Question

A \(5-g\) sample of aluminum and a \(5-g\) sample of iron are heated in a boiling water bath in separate test tubes. The test tubes are then placed together into a beaker containing ice. Which metal will lose the most heat in cooling down?

Short answer

The aluminum sample will lose more heat in cooling down compared to the iron sample. This is because aluminum has a higher specific heat capacity (\(0.897 J/g^{\circ}C\)) than iron (\(0.450 J/g^{\circ}C\)), resulting in a higher heat transfer for aluminum (448.5 J) than for iron (225 J) during the cooling process.

Step-by-step solution

Recall the formula for heat transfer

The formula to calculate heat transfer is given by: Q = mcΔT where Q is the heat transfer, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature. In this case, both aluminum and iron are being heated in boiling water, which is at 100°C, and then cooled down in ice, which is at 0°C. So, the change in temperature, ΔT, for both metals is (100 - 0) = 100°C.

Find the specific heat capacity of each metal

Now, we need to find the specific heat capacity, c, of aluminum and iron. These values can be found in a reference table or textbook. For aluminum and iron, the specific heat capacities are: c (aluminum) = \( 0.897 J/g^{\circ}C\) c (iron) = \( 0.450 J/g^{\circ}C\)

Calculate the heat transfer of each metal

We can now calculate the heat transfer, Q, for each metal using the following formula: Q = mcΔT For aluminum: Q(Al) = (5 g) * (\(0.897 J/g^{\circ}C\)) * (100°C) ≈ 448.5 J For iron: Q(Fe) = (5 g) * (\(0.450 J/g^{\circ}C\)) * (100°C) ≈ 225 J

Compare the heat transfer of each metal

Now that we have calculated the heat transfer of both metals during the cooling process, we can compare the values to determine which metal will lose the most heat in cooling down: The heat transfer of aluminum: Q(Al) ≈ 448.5 J The heat transfer of iron: Q(Fe) ≈ 225 J Since 448.5 J > 225 J, aluminum will lose more heat in cooling down compared to iron.

Key concepts

Specific Heat Capacity

Specific heat capacity is a fundamental property of materials that determines how much heat energy a substance can hold. Simply put, it's a measure of a substance's resistance to changing temperature. The higher the specific heat capacity, the more energy is needed to increase the temperature of the substance.
Imagine specific heat capacity as a sponge's ability to absorb water. A large sponge holds more water-like heat compared to a smaller container - that’s similar to how materials with high specific heat capacity can hold more heat.

• **Aluminum's specific heat capacity** is relatively high at 0.897 J/g°C, which means it takes more energy to change its temperature compared to many other metals.
• **Iron's specific heat capacity** is 0.450 J/g°C, indicating it requires less energy compared to aluminum to increase in temperature.

This difference in specific heat capacities explains why, under similar conditions like heating, some substances heat up or cool down slower or faster than others.

Temperature Change

Temperature change is the variation in temperature a material undergoes when it either gains or loses thermal energy. In this exercise, both aluminum and iron experience a temperature change when heated from 0°C to 100°C in boiling water and then cooled back to 0°C in ice.
A constant temperature difference of 100°C for both metals means every gram of either metal undergoes the same initial temperature change. This implies that the difference in heat loss or gain is influenced more by their specific heat capacities, rather than the temperature change itself.

• This explains why, with the same mass and temperature change, metals with higher specific heat capacities like aluminum will lose more heat when cooled.
  
• The formula ⁡\(∆T = T_{final} - T_{initial}\) encapsulates this change, where both metals exhibit a ⁡100°C differential in this case.

Thermal Energy

Thermal energy is the internal energy possessed by a system due to its temperature. It’s the total of all kinetic energy of atoms in a substance. When heat is transferred, thermal energy is either gained or lost by the substance.
This exercise showcases thermal energy in action as heat is transferred from metals to the surrounding environment when cooled down after heating. The amount of thermal energy each metal holds or releases is determined by their mass and specific heat capacity.

• For **aluminum**, with its greater specific heat capacity, it absorbs and subsequently releases more thermal energy compared to iron when cooled.
• The heat transfer equation, \(Q = mcΔT\), shows that larger values of specific heat capacity result in larger values of thermal energy change, illustrated by aluminum's greater heat loss of ⁡448.5 J compared to iron’s ⁡225 J.

Understanding these principles helps demystify why aluminum and iron behave differently under the same conditions and why materials often react in unique ways to heating and cooling processes. By mastering thermal energy concepts, you gain insights into the vast world of temperature dynamics and energy exchanges.