A layer of ice of `0^(@)C` of thickness `x_(1)` is floating on a pond. If the atmospheric temperature is `-7^(@)C`. Show that the time taken for thickness of the layer of ice to increase from `x_(1)` to `x_(2)` is given by
`t=(pL)/(2kT)(x(2)/(2)-x_(1)^(2))`
where p is the density of ice, k its thermal conductivity and L is the latent heat of fusion of ice.

Explanation

When the temperature of the air is less than `0^(@)C`, the cold air near the surface of the pond takes heat (latent) from the water which freezes in the fonm of layers. Consequently, the thickness of the ice layer keeps increasing with time. Let `x` be the thickness of the ice layer at a certain time. If the thickness is increased by dx in time dt, thn the amount of heat flowing through the salb in time dt is given by

Where A is the area of the layer of ice and `-7^(@)C` is the temperature of the surounding air. If dm is the mass of water frozen into ice, then `Q=dmxxL`. But `dm=Ap dx`, where p is the density of ice. Hence
`Q=Ap Ldx` (ii)
Equating (i) and (ii). we have
`(KATdt)/(x)=ApLdx` or `dt=(pL)/(kT)xds`
Integrating, we have
`underset(0)overset(t)intdt=(pL)/(kT)underset(x_(1))overset(x_(2))int` or `t=(pL)/(kT)|(x^(2))/(2)|_(x_(1))^(x_(2))=(pL)/(2kT)(x_(2)^(2)-x_(1)^(1))`