Answer :

**Given:**Temperature of the water: T

_{1}= 0 °C

Temperature of the atmosphere: T

_{2}= -10 °C

Change in temperature : ΔT = T

_{1}-T

_{2}= 10 °C

Length of the ice formed : l = 10 cm = 0.1 m

Density of water: ρ = 1000 kg m

^{–3}

Latent heat of fusion of ice: L = 3.36 × 10

^{5}J kg

^{–1}.

Thermal conductivity of ice: K0. = 1.7 W m

^{–1}°C

^{–1}.

**Formula used:**

Rate of amount of heat flowing or heat current is given as:

Here, Δθ is the amount of heat transferred, ΔT is the temperature difference, K is the thermal conductivity of the material, A is the area of cross section of the material and x is the thickness or length of the material.

Also,

Δθ = Q =L× m

Here, Q is the amount of heat absorbed or released, L is the Latent heat and m is the mass of the substance.

And,

Mass = Density × Volume

∴ m = ρ V = ρAl

Here ρ is the density of water, A is the area and l is the length.

(a)

Let change in thickness be Δx,thus rate of change of thickness is:

Hence, the thickness of the ice increases at a rate of 5.05× 10

^{-7}m/s.

(b)

Consider time required to form thin layer of ice dx is dt.

Using above formulations,

Mass of dx: dm = Adx × ρ

And heat absorbed by the thin layer,

dQ = dm× L

Now, rate of Heat transfer due to thin layer becomes:

Integrating on both sides and setting the limit of ice formed x: 0 to 0.1

Hence, it took 27.45 hours to form 10 cm thick ice.

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