Derive relation for thermal conductivity of a substance?

Difficulty: Hard

Thermal conductivity:
Let two opposite faces each of cross-sectional area A be heated to a temperature $T_{1}$. Heat Q flows along its length L to opposite face at temperature $T_{2}$ in t seconds.

The amount of heat that flows in unit time is called the heat flow rate.

Thus, Rate of flow of heat = $\frac{Q}{t}$                          (i)

It is observed that the rate at which heat flows through a solid object depends upon various factors.

CROSS-SECTIONAL AREA OF THE SOLID:
Larger cross-sectional area A of a solid contains a larger number of molecules and free electrons on each layer parallel to its cross-sectional area; hence, greater will be the flow of heat through the solid.

Thus, Rate of flow of heat $\frac{Q}{i}\propto A$                            (ii)

LENGTH OF THE SOLID:
The larger the length between the hot and cold ends of the solid, the more time it will take to conduct heat to the colder end, and the smaller will be the heat flow rate.

Thus, Rate of flow of heat $\frac{Q}{t}\propto \frac{1}{L}$                       (iii)

TEMPERATURE DIFFERENCE BETWEEN ENDS:

Greater is the temperature difference $T_{1} – T_{2}$ between hot and cold faces of the solid, greater will be the rate of flow of heat.

Thus, Rate of flow of heat $\frac{Q}{t} \propto ( T_{1} – T_{2} )$              (iv)

Combining the above factors I, ii, iii, and iv, we get

$\frac{Q}{t}\propto \frac{(〖A(T〗_{1} – T_{2}))}{L}$
Rate of flow of heat = $\frac{Q}{t}$ = $\frac{(〖kA(T〗_{1} – T_{2}))}{L}$

Here k is the proportionality constant called thermal conductivity of the solid.
k = $\frac{Q}{t}\times \frac{L}{(〖A(T〗_{1} – T_{2}))}$

Coefficient of thermal conductivity:
Thus, the thermal conductivity of a substance can be defined as:
The rate of flow of heat across the opposite faces of a meter cube of a substance maintained at a temperature difference of one kelvin is called the thermal conductivity of that substance.

Unit of thermal conductivity:
W $m^{-1} K^{-1}$ Or J $m ^{-1} K ^{-1} s^{-1}$