Question

A \(100.0\) -g block of iron and a \(100.0\) -g block of aluminum are both initially at \(25.0^{\circ} \mathrm{C}\). Both are then warmed to \(100.0^{\circ} \mathrm{C}\). Does one block require more heat energy than the other to reach \(100.0{ }^{\circ} \mathrm{C}\) ? If so, how much more?

Short answer

The aluminum block requires an additional 3337.5 J of heat energy to reach 100.0 °C compared to the iron block. This is because the aluminum has a higher specific heat capacity, which means it takes more heat energy to change its temperature by the same amount compared to iron.

Step-by-step solution

Determine the given values

We are given the following values: - Mass of iron (\(m_{Fe}\)) = 100.0 g - Mass of aluminum (\(m_{Al}\)) = 100.0 g - Initial temperature (\(T_i\)) = 25.0 °C - Final temperature (\(T_f\)) = 100.0 °C - Specific heat capacity of iron (\(c_{Fe}\)) = 0.450 J/g°C - Specific heat capacity of aluminum (\(c_{Al}\)) = 0.897 J/g°C

Calculate the change in temperature

To calculate the change in temperature for both the iron and the aluminum blocks, we subtract the initial temperature from the final temperature: \(\Delta T = T_f - T_i = 100.0 °C - 25.0 °C = 75.0 °C\)

Calculate the heat energy for the iron block

Now, we will use the values for the mass, specific heat capacity, and change in temperature of the iron block to compute the heat energy required to warm it to 100.0 °C: \(q_{Fe} = m_{Fe} \times c_{Fe} \times \Delta T = (100.0 \,\text{g}) \times (0.450 \,\text{J/g°C}) \times (75.0 \, °C) = 3375 \, \text{J}\)

Calculate the heat energy for the aluminum block

Similarly, we will compute the heat energy required to warm the aluminum block to 100.0 °C: \(q_{Al} = m_{Al} \times c_{Al} \times \Delta T = (100.0 \,\text{g})\times (0.897\,\text{J/g°C}) \times (75.0\,°C) = 6712.5 \,\text{J}\)

Compare the heat energies and determine the difference

It is clear from the calculations that the aluminum block requires more heat energy to reach 100.0 °C than the iron block: \(6712.5\,\text{J} > 3375\,\text{J}\) To find out how much more heat energy the aluminum block requires, we subtract the heat energy for the iron block from that of the aluminum block: \(\Delta q = q_{Al} - q_{Fe} = 6712.5\,\text{J} - 3375\,\text{J} = 3337.5\,\text{J}\) Therefore, the aluminum block requires an additional 3337.5 J of heat energy to reach 100.0 °C compared to the iron block.

Key concepts

Heat Energy Calculation

Heat energy calculation is fundamental to understanding how different materials require varying amounts of energy to change their temperatures. The formula to calculate heat energy ( q ) absorbed or released by an object is q = mcΔT . Here, m represents the mass of the object, c is the specific heat capacity, and ΔT is the change in temperature.
To increase the temperature of a given object, the amount of heat required depends on the product of these three factors:

• Mass: Heavier objects need more heat energy for the same temperature change.
• Specific Heat Capacity: This property indicates how much heat energy a material needs to change its temperature by 1°C. Higher values mean more energy is required.
• Temperature Change: Larger ΔT values mean more energy is needed for a greater temperature increase.

Utilizing these concepts, we can effectively calculate how much heat is needed for different materials under the same conditions.

Thermal Properties of Metals

Metals possess distinct thermal properties, prominently their specific heat capacity, that affect how they respond to heat energy. Specific heat capacity is crucial in determining how much temperature change occurs upon adding a certain amount of energy. For example, iron has a specific heat capacity of 0.450 J/g°C, while aluminum has a specific heat capacity of 0.897 J/g°C.
The larger specific heat capacity of aluminum indicates that it needs more energy to achieve the same temperature increase as iron. This difference in specific heat capacity explains why, in the same heating conditions, aluminum requires more energy than iron to reach the same final temperature.
Understanding these thermal properties not only sheds light on the behavior of different metals under heat but also guides practical applications and material choices in engineering and design.

Temperature Change

The concept of temperature change is central to the study of thermodynamics, especially when discussing heat energy transfer between objects. Temperature change ( ΔT ) is simply the difference between the final temperature and the initial temperature of a substance. It is calculated as ΔT = T_f - T_i , where T_f is the final temperature, and T_i is the initial temperature.
A positive ΔT means the object has absorbed heat and its temperature increased, while a negative ΔT means it released heat and its temperature decreased.

• This straightforward calculation is essential for the accurate determination of heat energy transfer.
• In our exercise, both the iron and aluminum blocks undergo the same ΔT of 75°C, highlighting that the masses and specific heat capacities primarily influence the differing energy requirements.

By understanding temperature change, we can predict how substances will behave when exposed to different thermal environments, which is crucial across various scientific and industrial applications.