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Chapter 11: Problem 55

The energy gap, \(\Delta E\), for silicon is \(110 \mathrm{kJ} / \mathrm{mol}\). What is the minimum wavelength of light that can promote an electron from the valence band to the conduction band in silicon? In what region of the electromagnetic spectrum is this light?

Short Answer

Expert verified
The minimum wavelength of light that can promote an electron from the valence band to the conduction band in silicon is found by substituting given values into Planck's equation. The region of the electromagnetic spectrum that this light falls in can be determined by comparing the obtained wavelength with known ranges for different regions of the electromagnetic spectrum.

Step by step solution

01

Convert energy to appropriate units

First, let's convert the energy gap from kilojoules per mole to joules per mole by multiplying by \(1000\), so \(\Delta E = 110000 \, \mathrm{J} / \mathrm{mol}\). Next, since we know the energy is for one mole of photons, we can find the energy for a single photon by dividing by Avogadro's number \((6.022 \times 10^{23} \, \mathrm{mol}^{-1})\). Therefore, we get \(E_{\mathrm{photon}} = (110000 \, \mathrm{J} / \mathrm{mol}) / ( 6.022 \times 10^{23} \, \mathrm{mol}^{-1})\).
02

Use Planck's equation to determine wavelength

Next use Planck's equation, \(E=hc/\lambda\), rearranged to solve for wavelength: \(\lambda = hc/E\). Here, \(h = 6.626 \times 10^{-34} \, \mathrm{J} \cdot \mathrm{s}\) is Planck's constant and \(c = 2.998 \times 10^{8} \, \mathrm{m/s}\) is the speed of light. Substitute the values for \(h\), \(c\) and \(E_{\mathrm{photon}}\) to get the value for \(\lambda\) in meters.
03

Determine the region of the electromagnetic spectrum

The obtained wavelength will indicate the region of the electromagnetic spectrum this light falls in. For instance, if it falls between \(400-700 \, \mathrm{nanometers}\), it is in the visible light range.

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