A highly thin conducting spherical shell of radius $R_{0}$ is completely filled with water at $0^{\circ} C$ . Outside surrounding temperature is $- {T_{0}}^{\circ} C$ . Latent heat of fusion of water is $L_{0}$ and thermal conductivity of ice is $K_{i c e}$ . Assume heat is flowing symmetrically in radial direction. Choose the CORRECT option(s). Density of water is $ρ_{0}$ . Neglect the expansion of water on freezing.

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Time in which thickness of ice will become $\frac{R_{0}}{2}$ , is given by $\frac{ρ_{0} L_{0} R_{0}^{2}}{12 K_{i c e} T_{0}}$

Time in which complete water will freeze, is given by $\frac{ρ_{0} L_{0} R_{0}^{2}}{4 K_{i c e} T_{0}}$

Total heat released by water in freezing, is given by $\frac{4 π ρ_{0} L_{0} R_{0}^{3}}{3}$

Time in which complete water will freeze, is given by $\frac{ρ_{0} L_{0} R_{0}^{2}}{6 K_{i c e} T_{0}}$

answer is A, C, D.

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Detailed Solution

Resistance $= \frac{d x}{K 4 π x^{2}} = \frac{1}{4 π K} [\frac{R_{2} - R_{1}}{R_{1} R_{2}}]$

$∴ \frac{d Q}{d t} = \frac{T_{0} 4 π K R_{0} x}{R_{0} - x} ρ_{0} 4 π x^{2} \frac{d x}{d t} L_{0} = \frac{K 4 π R_{0} x T_{0}}{R_{0} - x} \int_{0}^{R_{0}} \frac{ρ_{0} x (R_{0} - x) d x L_{0}}{K R_{0} T} = \int_{0}^{t} d t \frac{ρ_{0} L_{0} R_{0}^{2}}{6 K T_{0}} = t$