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}

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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:
$Q=-KA\left( \dfrac{dT}{dx} \right)$

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”.