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unscientific

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## Homework Statement

Part (a): Derive Clausius-Clapeyron Equation. Find latent heat of fusion of ice.

Part (b): Find rate of formation of ice

Part (c): What is the maximum thickness of ice formed?

## Homework Equations

## The Attempt at a Solution

__Part (a)__I have derived the relation. Using the values, I found that the latent heat of fusion of ice is ##L = 3.44 \times 10^5 J kg^{-1}## which seems right.

__Part (b)__Using heat flux ##J = k (\frac{\partial T}{\partial z})##, consider in time ##\delta t## amount of ice ##\Delta m## is formed.

[tex]\Delta Q = (\Delta m)L = (JA) \Delta t[/tex]

[tex] \rho (\Delta z)L = \frac{k(T_2-T_1)}{z} \Delta t[/tex]

[tex]\frac{dz}{dt} = \frac{k(T_2-T_1)}{\rho L z}[/tex]

Taking ##k = 2.3##, ##T_2-T_1 = 0.5##, ##\rho = 1000##, ##L = 3.44 \times 10^5##:

I find that ##\frac{dz}{dt} = 3.34 \times 10^{-7} m s^{-1}##, so ice forms awfully slow.

__Part (c)__In steady state, ## \frac{dQ_1}{dt} = \frac{dQ_2}{dt}##.

Therefore, we equate the heat fluxes:

[tex]k_{water} \left(\frac{2}{1-z_f}\right)A = k_{ice}\frac{0.5}{z_f}A[/tex]

[tex]z_f = \frac{k_{ice}}{4k_{water} + k_{ice}} = 0.51 m [/tex]

It's hard to believe that over half the lake would be frozen, when the rate of formation of ice above (without the interference of the bottom of the lake) is merely ##3.34 \times 10^{-7} m s^{-1}##.