question_answer

Two vessels of different materials are identical in size and wall-thickness. They are filled with equal quantities of ice at\[\mathbf{0}{}^\circ \text{ }\mathbf{C}\]. If the ice melts completely in 10 and 25 minutes respectively, compare the coefficients of thermal conductivity of the materials of the vessels.

Answer:

                Let \[{{K}_{1}}\] and \[{{K}_{2}}\] be the coefficients of thermal conductivity of the materials and \[{{t}_{1}}\] and \[{{t}_{2}}\] be the times in which ice melts in the two vessels. As the same quantity of ice melts in the two vessels, the quantity of heat flowed into the vessels must be same. \[\therefore \]  \[Q=\frac{{{K}_{1}}A({{T}_{1}}-{{T}_{2}}){{t}_{1}}}{x}=\frac{{{K}_{2}}A({{T}_{1}}-{{T}_{2}}){{t}_{2}}}{x}\] or \[{{K}_{1}}{{t}_{1}}={{K}_{2}}{{t}_{2}}\]            \[\therefore \] \[\frac{{{K}_{1}}}{{{K}_{2}}}=\frac{{{t}_{2}}}{{{t}_{1}}}=\frac{25\min }{10\min }=\mathbf{5:2}.\]