2 boxes are stacked below the header Set A. A box labeled Top Gear contains 4 high energy molecules, each marked 80. Below is a box labeled Bottom Gear containing 4 low energy molecules, each marked 40. 2 more boxes are stacked below the header Set B. The box labeled Top Gear contains 4 high energy molecules, each marked 80. Below is a box labeled Bottom Gear containing 9 low energy molecules, each marked 40.

Question

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3. Louis has two sets of two gears (Set A and Set B) that he is using to build two different machines. He has all the gears laying out separately, as shown in the diagram above. Then he puts the gears into the machines so that the gears in Set A are touching each other and the gears in Set B are touching each other. Use the information in the diagram to answer the question.

After the gears have been touching for a while, which of the two bottom gears will be cooler, and why?

Responses

a
The smaller bottom gear will be cooler than the larger bottom gear, because less energy has to transfer for the molecules to reach the same temperature as the molecules of the top gear.
The smaller bottom gear will be cooler than the larger bottom gear, because less energy has to transfer for the molecules to reach the same temperature as the molecules of the top gear.

b
The larger bottom gear will be cooler than the smaller bottom gear, because the energy that transferred to it was spread out over more molecules.
The larger bottom gear will be cooler than the smaller bottom gear, because the energy that transferred to it was spread out over more molecules.

c
The larger bottom gear will be cooler than the smaller bottom gear. It started with more total energy, so less energy had to transfer for both gears in Set B to reach the same total energy.
The larger bottom gear will be cooler than the smaller bottom gear. It started with more total energy, so less energy had to transfer for both gears in Set B to reach the same total energy.

d
Both gears will be the same temperature, because both of the top gears had the same amount of energy to transfer and the molecules in the bottom gears start with the same energy.

Answer 1

To determine which of the two bottom gears will be cooler after the gears have been touching for a while, we need to analyze the energy distribution in both sets of gears.

In Set A:

• The Top Gear has 4 high-energy molecules with energy marked as 80.
• The Bottom Gear has 4 low-energy molecules with energy marked as 40.

Total initial energy for Set A:

• Top Gear: \(4 \times 80 = 320\)
• Bottom Gear: \(4 \times 40 = 160\)

In Set B:

• The Top Gear has 4 high-energy molecules with energy marked as 80 (same as Set A).
• The Bottom Gear has 9 low-energy molecules with energy marked as 40.

Total initial energy for Set B:

• Top Gear: \(4 \times 80 = 320\)
• Bottom Gear: \(9 \times 40 = 360\)

When the top gears (high energy) are touching the bottom gears (low energy), energy will transfer from the top to the bottom gears until thermal equilibrium is reached.

In Set A, the bottom gear has a total energy of 160, and in Set B, the bottom gear has a total energy of 360.

As a result, the energy that needs to be transferred to raise the temperature of the bottom gears will be different:

• The bottom gear in Set A will absorb more energy proportionally compared to the bottom gear in Set B.
• The bottom gear in Set B has a larger initial energy (360) and thus, it would be spread out over more molecules (9 molecules in Set B, versus 4 in Set A).

Given this information, we can evaluate the responses:

Response a is incorrect because it refers to size rather than total energy.

Response b is correct. The larger bottom gear (Set B) will be cooler than the smaller bottom gear (Set A) because the energy transferred to it was spread out over more molecules, meaning it doesn't raise the temperature as much.

Response c is incorrect because it mistakenly attributes energy started as "more total energy," which is not how temperature equilibrium works in this context.

Response d is incorrect because it ignores the differences in energy transfer potential due to the number of molecules.

Thus, the best response to the question is:

b The larger bottom gear will be cooler than the smaller bottom gear, because the energy that transferred to it was spread out over more molecules.