Question

Two vessels of different materials are similar in size in every respect. The same quantity of ice filled in them gets melted in 20 minutes and 40 minutes respectively. Find the ratio of thermal conductivities of the materials.
A. 5:6
B. 6:5
C. 3:1
D. 2:1

Answer 1

Hint:In order to solve this problem we need to understand the thermal conductivity. The rate at which heat is transferred by conduction through a unit cross-section area of a material is known as thermal conductivity.

Formula Used:
To find the thermal conductivity the formula is,
\[\dfrac{Q}{t} = KA\dfrac{{\Delta T}}{L}\]
Where, A is a cross-sectional area, \[\Delta T\] is the temperature difference between two ends and L is the length of the cylinder.

Complete step by step solution:
The two vessels of different materials are similar in size in every respect means the area and length are the same. We need to find the ratio of thermal conductivity. It says the same quantity of ice gets melted, that is here Q and \[\Delta T\] remains the same. That is,
\[K \propto \dfrac{1}{t}\]
\[\Rightarrow \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{t_2}}}{{{t_1}}}\]

Substitute the value of \[{t_1} = 20\min \] and \[{t_2} = 40\min \]we get,
\[\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{40}}{{20}}\]
\[\therefore \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{2}{1}\]
That is, \[{K_1}:{K_2} = 2:1\]
Therefore, the ratio of thermal conductivities of the materials is 2:1.

Hence, option D is the correct answer.

Note:The reciprocal of thermal conductivity is known as thermal resistivity. The materials which have a high thermal conductivity are used in heat sinks whereas materials with low values are used as thermal insulators. In order to measure thermal conductivities of materials there exist several methods, which are broadly classified into two types of techniques one is transient and other is steady-state techniques.