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Chapter 12: Problem 111

Greenockite (CdS) was utilized as a pigment known as vermillion. It has a band gap of \(2.6 \mathrm{eV}\) near room temperature for the bulk solid. What wavelength of light (in \(\mathrm{nm}\) ) would a photon of this energy correspond to?

Short Answer

Expert verified
The wavelength of light that corresponds to the given photon energy of Greenockite (CdS) is \(477\, \text{nm}\).

Step by step solution

01

Write down the given information and the formula we will use

We are given: - Bandgap energy: \(E = 2.6 eV\) - We will need the speed of light constant: \(c = 3.00\times10^8 m/s\) We want to find: - Wavelength of light: \(\lambda\) The formula relating energy, wavelength, and the speed of light is: \[E = \dfrac{hc}{\lambda}\] where \(h\) is the Planck's constant \(h = 6.626\times10^{-34} Js\), \(c\) is the speed of light, and \(\lambda\) is the wavelength.
02

Convert energy from electron-volt to joules

To find the wavelength, we first need to convert the given bandgap energy from eV to Joules. The conversion factor is: \[1\,\text{eV} = 1.6\times10^{-19}\,\text{J}\] Thus, converting the energy: \[E = 2.6\,\text{eV} \times \dfrac{1.6\times10^{-19}\,\text{J}}{1\,\text{eV}} = 4.16\times10^{-19}\,\text{J}\]
03

Use the formula to find the wavelength

Substitute the given values and constants into the formula and solve for the wavelength: \[\lambda = \dfrac{hc}{E} = \dfrac{(6.626\times10^{-34}\,\text{Js})(3.00\times10^8\,\text{m/s})}{4.16\times10^{-19}\,\text{J}}\]
04

Calculate the wavelength and convert it to nanometers

Calculate the wavelength and convert it to nanometers: \[\lambda = \dfrac{(6.626\times10^{-34}\,\text{Js})(3.00\times10^8\,\text{m/s})}{4.16\times10^{-19}\,\text{J}} = 4.77 \times 10^{-7} \,\text{m}\] To convert the wavelength to nanometers, multiply by \(10^9\) : \[\lambda = 4.77\times10^{-7}\,\text{m} \times \dfrac{10^9\,\text{nm}}{1\,\text{m}} = 477\,\text{nm}\] So, the wavelength of light that corresponds to the given photon energy of Greenockite (CdS) is \(477\, \text{nm}\).

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