Problem 51 - Master Resource Book in JEE Main Physics

Question

A substance of mass \(m \mathrm{~kg}\) requires a power input of \(P\) watts to remain in the molten state at its melting point. When the power is turned off, the sample completely solidifies in time \(t\) sec. What is the latent heat of fusion of the substance? (a) \(\frac{P m}{t}\) (b) \(\frac{P t}{m}\) (c) \(\frac{m}{P t}\) (d) \(\frac{t}{P m}\)

Step-by-Step Solution

Answer

(b) \(\frac{P t}{m}\)

Step 1: Understanding the Relationship Between Power and Energy

Power (P) is defined as the rate at which energy (E) is transferred or converted. It can be expressed using the formula: \[ P = \frac{E}{t} \]Here, t is the time in seconds. Rearrange this to find the energy: \[ E = P \cdot t \]

Step 2: Latent Heat of Fusion Definition

Latent heat of fusion (L) is the amount of energy required to change 1 kg of a substance from the solid phase to the liquid phase at constant temperature. It can be calculated using the formula: \[ E = m \cdot L \]where E is the energy required (or lost) and m is the mass of the substance.

Step 3: Equating Energy Expressions

Since the energy required to keep the substance molten (P \cdot t) is the same amount of energy released when it solidifies completely, we equate the formulas for energy: \[ P \cdot t = m \cdot L \]

Step 4: Solving for Latent Heat of Fusion

Rearrange the equation \( P \cdot t = m \cdot L \) to solve for L (latent heat of fusion):\[ L = \frac{P \cdot t}{m} \]This shows that the latent heat of fusion is proportional to the power and time, and inversely proportional to the mass of the substance.

Key Concepts

Power and Energy Relationship

Power is an essential concept in physics and engineering that describes the rate of energy transfer or conversion. It is measured in watts (W), where one watt equates to one joule per second. This means that power quantifies how quickly energy is being used or moved, which is crucial in understanding physical processes.

• Power is given by the formula: \( P = \frac{E}{t} \), where \( E \) is energy in joules and \( t \) is time in seconds.
• To find the total energy transferred, the formula rearranges to \( E = P \cdot t \).

By understanding the relationship between power and energy, we can decipher how much energy is being consumed or produced over a given time period. This knowledge is instrumental in processes like heating, cooling, and in this case, maintaining a substance in a molten state.

Melting Point

The melting point is a fundamental property of a substance that defines the specific temperature at which a solid turns into a liquid under atmospheric pressure. At this precise temperature, the substance transitions between phases, and all the heat energy added goes into changing the state rather than increasing the temperature.

• Each substance has a unique melting point which must be reached to initiate the melting process.
• At the melting point, thermal energy disrupts the structured molecular lattice of the solid, allowing it to adopt the looser structure of a liquid.

The notion of melting point is crucial when discussing phase changes, particularly when considering the energy required to maintain substances in particular states or facilitate transitions.

Solidification Process

When a substance undergoes solidification, it changes from a liquid to a solid, releasing energy in the form of heat. This energy release occurs because the molecules arrange themselves into a more orderly structure, typically a crystalline lattice. Solidification is the reverse of melting, and the same amount of energy involved in melting is what you'll find involved in solidification.

• During solidification, the latent heat of fusion is released into the environment.
• This process requires no change in temperature, even though the substance loses energy.

Understanding solidification is crucial in areas such as materials science and engineering, where controlling the phase of substances is vital for product manufacture and quality control.

Rate of Energy Transfer

The rate of energy transfer refers to how swiftly energy moves from one place to another or converts from one form to another. It's inherently linked to power, since power itself is the measurement of this rate. In the context of phase changes, the rate of energy transfer affects how swiftly processes like melting and solidification happen.

• A higher rate of energy transfer implies faster changes, which can be critical in reducing process times in industrial applications.
• It influences the efficiency and control over physical and chemical processes.
• The speed at which a substance heats up, melts, solidifies, or freezes all depend on the rate of energy transfer.

By managing the rate of energy transfer, one can ensure optimal performance in various applications, such as cooling systems, heat exchangers, and refrigeration units.