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Chapter 12: Problem 96

GaP has a band gap of \(2.26 \mathrm{eV}\). If GaP is illuminated with ultraviolet light, it emits light equal to the band gap energy. (a) What color is the emitted light? (b) Would appropriately sized GaP quantum dots be able to emit blue light? (c) What about red light?

Short Answer

Expert verified
(a) The emitted light is green-yellow, as the wavelength is around 551 nm. (b) No, appropriately sized GaP quantum dots cannot emit blue light, as the wavelength of GaP is larger than the maximum wavelength for blue light. (c) No, appropriately sized GaP quantum dots cannot emit red light, as the wavelength of GaP is smaller than the minimum wavelength for red light.

Step by step solution

01

(Step 1: Convert energy to wavelength)

First, we need to convert the given band gap energy of GaP to the corresponding wavelength. To do this, we use Planck's constant and the relation between energy and wavelength: \[ E = \frac{hc}{\lambda} \] Where, \(E\) = energy (in eV) \(h\) = Planck's constant (\(4.14 × 10^{-15}\) eVs) \(c\) = speed of light (\(3 × 10^{8} \,\text{m/s}\)) \(\lambda\) = wavelength (in meters) We need to find the wavelength (\(\lambda\)), so rearrange the formula to solve for \(\lambda\): \[ \lambda = \frac{hc}{E} \] Now plug in the given energy and constants to calculate the wavelength: \[ \lambda = \frac{(4.14 × 10^{-15}\,\text{eVs})(3 × 10^{8}\,\text{m/s})} {2.26\,\mathrm{eV}} \]
02

(Step 2: Calculate the wavelength)

After plugging in the values and performing the calculation, we obtain the wavelength of emitted light: \[ \lambda = 5.51 × 10^{-7}\,\text{m} \] We will now convert this wavelength to nanometers (nm) for easier comparison to visible light wavelengths: \[ \lambda = 5.51 × 10^{-7}\,\text{m} \times \frac{10^9\,\text{nm}}{1\,\text{m}} = 551\,\text{nm} \]
03

(Step 3: Determine the emitted light color)

Now we can determine the color of light corresponding to the calculated wavelength. The wavelengths of visible light range from approximately 380 nm (violet) to 750 nm (red). Based on this range and the calculated wavelength, we see that GaP emits light in the green-yellow region of the spectrum (around 551 nm). Answer: (a) The emitted light is green-yellow.
04

(Step 4: Check GaP quantum dots for blue light emission)

Blue light has a wavelength around 450 nm to 495 nm. To check if GaP quantum dots can emit blue light, we compare this range with the wavelength of light emitted by GaP. Since the emitted wavelength of GaP is around 551 nm, which is larger than the maximum wavelength for blue light, GaP quantum dots cannot emit blue light. Answer: (b) No, appropriately sized GaP quantum dots cannot emit blue light.
05

(Step 5: Check GaP quantum dots for red light emission)

Red light has a wavelength between 620 nm to 750 nm. To check if GaP quantum dots can emit red light, we compare this range with the wavelength of light emitted by GaP. Since the emitted wavelength of GaP is around 551 nm, which is smaller than the minimum wavelength for red light, GaP quantum dots cannot emit red light. Answer: (c) No, appropriately sized GaP quantum dots cannot emit red light.

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Most popular questions from this chapter

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