Question

A \(25.0 \mathrm{~g}\) sample of molten iron releases 1230 cal as it cools from \(2000^{\circ} \mathrm{C}\) to its freezing point, \(1535^{\circ} \mathrm{C}\). It then releases an additional 1590 cal as it solidifies. How much energy is required to heat the solid iron from \(1535^{\circ} \mathrm{C}\) to \(2000{ }^{\circ} \mathrm{C} ?\)

Short answer

2820 cal is required to heat the solid iron from 1535°C to 2000°C.

Step-by-step solution

Determine Energy Released During Cooling

First, calculate the total energy released by the molten iron as it cools from \(2000^{\circ} \mathrm{C}\) to its freezing point and then as it solidifies. The energy released during cooling is given as \(1230\text{ cal}\), and the additional energy released during solidification is \(1590\text{ cal}\). Thus, total energy released is \(1230\text{ cal} + 1590\text{ cal} = 2820\text{ cal}\).

Calculate Energy Required for Heating

The energy required to heat the solid iron back to \(2000^{\circ} \mathrm{C}\) is equal to the total energy released during cooling and solidification. Thus, the energy required is \(2820\text{ cal}\).

Key concepts

Latent Heat

Latent heat is an essential concept in understanding phase changes. It is the amount of heat required to change the phase of a substance without changing its temperature. Think of it as the hidden heat being absorbed or released during processes like melting or freezing.
When iron melts, it absorbs latent heat, and when it freezes, it releases that same latent heat.
The exercise mentioned involves calculating latent heat because the sample releases additional energy as it solidifies.

• In our case, 1590 cal represents the latent heat released when changing from a liquid to a solid.
• This concept is crucial in calculating the energy transformations that occur during phase changes.

Latent heat is typically measured in units such as calories or joules, and knowing the specific type of phase change helps in determining the energy exchanged.

Specific Heat Capacity

Specific heat capacity is another vital concept when discussing heat transfer. It measures how much heat energy is needed to raise the temperature of a unit mass of a substance by 1°C.
This property varies between materials, making it a crucial factor in heat transfer calculations.

• In this exercise, specific heat capacity would be relevant when calculating how much energy is needed to change the temperature of iron from its freezing point back to its higher temperature.
• Every material has a unique specific heat capacity, and for iron, it dictates how much energy is released or absorbed as its temperature changes.

The formula to find how much heat energy is needed to increase or decrease the temperature is given by: \( Q = m \, c \, \Delta T \), where:

• \( Q \) is the heat energy,
• \( m \) is the mass,
• \( c \) is the specific heat capacity, and
• \( \Delta T \) is the change in temperature.

This formula helps in understanding how temperature changes impact heat requirements, crucial for calculations involving temperature shifts in materials.

Phase Change

Phase change involves the transformation of a substance from one state of matter to another, such as from a solid to a liquid or a liquid to a solid. This change occurs without altering the chemical composition of the substance.

• In the given problem, the molten iron transitions from a liquid to a solid, which is an example of a phase change.
• During this transition, energy is exchanged in the form of heat, relating directly to the latent heat of the substance.

Phase changes can involve different forms such as melting, freezing, evaporation, and condensation.
During a phase change, even though the heat energy is absorbed or released, the temperature of the substance remains constant until the complete transformation occurs.
Recognizing how phase changes work helps in understanding why temperature doesn't change when a substance is undergoing a phase change despite heat being added or removed.