Physics · Thermal Properties Of Matter · NEET
The already-formed ice is the wall that heat must pass through to escape to the cold air. When the ice is thin, heat escapes fast, so water freezes fast. As x grows, the path L = x gets longer, the temperature gradient (T/x) drops, and the heat current K*A*T/x falls. Since dx/dt = K*T/(x*rho*L), a bigger x directly means a smaller dx/dt. The ice insulates the water beneath it.
New ice forms only at the bottom face, where water at 0 C turns into ice at 0 C. This is a change of state at constant temperature, so the heat released per kilogram is the latent heat of fusion L, not c*deltaT. In time dt a thin layer of thickness dx and mass rho*A*dx freezes and releases rho*A*dx*L of heat, which is carried up by conduction.
Ice floats and seals the surface (anomalous expansion of water: water is densest at 4 C). The bottom of the ice touches liquid water at 0 C, and the top touches the cold air at temperature T (below 0). So the temperature difference across the ice sheet is (0 - (-T)) = T. Only the air side gets colder; the water side stays pinned at the freezing point while liquid remains.
Start from dx/dt = K*T/(x*rho*L). Separate variables: x*dx = (K*T/(rho*L))*dt. Integrate from 0 to x on the left and 0 to t on the right: x^2/2 = (K*T/(rho*L))*t. So t = (rho*L*x^2)/(2*K*T). Time to grow the layer scales with the SQUARE of the thickness, which is why the first cm forms quickly but the next cm takes far longer.
Because t is proportional to x^2, the time to thicken from x1 to x2 is t = (rho*L)/(2*K*T) * (x2^2 - x1^2). Notice it depends on the difference of squares, not the plain difference, so doubling the thickness takes about four times the total time from the start.
A deep pond is covered by a frozen ice layer of thickness x while the outside air is at a steady -26 C (water below is at 0 C). The ice has thermal conductivity K, density rho and specific latent heat of fusion L. The rate of increase of the thickness of the ice layer at this instant is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
From the top. Water is densest at 4 C, so the coldest surface water (below 4 C) is lighter and stays on top, freezes first, and the ice floats and insulates the water below. This anomalous expansion of water is why fish survive winter under the ice.
dx/dt = K*T/(x*rho*L), where x is ice thickness, K thermal conductivity of ice, T the number of degrees the air is below 0 C, rho the density of ice and L the latent heat of fusion.
Time to reach thickness x follows t = rho*L*x^2/(2*K*T), so time grows with the square of thickness. The thicker ice is a longer insulating path, so heat escapes slowly and freezing slows down.
No. dx/dt is largest when the ice is thin and decreases as x increases, because dx/dt is proportional to 1/x. The ice sheet grows quickly at first and then more and more slowly.
