The observation for three materials shows: Material A absorbs all the visible light (opaque) Material B is transparent to all the visible light (transparent) Material C absorbs blue light but is transparent to red light. (Blue light has wavelength 0.4 �m, red light has wavelength 0.7 �m, and all other visible lights have wavelengths in between.) (5.a) Draw corresponding energy band diagram for each sample. (5.b) Rank materials A, B, and C on scale of increasing insulator behavior: (5.c) State the reasons why you ranked as you did in (5.b). (5.d) Predict trend of resistivity as function of increasing temperature for the three samples

To answer this question, we need to understand the concept of energy band diagrams, insulator behavior, and resistivity as a function of temperature.

1. Drawing energy band diagrams:
Energy band diagrams represent the energy levels available for electrons in a material. The width of the band represents the range of allowed energy levels, and the position of the band determines the energy available to electrons. In this case, we have three materials: A, B, and C.

- Material A absorbs all visible light, which indicates that it does not allow light to pass through. In terms of energy band diagrams, this suggests that the material has a completely filled valence band and an empty conduction band, creating a large energy gap between them (indicating that it is an insulator).
- Material B is transparent to all visible light, implying that it allows light to pass through without significant absorption. This suggests that material B has a small or no energy gap between the valence and the conduction band, enabling electrons to move freely between them (indicating that it is a conductor or a semiconductor).
- Material C absorbs blue light but is transparent to red light. This implies that material C has a smaller energy gap compared to material A but still has a significant gap. It absorbs blue light because the energy of blue photons matches the energy required to promote electrons from the valence to the conduction band, while red light has lower energy and therefore does not have enough energy to promote electrons across the energy gap.

2. Ranking materials A, B, and C in terms of increasing insulator behavior:
Based on the information provided, we can rank the materials as follows:
Material B (Transparent to all visible light) > Material C (Absorbs blue light, transparent to red light) > Material A (Absorbs all visible light).

3. Reasons for the ranking:
Material B allows all visible light to pass through, indicating that it has a relatively small or no energy gap between the valence and the conduction band, making it the least insulating material.
Material C absorbs blue light but is transparent to red light, indicating that it has a larger energy gap compared to material B, placing it in the middle for insulator behavior.
Material A absorbs all visible light, suggesting that it has the largest energy gap between the valence and the conduction band, making it the most insulating material.

4. Predicting the trend of resistivity as a function of increasing temperature:
In general, as temperature increases, the resistivity of a material tends to increase. This is because at higher temperatures, the lattice vibrations in the material increase, leading to more collisions between electrons and the lattice. These collisions impede the flow of electrons, resulting in greater resistance and higher resistivity.

Considering this trend, we can predict the following:

- Material A, which has the largest energy gap and behaves as the most insulating material, is likely to have a higher resistivity compared to the other materials as temperature increases.
- Material B, which is transparent to all visible light and behaves as the least insulating material, is likely to have the lowest resistivity among the three materials, even at higher temperatures.
- Material C, which absorbs blue light but is transparent to red light, falls in between materials A and B in terms of insulating behavior. Its resistivity is expected to increase with temperature, but it may be lower than that of material A and higher than that of material B.

(5.a) I cannot draw the diagrams for you, but I can describe the energy band diagrams for each sample:

1. Material A (opaque): The energy band diagram for Material A would show a completely filled valence band and an empty conduction band, indicating that it is an insulator with a wide band gap.

2. Material B (transparent): The energy band diagram for Material B would show a partially filled valence band and an empty or partially filled conduction band, indicating that it is a semiconductor with a smaller band gap compared to Material A.

3. Material C (absorbs blue light but transparent to red light): The energy band diagram for Material C would show that it has a partially filled valence band and a partially filled conduction band, indicating that it is a semiconductor with a band gap smaller than Material A and larger than Material B.

(5.b) Ranking the materials on the scale of increasing insulator behavior:

1. Material A: Most insulating behavior as it absorbs all visible light.
2. Material B: Less insulating behavior than Material A but still transparent to all visible light.
3. Material C: Least insulating behavior as it absorbs some blue light but is transparent to red light.

(5.c) Reasons for ranking as stated in (5.b):

Material A is ranked first because it completely absorbs all visible light, indicating a larger band gap and hence stronger insulating behavior.

Material B is ranked second because it is transparent to all visible light, indicating a smaller band gap compared to Material A.

Material C is ranked last because it absorbs some blue light, indicating a smaller band gap compared to Material A and B. However, it is still transparent to red light, suggesting a larger band gap than some other materials.

(5.d) Prediction for the trend of resistivity as a function of increasing temperature for the three samples:

For all three samples (A, B, and C), the resistivity is expected to increase with increasing temperature. This is a common behavior in many materials, known as a positive temperature coefficient of resistivity. As the temperature rises, the atoms or ions within the material vibrate more, causing more collisions and hindering the flow of current, leading to increased resistivity.