Question

There are two identical vessels filled with equal amounts of ice. The vessels are of different metals. If the ice melts in the two vessels in 20 and 35 minutes respectively. Find the ratio of the coefficients of thermal conductivity of the two metals.
A. 4:7
B. 7:4
C. 16:49
D. 49:16

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 heat flow the formula is,
\[\dfrac{Q}{t} = KA\dfrac{{\Delta T}}{L}\]
Where, A is a cross-sectional area of metal rod, \[\Delta T\] is the temperature difference between two ends of the metal rod, L is the length of the metal rod and K is the thermal conductivity.

Complete step by step solution:
Consider two identical vessels that are filled with equal amounts of ice and the vessels are of different metals. If the ice melts in the two vessels in 20 minutes and 35minutes. We need to find the ratio of the coefficients of thermal conductivity of the two metals. The heat flow is,
\[\dfrac{Q}{t} = KA\dfrac{{\Delta T}}{L}\]
For the first vessel, the heat flow is,
\[\dfrac{Q}{{{t_1}}} = {K_1}A\dfrac{{\Delta T}}{L}\]
\[\Rightarrow Q = \dfrac{{{K_1}A\Delta T{t_1}}}{L}\]……. (1)

For the second vessel, the heat flow is,
\[Q = \dfrac{{{K_2}A\Delta T{t_2}}}{L}\]…… (2)
Equate the two equations (1) and (2) we obtain,
\[{K_1}{t_1} = {K_2}{t_2}\]
\[\Rightarrow \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{{t_2}}}{{{t_1}}}\]
Here, \[{t_1} = 20\,min\] and \[{t_2} = 35\,min\]

Substitute the value in the above equation we get,
\[\dfrac{{{K_1}}}{{{K_2}}} = \dfrac{{35}}{{20}}\]
\[\therefore \dfrac{{{K_1}}}{{{K_2}}} = \dfrac{7}{4}\]
Therefore, the ratio of the coefficients of thermal conductivity of the two metals is 7:4

Hence, option B is the correct answer.

Note:Here in the given problem it is important to remember the equation for the heat flow and using the formula of heat flow we are going to find the ratios of thermal conductivity of the two given metals.