Problem 69 - Chemistry The Central Science

Question

Silicon has a band gap of \(1.1 \mathrm{eV}\) at room temperature. (a) What wavelength of light would a photon of this energy correspond to? (b) Draw a vertical line at this wavelength in the figure shown, which shows the light output of the sun as a

Step-by-Step Solution

Answer

The wavelength of light corresponding to Silicon's band gap energy of 1.1 eV is approximately 1130 nm. Locate 1130 nm on the x-axis of the provided figure and draw a vertical line at that position.

Step 1: (Step 1: Write down the Planck-Einstein relation)

The Planck-Einstein relation is used to convert energy of light into its corresponding wavelength, and is given by: \[E = h\frac{c}{\lambda}\] Where \(E\) is the energy of the light, \(h\) is Planck's constant (approximately \(6.63 \times 10^{-34} \, \mathrm{Js}\)), \(c\) is the speed of light (approximately \( 3 \times 10^8\, \mathrm{m/s}\)) and \(\lambda\) is the wavelength.

Step 2: (Step 2: Convert energy from eV to Joules)

The given energy of the photon is 1.1 eV, but it needs to be converted to Joules before using it in the Planck-Einstein relation. To convert eV to Joules, we multiply the energy in eV by the elementary charge, which is approximately \(1.6 \times 10^{-19} \, \mathrm{C}\). \[E = 1.1 \, \mathrm{eV} \times 1.6 \times 10^{-19}\, \mathrm{C} \approx 1.76 \times 10^{-19} \, \mathrm{J}\]

Step 3: (Step 3: Calculate the wavelength using Planck-Einstein relation)

We can now use the Planck-Einstein relation to calculate the wavelength: \[\lambda = \frac{hc}{E}\] Plug in the values of \(h\), \(c\), and \(E\): \[\lambda = \frac{(6.63 \times 10^{-34}\, \mathrm{Js}) (3 \times 10^8\, \mathrm{m/s})}{1.76 \times 10^{-19}\, \mathrm{J}}\] Solve for the wavelength: \[\lambda \approx 1.13 \times 10^{-6} \, \mathrm{m} = 1130 \, \mathrm{nm}\]

Step 4: (Step 4: Draw a vertical line at the calculated wavelength)

Now, locate the point corresponding to 1130 nm on the x-axis of the provided figure, which represents the wavelength axis. Draw a vertical line at 1130 nm, from the x-axis to the curve or any desired height. The vertical line at 1130 nm represents the wavelength of light associated with the band gap energy of Silicon.

Key Concepts

Planck-Einstein Relation

The Planck-Einstein relation is a fundamental concept that bridges the world of quantum physics with the macroscopic behavior of light. It connects the energy of a photon to its electromagnetic wave properties, specifically its wavelength. In other words, it showcases how the energy contained in a single quantum of light is inversely proportional to the wavelength of that light.

Here's the formula that captures this essence: \[E = h\frac{c}{\lambda}\]Where:

• \(E\) represents the energy of the photon
• \(h\) is Planck's constant, which is approximately \(6.63 \times 10^{-34} \, \mathrm{Js}\)
• \(c\) is the speed of light in a vacuum, roughly \(3 \times 10^8 \, \mathrm{m/s}\)
• \(\lambda\) is the wavelength of the light

This equation is crucial for understanding how light of certain energies interacts with materials, like how a photon with energy equivalent to the band gap of silicon can excite an electron from the valence band to the conduction band, a key concept in semiconductor physics and, by extension, electronics.

By rearranging the formula, one can solve for any of the variables given the others. For instance, given the energy, you can find the corresponding wavelength, or vice versa. This allows scientists and engineers to design systems that can manipulate and detect light of specific energies, such as solar cells, LEDs, and lasers.

Wavelength Calculation

Calculating the wavelength of light from its energy involves using the Planck-Einstein relation. Here's how you can perform this calculation:
To find the wavelength \(\lambda\), rearrange the formula to solve for \(\lambda\): \[\lambda = \frac{hc}{E}\]Plug in the known values for Planck's constant \(h\), the speed of light \(c\), and the energy \(E\) of the photon. For our example involving silicon's band gap energy, after converting the energy from electron volts to joules, you can substitute those values into the equation.The resulting calculation will give you the wavelength in meters, which, in the context of homework problems or practical applications, might be more conveniently expressed in nanometers (nm), where \(1 \, \mathrm{nm} = 1 \times 10^{-9} \, \mathrm{m}\). This conversion is particularly helpful when dealing with light in the visible spectrum or other applications where nm is a standard unit of measure.

Energy Conversion from eV to Joules

When it comes to working with photon energies, it's quite common to find values expressed in electron volts (eV), a unit favored in atomic and quantum physics due to its convenient scale when dealing with subatomic particle energies. However, for calculations involving the Planck-Einstein relation, we need to convert this energy to Joules (J), the standard unit of energy in physics.

The conversion is a straightforward multiplication:
\[E_{\mathrm{Joules}} = E_{\mathrm{eV}} \times e\]Where \(E_{\mathrm{eV}}\) is the energy in electron volts and \(e\) is the elementary charge, valued at approximately \(1.6 \times 10^{-19} \, \mathrm{Coulombs}\) (C). One electron volt is defined as the amount of energy gained or lost by an electron as it moves across an electric potential difference of one volt, which translates to the equation above.

By performing this conversion, we ensure that our energy value is compatible with the SI (International System of Units), making it possible to apply the Planck-Einstein relation, and thus providing a clear pathway from understanding subatomic particle energies to recognizing their corresponding wavelengths in the macroscopic world.