinetic molecular theory is not applicable to conduction electrons in sodium
If KE = |CE| = mv2/2 (equation 2), or
. Solving this equation gives v = 1.652 x 106 m/s.
The mean kinetic energy is comparable to the mean Coulombic interaction energy. It can be explained using the virial theorem (virial means a function relating to a system of forces and their points of application), which relates the time average of the total kinetic energy with that of the total Coulombic energy (potential energy). The virial theorem does not depend on the notion of temperature. Mathematically, the virial theorem can be written as:
(equation 3)
A simple explanation to the virial theorem can be found here. Experimentally, the conductivity of most non-magnetic metals is found to be "linearly" dependent on temperature, whereas according to the kinetic molecular theory of matter the conductivity depends on square root of the absolute temperature (mv2/2 = 3KT/2). The linear dependence of conductivity on temperature can be explained only when the mean speed of the electrons is constant, which can be done only by taking equations 1, 2, and 3.
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The electrical conductivity of sodium can be calculated using the formula
, where n = 2.544 x 1022 electrons per cm3 and the electron drift mobility is 53 cm2(V-s)-1. Solving this equation gives the electrical conductivity to be 2.157 x 107 (
-m)-1, which is within 3% of the experimental value.
