Question

Consider the following data:$$\begin{array}{lcc}\text { Metal } & \text { Al } & \text { Cu } \\\\\hline \text { Mass }(\mathrm{g}) & 10 & 30 \\\\\text { Specific heat }\left(\mathrm{J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right) & 0.900 & 0.385 \\\\\text { Temnerature }{ }^{\circ}{ }^{\circ} \mathrm{C} \text { ) } & 40 & 60\end{array}$$ When these two metals are placed in contact, which of the following will take place? (a) Heat will flow from \(\mathrm{Al}\) to Cu because \(\mathrm{Al}\) has a larger specific heat. (b) Heat will flow from \(\mathrm{Cu}\) to \(\mathrm{Al}\) because \(\mathrm{Cu}\) has a larger mass. (c) Heat will flow from \(\mathrm{Cu}\) to \(\mathrm{Al}\) because \(\mathrm{Cu}\) has a larger heat capacity (d) Heat will flow from Cu to Al because Cu is at a higher temperature. (e) No heat will flow in either direction.

Short answer

(d) Heat will flow from Cu to Al because Cu is at a higher temperature.

Step-by-step solution

Identify the Variables

Given mass of Aluminum (Al) is 10 grams, and its specific heat is 0.900 J/g°C. The initial temperature of Al is 40°C. For Copper (Cu), the mass is 30 grams, its specific heat is 0.385 J/g°C, and its initial temperature is 60°C.

Calculate Heat Capacity

The heat capacity of a substance can be calculated as the product of its mass and its specific heat. Calculate for both metals: \[ \text{Heat Capacity of Al} = 10 \, \text{g} \times 0.900 \, \text{J/g°C} = 9 \, \text{J/°C} \] \[ \text{Heat Capacity of Cu} = 30 \, \text{g} \times 0.385 \, \text{J/g°C} = 11.55 \, \text{J/°C} \]

Determine Heat Flow Direction

Heat flows from a body at a higher temperature to one at a lower temperature. Copper (Cu) is at 60°C, while Aluminum (Al) is at 40°C. Therefore, heat will flow from Cu to Al.

Evaluate the Answer Choices

Analyze the listed options based on your calculations: - (a) Incorrect: Al has a larger specific heat but is at a lower temperature. - (b) Incorrect: Although Cu has a larger mass, temperature dictates heat flow. - (c) Incorrect: Even with larger heat capacity, Cu's higher temperature is key. - (d) Correct: Heat flows from Cu to Al because Cu is at a higher temperature. - (e) Incorrect: Heat will flow from Cu to Al.

Key concepts

Specific Heat

Specific heat is a property of a substance that tells us how much heat is needed to raise the temperature of a unit mass of the substance by one degree Celsius. It's an important concept when studying heat transfer because it helps us understand how different materials react to adding or removing heat.
For example, in the case of Aluminum and Copper from the problem, the specific heat of Aluminum is 0.900 J/g°C, meaning it takes 0.9 joules of energy to raise the temperature of 1 gram of Aluminum by 1°C.
Copper, on the other hand, has a specific heat of only 0.385 J/g°C. This means Copper heats up faster than Aluminum since it requires less energy to increase its temperature by the same amount.
Recognizing how specific heat affects temperature changes in materials can help determine how heat will flow between substances.

Heat Capacity

Heat capacity is similar to specific heat but considers the amount of the substance you have. In simple terms, it's specific heat multiplied by the mass of the substance.
The formula for heat capacity can be written as:

• Heat Capacity = Mass \( \times \) Specific Heat

In our exercise, the heat capacity of Aluminum is calculated as 10 g \( \times \) 0.900 J/g°C, which equals 9 J/°C.
For Copper, we multiply 30 g by 0.385 J/g°C to get 11.55 J/°C.
These values indicate how much heat energy a material can store. A larger heat capacity means a substance can absorb more heat before its temperature increases noticeably.
In our example, Copper has a higher heat capacity than Aluminum, but ultimately, the direction of heat flow is determined by temperature difference.

Temperature Difference

The temperature difference between two objects plays a crucial role in determining the direction of heat flow. Heat naturally flows from a hotter object to a cooler one, seeking equilibrium.
In this exercise, Copper starts at a higher temperature (60°C) compared to Aluminum (40°C).
Because heat moves from high to low temperature, it will flow from Copper to Aluminum until both reach a common temperature. This fundamental principle is sometimes referred to as Newton's Law of Cooling.
Understanding temperature difference is essential for predicting heat flow in real-world applications, like when mixing substances or designing thermal systems.

Mass

Mass is the amount of matter in a substance, and it influences the total heat energy a substance can absorb or release. However, it's important to differentiate between mass and the role it plays in heat transfer compared to temperature difference and heat capacity.
In our problem, Copper has a greater mass (30 grams) than Aluminum (10 grams). This greater mass contributes to Copper's greater heat capacity even though its specific heat is less than that of Aluminum.
While mass does affect heat capacity, the actual direction of heat flow is primarily determined by the temperature difference between the bodies.
Therefore, while mass is an important factor in calculating heat capacity, it alone does not determine the direction of heat flow.