Question

A 10.0 g sample of mercury absorbs 110 cal as it is heated from \(25^{\circ} \mathrm{C}\) to its boiling point at \(356^{\circ} \mathrm{C} .\) It then requires an additional 697 cal to vaporize. How much energy is released as the mercury vapor cools from \(356^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C} ?\)

Short answer

807 cal of energy is released.

Step-by-step solution

Understanding the problem

We need to find out how much energy is released when mercury vapor cools from its boiling point at \( 356^{\circ} \mathrm{C} \) back to \( 25^{\circ} \mathrm{C} \). When heating mercury, 110 cal was absorbed to reach the boiling point, and 697 cal was used to vaporize. Cooling should release the same amount of energy in reverse.

Releasing Energy to Condense

When mercury vapor cools back down, the energy that was used to vaporize it must be released. This means 697 cal is released during condensation.

Releasing Energy to Cool Down

After condensing, cooling from \( 356^{\circ} \mathrm{C} \) to \( 25^{\circ} \mathrm{C} \) releases 110 cal, which was initially required to heat the liquid mercury to the boiling point. We account for this energy release separately.

Calculating Total Energy Released

The total energy released during cooling includes both the energy released during condensation and while cooling the liquid mercury, which is \( 697 \text{ cal} + 110 \text{ cal} = 807 \text{ cal} \).

Conclusion

The total energy released when the mercury vapor cools from \( 356^{\circ} \mathrm{C} \) to \( 25^{\circ} \mathrm{C} \) is 807 cal, as it includes both condensing and temperature reduction phases.

Key concepts

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics, involving the movement of thermal energy from one object or substance to another. In the given exercise, heat transfer occurs when the mercury sample absorbs energy as it is heated from room temperature to its boiling point. This process involves transferring kinetic energy to the mercury atoms, causing them to vibrate more vigorously and raising the temperature of the substance.

Heat transfer can occur through three primary mechanisms:

• Conduction: This is heat transfer through direct contact. In our scenario, this would occur as heat moves through the mercury itself.
• Convection: While not directly applicable in this exercise, convection involves the movement of heat through fluids (liquids or gases) caused by circulation.
• Radiation: This involves heat transfer through electromagnetic waves, such as heat from the sun reaching the Earth.

In our scenario, the primary mechanism is conduction, as the heat travels through the mercury. The specific value of heat absorbed by the mercury for raising its temperature is provided: 110 cal, which facilitates this understanding.

Phase Change

Phase changes are transformations from one state of matter to another, such as from solid to liquid or liquid to gas. During a phase change, the temperature of a substance remains constant, even though heat energy is still being exchanged. For instance, the mercury in the exercise moves from a liquid state to a gaseous state at a constant temperature when it boils at 356°C.

When mercury vapor is formed, additional energy is necessary to overcome the intermolecular forces holding the liquid together. This energy is known as the heat of vaporization. The exercise specifies that 697 cal is required to accomplish this phase change. During the cooling phase of this solution, as the mercury condenses back to a liquid at the same temperature (356°C), the same energy, 697 cal, is released.

This energy release upon condensation plays a key role when calculating the total energy released as the vapor cools down. The energy involved in phase changes is significant and must always be considered alongside sensible heat changes in energy calculations.

Energy Calculation

Energy calculation is a critical step in understanding thermodynamics exercises like the one provided. Total energy considerations must include all stages of heat transfer and phase change. Here’s how it plays out in our exercise:

Firstly, calculate the energy absorbed by the mercury while heating it from 25°C to its boiling point at 356°C. This involves a total of 110 cal, as given in the problem. Once the boiling point is reached, an additional 697 cal are needed for mercury to undergo a phase change into vapor.

When mercury cools, energy is released first during condensation, where the phase reverts from gas to liquid, releasing 697 cal. Next, as the liquid mercury cools back to 25°C, it releases another 110 cal.

The final step is adding these energy releases: 697 cal from condensation and 110 cal from cooling. This sums up to a total of 807 cal of energy released.

In energy calculations, always ensure you consider each part of the process separately: the energy needed for temperature changes and the energy involved in phase changes. This structured approach ensures clarity and accuracy.