A lake is freezing in a region where the steady outside air temperature is - ^ C . The density of ice is and its specific latent heat of fusion is L . A graph is plotted showing the square of the ice thickness ( x^2 ) on the y-axis against time ( t ) on the x-axis. If the graph is a straight line passing through the origin with a slope m , the thermal conductivity of the ice is given by:

Step-by-step solution

The rate of heat conduction through the ice layer is equal to the rate of heat released during freezing. KA x = dm dt L Since dm = A dx , we have: KA x = A L dx dt dx dt = K L x Integrating both sides with respect to time: x , dx = K L , dt x^2 2 = K L t x^2 = ( 2K L ) t Comparing this with the equation of a straight line passing through the origin, y = mt , the slope m of the x^2 versus t graph is: m = 2K L Rearranging for the thermal conductivity K , we get: K = m L 2 Answer: m L 2