Problem 55 - General Chemistry: Principles and Modern Applications

Question

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?

Step-by-Step Solution

Answer

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 1: 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 \(E_{\mathrm{photon}} = (110000 \, \mathrm{J} / \mathrm{mol}) / ( 6.022 \times 10^{23} \, \mathrm{mol}^{-1})\).

Step 2: 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.

Step 3: 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.

Key Concepts

Energy Gap in Silicon

The energy gap, often symbolized as \(\Delta E\), in semiconductors like silicon is a crucial concept. It is the energy required to move an electron from the valence band, where it is bound to an atom, to the conduction band, where it can move freely and conduct electricity. In silicon, this energy gap is given as \(110,000 \, \text{J/mol}\).
This particular value happens to be a characteristic property of silicon, influencing its electrical behavior and application in devices such as diodes and transistors. Understanding this energy gap helps in designing materials suitable for electronics based on the energy required to excite electrons.
To find how much energy is needed for one electron, we convert this molar energy (for Avogadro's number of electrons) into energy per electron, using Avogadro's constant, \(6.022 \times 10^{23} \, \text{mol}^{-1}\). This conversion makes it applicable at the atomic scale, necessary for calculating photon-related transitions.

Photon Wavelength Calculation

To determine the wavelength of light necessary to promote an electron across the energy gap, we use the relationship between energy and wavelength given by Planck's equation: \(E = \frac{hc}{\lambda}\).
In this equation, \(h = 6.626 \times 10^{-34} \, \text{J} \cdot \text{s}\) is Planck's constant and \(c = 2.998 \times 10^{8} \, \text{m/s}\) is the speed of light in a vacuum. The energy \(E\) in this context is the energy of a single photon, calculated from the given energy gap for silicon.
By rearranging the formula to solve for wavelength \(\lambda\), we get \(\lambda = \frac{hc}{E_{\text{photon}}}\).
Substituting the known constants and the calculated photon energy into this equation gives the wavelength of light required to excite an electron from the valence band to the conduction band in silicon. This wavelength is key to identifying the type of light used in practical applications.

Electromagnetic Spectrum Regions

The electromagnetic spectrum spans a vast range of wavelengths, from very short gamma rays to long radio waves. Each type of radiation within the spectrum is categorized by its wavelength range.
Upon calculating the wavelength necessary to excite silicon's energy gap, we can determine which part of the electromagnetic spectrum it falls into. For example:

• If the calculated wavelength is between 400 to 700 nanometers, the light is visible to the human eye, generally perceived as visible light.
• Wavelengths shorter than this range fall into ultraviolet or even X-ray regions.
• Longer wavelengths transition into the infrared region.

Knowing the exact region helps in developing optical and electronic devices that utilize specific light types, such as solar cells or sensors based on silicon, optimizing their performance based on the incoming light spectrum.