Problem 68 - Master Resource Book in JEE Main Physics

Question

There is formation of layer of snow \(x \mathrm{~cm}\) thick on water, when the temperature of air is \(-\theta^{\circ} \mathrm{C}\) (less than freezing point). The thickness of layer increases from \(x\) to \(y\) in the time \(t\), then the value of \(t\) is given by (a) \(\frac{(x+y)(x-y) \rho L}{2 k \theta}\) (b) \(\frac{(x-y) \rho L}{2 k \theta}\) (c) \(\frac{(x+y)(x-y) \rho L}{k \theta}\) (d) \(\frac{(x-y) \rho L k}{2 \theta}\)

Step-by-Step Solution

Answer

The correct formula for \(t\) is option (a), \(\frac{(x+y)(x-y) \rho L}{2 k \theta}\).

Step 1: Understand the heat transfer concept

When a snow layer forms and grows on water, it's based on the principle of heat transfer and phase change. The heat conducted through the snow layer is equal to the latent heat of fusion needed to freeze the water beneath it.

Step 2: Write the heat transfer equation

The heat conducted through the snow layer is given by Fourier's Law: \[ \text{Rate of heat transfer} = \frac{k \cdot A \cdot \theta}{x} \]where \(k\) is the thermal conductivity, \(A\) is the area, \(\theta\) is the temperature difference, and \(x\) is the thickness of the layer.

Step 3: Equate heat transfer with latent heat

The heat transferred is used to convert water to ice, so:\[ \frac{k \cdot A \cdot \theta}{x} = \frac{m \cdot L}{t} \]where \(m\) is the mass of water frozen, \(L\) is the latent heat of fusion, and \(t\) is the time. The mass \(m\) can be written as \(\rho \cdot A \cdot (y - x)\) since \(\rho\) is the density of water.

Step 4: Substitute and rearrange the equation

Substitute \(m = \rho \cdot A \cdot (y - x)\) into the heat equation and rearrange to solve for \(t\):\[ \frac{k \cdot A \cdot \theta}{x} = \frac{\rho \cdot A \cdot (y - x) \cdot L}{t} \]Simplify to express \(t\):\[ t = \frac{x \cdot \rho \cdot L \cdot (y - x)}{k \cdot \theta} \]

Step 5: Simplify for an average thickness

For ease, average the thickness over time from \(x\) to \(y\), i.e., use \(\frac{x+y}{2}\) in place of \(x\). The equation becomes:\[ t = \frac{(x+y) \cdot (y - x) \cdot \rho \cdot L}{2 k \cdot \theta} \]This matches option (a).

Key Concepts

Fourier's Law

When it comes to understanding how heat moves through snow, Fourier's Law is your guide. This principle describes how heat transfers through materials. Picture it as a bridge that allows heat to flow. The key equation here is: \[ \text{Rate of heat transfer} = \frac{k \cdot A \cdot \theta}{x} \]

• \( k \): Thermal conductivity - a measure of how well the material conducts heat.
• \( A \): The area through which heat is flowing.
• \( \theta \): The temperature difference driving the flow.
• \( x \): Thickness of the snow layer.

The formula highlights that greater temperature differences or larger surface areas lead to more heat being transferred. Also, a thicker snow layer means less heat is able to pass through.

Latent Heat of Fusion

The latent heat of fusion is an essential concept in understanding snow formation. It refers to the amount of heat needed to change one kilogram of water at its freezing point into ice, without changing its temperature. This is a crucial part because, as snow forms on water, the freezing process involves converting water to ice.

• During the phase change from water to ice, energy is released as a result of the change in state, even though the temperature does not change.
• This energy, the latent heat of fusion, is required to break the molecular bonds in water.

In the equations, the latent heat of fusion is denoted by \( L \) and is essential in calculating how long it takes for new snow to form from water.

Thermal Conductivity

Thermal conductivity is a property that dictates how well a material can conduct heat. Think of it as the material's talent at allowing heat to flow through it. In the context of snow, this is especially important because it affects how quickly heat can escape from the water beneath.

• Low thermal conductivity means heat flows slowly, thus a layer like snow retains heat effectively.
• High thermal conductivity would mean heat escapes quickly through the snow.

In the provided exercise, thermal conductivity is represented by \( k \). It's critical to understanding how the thickness of snow impacts the rate of heat transfer, influencing how fast the layer of snow grows.

Phase Change Calculations

Calculating phase change involves figuring out how much heat is needed for substances to change their state. Here, we are concerned with how much heat is required to turn water into ice. To find this, we use the relationship between heat transfer and latent heat, where the heat flowing through the snow impacts the phase change below it. The equation given by: \[ \frac{k \cdot A \cdot \theta}{x} = \frac{m \cdot L}{t} \] Where the amount of water frozen \( m \) is calculated as \( \rho \cdot A \cdot (y - x) \). This ultimately tells us how long it takes for layers of snow to form. Substituting and rearranging gives: \[ t = \frac{x \cdot \rho \cdot L \cdot (y - x)}{k \cdot \theta} \] This tells us the time \( t \) it takes for an additional layer of snow to form, with \( \rho \) being the density of water, \( L \) the latent heat, and \( \theta \) the temperature difference. Understanding and applying these calculations help in predicting and explaining snow formation.