The coefficient of thermal conductivity of copper, mercury and glass respectively are ${K_{C,}}{K_m}$ and ${K_g}$ such that ${K_C} > {K_m} > {K_g}$. If the same quantity of heat is flowing per second per unit area through each and corresponding temperature gradient are ${X_c},{X_m}$ and ${X_g}$. Then:
(A) ${X_c} = {X_m} = {X_g}$
(B) ${X_c} > {X_m} > {X_g}$
(C) ${X_c} < {X_m} < {X_g}$
(D) ${X_m} < {X_c} < {X_g}$

Hint We know that the law of conduction of heat is also known as the Fourier’s law. We know that the heat transfer is classified into various mechanisms, which are thermal conduction, thermal conduction, thermal radiation, and the transfer of energy by the phase changes. This is the concept which is used to solve this answer. We should consider the fact that engineers also consider the transfer of mass of differing chemical species, either cold, or hot, to reach the required amount of heat transfer.

Complete step by step answer
We should know that from the basis law of conduction which is Fourier’s law that:
$Q = - KA\left( {\dfrac{{dT}}{{dx}}} \right)$
Where the $\dfrac{{dT}}{{dx}}$is the temperature gradient
So, K is inversely proportional to the temperature gradient.
Hence, we can say that:
${K_{metal}} > {K_{liquid}} > {K_{gas}}$
Now we can write that:
${X_{metal}} < {X_{liquid}} < {X_{gas}}$or we can say,
${X_{copper}} < {X_{mercury}} < {X_{glass}}$
Hence the ceramics material has greater conductivity as compared to that of gas. So it can said that the same quantity of heat is flowing per second per unit area through each and corresponding temperature gradient are ${X_c},{X_m}$ and ${X_g}$, then ${X_c} < {X_m} < {X_g}$.

Hence the correct answer is option C.

Note The law of heat conduction is known as the explanation which states that the rate of transfer of heat through a material is proportional to the negative gradient in the temperature and to the area, at the right angles to that of the gradient, through which the heat will flow.
The basic idea of the law states that the heat flux vector will be proportional to the negative vector gradient of the temperature. The process of conduction is greater in the solids because of the network of the relatively close fixed spatial relationships between the atoms to help the transfer of the energy between the vibration.