Question

The coefficient of thermal conductivity of copper, mercury and glass are respectively ${{\text{K}}_{\text{c}}}\text{,}{{\text{K}}_{\text{m}}}\text{and }{{\text{K}}_{g}}$ such that ${{\text{K}}_{\text{c}}}\prec {{\text{K}}_{\text{m}}}\prec {{\text{K}}_{\text{g}}}$ ​. If the same quantity of heat is to flow per sec per unit area of each and corresponding temperature gradients are ${{\text{X}}_{\text{c}}}{,}{{\text{X}}_{\text{m}}}\text{ and }{{\text{X}}_{\text{g}}}$ then.

$\begin{align}
  & \text{A}\text{. }{{\text{X}}_{\text{c}}}\text{=}{{\text{X}}_{\text{m}}}\text{=}{{\text{X}}_{\text{g}}} \\
 & \text{B}\text{. }{{\text{X}}_{\text{c}}}\succ {{\text{X}}_{\text{m}}}\succ {{\text{X}}_{\text{g}}} \\
 & \text{C}\text{. }{{\text{X}}_{\text{c}}}\prec {{\text{X}}_{\text{m}}}\prec {{\text{X}}_{\text{g}}} \\
 & \text{D}\text{. }{{\text{X}}_{\text{m}}}\prec {{\text{X}}_{\text{c}}}\prec {{\text{X}}_{\text{g}}}
\end{align}$

Answer 1

Hint: We will use the basic law of conduction that is Fourier’s law. This type of question can be solved by observing the relation in the coefficient of thermal conductivity to the temperature gradient and its variation respectively. we can simply observe the relationship and reach the correct solution.

Formula used:
\[Q=-KA\left( \dfrac{dT}{dx} \right)\]

Complete answer:
We know from the basic law of conduction that is Fourier’s law we know that

\[Q=-KA\left( \dfrac{dT}{dx} \right)\]
Where \[\left( \dfrac{dT}{dx} \right)\] = Temperature gradient
So from above, we can conclude that K is inversely proportional to the temperature gradient.
as the temperature gradient increases the coefficient of thermal conductivity decreases and vice versa.
Hence we know that:

\[\begin{align}
& {{\text{K}}_{\text{metal}}}\succ {{\text{K}}_{\text{liquid}}}\succ {{\text{K}}_{\text{gas}}} \\ & \text{ }\!\!~\!\!\text{ }{{\text{X}}_{\text{metal}}}\prec {{\text{X}}_{\text{liquid}}}\prec {{\text{X}}_{\text{gas}}}\text{ } \\
 & \text{or} \\
 & {{\text{X}}_{\text{copper}}}\prec {{\text{X}}_{\text{mercury}}}\prec {{\text{X}}_{\text{glass}}} \\
\end{align}\]

Ceramic materials have higher conductivity as compared to gas .

So, the correct answer is “Option C”.

Additional Information:
The Fourier equation holds true for all matter solid, liquid, or gas. The vector expression indicates that heat flow is normal to an isotherm and is within the direction of decreasing temperature. Newton’s law of cooling and Ohm’s law are also a type of Fourier’s law.

Note:
Fourier’s law states that the negative gradient of temperature and therefore the time rate of warmth transfer is proportional to the world at right angles of that gradient through which the heat flows. Fourier’s law is the other name of the law of warmth conduction. Discrete and electrical analog of Fourier’s law can also be defined as Newton’s law of cooling and Ohm’s law. The derivation of Fourier’s law was explained with the assistance of an experiment which explained the speed of warmth transfer through a plane layer is proportional to the temperature gradient across the layer and heat transfer area.