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Question

Consider the shown case of a freezing lake due to negative environmental temperature (–θ∘C). Thickness (x) of ice layer is small in comparison to depth of lake. Rate of increase in x will be greater

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Solution

The correct options are

**C** if environmental temperature decreases

**D** for smaller thickness of ice layer

Let the air temperature be −θ∘ C and water temperature just below ice be −0∘ C. At a certain time let the thickness of ice be x and let it increase dy in time dt.

The latent heat released on melting has to be conducted away through the ice layers as the water freezes and therefore

Q=KA(ΔT)x=KA(0−(θ))x;⇒LAρ dydt=KAθx

Here L is Specific latent heat of fusion of water,

ρ is density of ice

A is area of cross section

⇒ dydt=KθLρx

From the equation we can see that Rate is directly proportional to θ while inversely proportional to the thickness of ice layer.

Let the air temperature be −θ∘ C and water temperature just below ice be −0∘ C. At a certain time let the thickness of ice be x and let it increase dy in time dt.

The latent heat released on melting has to be conducted away through the ice layers as the water freezes and therefore

Q=KA(ΔT)x=KA(0−(θ))x;⇒LAρ dydt=KAθx

Here L is Specific latent heat of fusion of water,

ρ is density of ice

A is area of cross section

⇒ dydt=KθLρx

From the equation we can see that Rate is directly proportional to θ while inversely proportional to the thickness of ice layer.

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