
In this part, we are asked to determine the number of free electrons donated by each In atom in the crystal. In other words, we are asked to determine the valency, which is the total number of conduction electrons per cm^{3} divided by the total number of atoms per cm^{3}. Given are the electrical resistivity = 8.37 x 10^{8} m (or 8.37 x 10^{6} cm) and the electron drift mobility = 6 cm^{2}V^{1}s^{1}. The reciprocal of resistivity gives the conductivity: 1/resistivity = conductivity () = 1/(8.37 x 10^{6}) = 11.947 x 10^{4} siemens per centimeter (or mho per centimeter).
We know that = en, where is the electrical conductivity and is the drift mobility of the electrons.
Plugging the values in the above equation gives the electron concentration (n) = 1.244 x 10^{23} per cm^{3}.
Now we need to determine the total number of Indium atoms per cm^{3}.
7.31 g of Indium has a volume of 1 cm^{3}.
114.82 g of Indium has 6.023 x 10^{23} atoms.
Therefore, 7.31 g of Indium has (7.31 x 6.023 x 10^{23})/114.82 = 3.835 x 10^{22} atoms
It means 1 cm^{3} of Indium has 3.835 x 10^{22} atoms
Thus, the valency of Indium is = 1.244 x 10^{23}/3.835 x 10^{22} = 3.24 (i.e., approximately 3)
Indium is placed in Group IIIB in the periodic table along with boron, aluminum, and gallium, which means its valency is 3. The above calculations attest the position of Indium in the periodic table.

The mean free path is the product of the mean speed of electrons (given to be 1.74 x 10^{8} cm/s) and the mean free time (i.e, the relaxation time ). The equation that relates the drift mobility and the mean free time is = e/m, where m is the mass of the electron and e the electron charge. Plugging the known values gives the mean free time = 3.413 x 10^{15} s. Kindly note that here we need to keep the unit of mobility in SI units, i.e., m^{2}V^{1}s^{1} and not cm^{2}V^{1}s^{1} because we are using the mass of electron and the electron charge in SI units.
Now we can determine the value of the mean free path = mean free time x mean speed = 3.413 x 10^{15} s x 1.74 x 10^{8} cm/s = 5.939 x 10^{7} cm or 5.939 nm.

To determine the thermal conductivity of Indium using its electrical conductivity, we need to use the WiedemannFranzLorenz law. The WiedemannFranzLorenz law states that the ratio of the thermal conductivity () and the electrical conductivity () of a metal is propotional to the temperature.
i.e., / is propotional to T
Or, / = LT, where L is the propotionality constant (referred to as the Lorenz number) given by the equation L = . The value of the Lorenz number is 2.44 x 10^{8} WK^{2}.
Plugging the values gives = 86.87 W (mK)^{1}. The experimental value is 81.6 W (mK)^{1}. Thus, we can say that the theoretical value is within 6% of the experimental value.


