question_answer

Two vessels A and B of different materials but having identical shape, size and wall- thickness are filled with ice and kept at the same place. Ice melts at the rate of \[\mathbf{100 g mi}{{\mathbf{n}}^{\mathbf{-}}}^{\mathbf{1}}\] and \[\mathbf{150 g mi}{{\mathbf{n}}^{\mathbf{-}}}^{\mathbf{1}}\] in A and B respectively. Assuming that heat enters the vessels through the walls only, calculate the ratio of thermal conductivities of their materials.

Answer:

                Let \[{{m}_{1}}\] and \[{{m}_{2}}\] be the masses of ice melted in same time \[t\text{ }(=\text{1 min})\]in vessels A and B respectively. Then the amounts of heat flowed into the two vessels will be \[{{Q}_{1}}=\frac{{{K}_{1}}A({{T}_{1}}-{{T}_{2}})t}{x}={{m}_{1}}L\]                           ?(i) \[{{Q}_{2}}=\frac{{{K}_{2}}A({{T}_{1}}-{{T}_{2}})t}{x}={{m}_{2}}L\]                           ?(ii) where L is latent heat of ice. Dividing (i) by (ii), we get \[\frac{{{K}_{1}}}{{{K}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}=\frac{100g}{150g}=\frac{2}{3}=\mathbf{2:3}\]