Electronic Properties of Materials: Conduction - Problems and Solutions


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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.

  1. Consider the collection of conduction electrons in the solid. If each Na atom donates one electron to the electron sea, estimate the mean separation between the electrons. (Note: If n is the concentration of particles, then the particles's mean separation d = 1/n1/3.)
  2. Estimate the mean separation between an electron (e-) and a metal ion (Na+), assuming that most of the time the electron prefers to be between two neighbouring Na+ ions. What is the approximate Coulombic interaction energy (in eV) between an electron and an Na+ ion?
  3. How does this electron/metal-ion interaction energy compare with the average thermal energy per particle, according to the kinetic molecular theory of matter? Do you expect the kinetic molecular theory to be applicable to the conduction electrons in Na? If the mean electron/metal-ion interaction energy is of the same order of magnitude as the mean KE of the electrons, what is the mean speed of electrons in Na? Why should the mean kinetic energy be comparable to the mean electron/metal-ion interaction energy?
  4. Calculate the electrical conductivity of Na and compare this with the experimental value of 2.1 x 107 (ohm-m)-1 and comment on the difference.
      





Solution  
  1. 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.

  2. 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:
    R = {a{sqrt{3}}}/4


    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:
    CE = {{-}e^2}/(4{pi}{epsilon_0}{R^2})    (equation 1)

    Solving the above equation gives CE = - 1.242 x 10-18 J or - 7.7628 eV.

  3. According to the kinetic molecular theory of matter, the average kinetic energy (KEthermal) for gas atoms is {3{kappa}T}/2. 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 v = sqrt{{2 CE}/m}. 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:
    KE = {{-1}/2}PE    (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.

  4. The electrical conductivity of sodium can be calculated using the formula sigma = en{mu_d}, 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 (Omega-m)-1, which is within 3% of the experimental value.