Problem 1 - Astronomy Today

Question

What is the energy (in electron volts-see More Precisely 4-1) of a 450 -nm blue photon? A 200 -nm ultraviolet photon?

Step-by-Step Solution

Answer

The energy of a 450 nm blue photon is approximately 2.76 eV, and a 200 nm ultraviolet photon is approximately 6.21 eV.

Step 1: Understand the Relationship between Wavelength and Photon Energy

The energy of a photon is related to its wavelength through the equation \(E = \frac{hc}{\lambda}\), where \(E\) is energy, \(h\) is Planck's constant \(6.626 \times 10^{-34} \text{ J}\cdot\text{s}\), \(c\) is the speed of light \(3.00 \times 10^8 \text{ m/s}\), and \(\lambda\) is the wavelength.

Step 2: Convert Wavelength from Nanometers to Meters

The given wavelengths need conversion from nanometers to meters: \(450 \text{ nm} = 450 \times 10^{-9} \text{ m}\) and \(200 \text{ nm} = 200 \times 10^{-9} \text{ m}\). This is necessary because the speed of light is in meters per second.

Step 3: Calculate Energy for the Blue Photon

Using the formula \(E = \frac{hc}{\lambda}\), substitute the values for the 450 nm blue photon: \[E = \frac{(6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \times (3.00 \times 10^8 \text{ m/s})}{450 \times 10^{-9} \text{ m}}\]Calculate this value to determine the energy in joules.

Step 4: Convert Energy from Joules to Electron Volts

1 electron volt (eV) equals \(1.602 \times 10^{-19} \text{ J}\). Convert the energy calculated from joules to electron volts by dividing the energy in joules by \(1.602 \times 10^{-19} \text{ J/eV}\).

Step 5: Calculate Energy for the Ultraviolet Photon

Similar to the blue photon calculation, use the same equation for the 200 nm ultraviolet photon: \[E = \frac{(6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \times (3.00 \times 10^8 \text{ m/s})}{200 \times 10^{-9} \text{ m}}\]Compute the energy in joules.

Step 6: Convert Energy for Ultraviolet Photon from Joules to Electron Volts

Convert this energy into electron volts using the same conversion factor as the blue photon. Divide the energy found in joules by \(1.602 \times 10^{-19} \text{ J/eV}\) to find the energy in electron volts.

Key Concepts

Wavelength and Energy Relationship

The connection between the wavelength of light and the energy of a photon is crucial in understanding how energy is transferred at the quantum level. This relationship is defined by the formula:
\[E = \frac{hc}{\lambda}\]where:

• \(E\) is the energy of the photon,
• \(h\) is Planck's constant,
• \(c\) is the speed of light, and
• \(\lambda\) is the wavelength of the light.

This equation shows that energy is inversely proportional to wavelength. This means the shorter the wavelength, the greater the energy of the photon.
For instance, ultraviolet light with a shorter wavelength than blue light, carries more energy. Such insights are essential for understanding phenomena like the photoelectric effect and photon absorption in materials.

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics. It relates the energy of photons to their frequency and is crucial for calculations that involve quantum phenomena. The value of Planck's constant is approximately:
\[h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}\]This small number signifies the very tiny energy changes that occur at the atomic level. It's named after Max Planck, who was instrumental in the development of quantum theory. Planck's constant allows us to calculate the energy of photons of different wavelengths, making it a key component in the \(E = \frac{hc}{\lambda}\) equation used for determining photon energy.

Photon Energy in Electron Volts

Energy calculation is often transitioned from joules, a standard unit, to electron volts for convenience, especially in atomic and particle physics. One electron volt (eV) is defined as:
\[1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}\]This conversion is crucial when working with small energy scales typical in quantum mechanics. When you calculate the energy of a photon in joules using the equation \(E = \frac{hc}{\lambda}\), you convert it to electron volts by dividing by \(1.602 \times 10^{-19} \text{ J/eV}\). This allows scientists to express the energy in a more convenient form for analysis.

Speed of Light

The speed of light in a vacuum is a significant constant that appears frequently in physics, especially when dealing with electromagnetic waves. It is denoted as:
\[c = 3.00 \times 10^8 \text{ m/s}\]This speed sets a fundamental limit on how fast information or matter can travel. In the formula for photon energy, the speed of light couples with Planck's constant to establish the relationship between energy and wavelength. This value underscores the remarkable speed photons travel, which translates directly to their energy potential depending on their wavelength.

Nanometer to Meter Conversion

To use the photon energy equation, it's important to ensure all units match, particularly the wavelength's unit. Wavelength is often given in nanometers (nm) when discussing light. However, the equation \(E = \frac{hc}{\lambda}\) requires the wavelength in meters:
One nanometer equals \(10^{-9}\) meters:
\[1 \text{ nm} = 10^{-9} \text{ m}\]When given a wavelength in nanometers, like 450 nm or 200 nm, it's essential to multiply by \(10^{-9}\) to convert to meters before inserting into the energy equation. This conversion ensures consistency with the speed of light unit, measured in meters per second, and is vital for accurate photon energy calculations.