Light, or electromagnetic radiation, travels in discrete, measurable packets of energy called photons. These photons are the fundamental units of light. Determining the energy contained within a single photon is necessary across fields like chemistry, physics, and biology. The energy a photon carries is defined by its wave properties, specifically its wavelength. This article provides the method for calculating the energy of a photon when only its wavelength is known.
The Relationship Between Wavelength and Energy
The wave-like nature of light means that every photon possesses a measurable wavelength, which is the physical distance between consecutive peaks of the wave. Wavelength is intimately connected to the photon’s frequency, which describes how many wave cycles pass a fixed point per second. These two properties are inversely related: a shorter wavelength means a higher frequency, and a longer wavelength corresponds to a lower frequency.
The energy carried by a photon is directly related to its frequency. A higher frequency wave oscillates more rapidly, which translates to a greater energy content for each photon.
This inverse relationship between wavelength and energy is easily seen in the visible light spectrum. Violet light, which has the shortest wavelengths (around 380–450 nanometers), consists of the most energetic photons visible to the human eye. Conversely, red light has the longest wavelengths (around 620–750 nanometers) and therefore carries the least energy per photon.
Defining the Physical Constants
Calculating photon energy requires the use of two universal physical constants that describe the fundamental nature of the universe. The first is the speed of light in a vacuum, symbolized by \(c\). This constant represents the maximum speed at which all electromagnetic radiation can travel.
For calculation purposes, the value of the speed of light is often approximated as \(3.00 \times 10^8\) meters per second (m/s). The second constant is Planck’s constant, denoted by \(h\), which establishes the relationship between a photon’s energy and its frequency.
Planck’s constant is \(6.626 \times 10^{-34}\) Joule-seconds (J·s). Using the standard International System of Units (SI), such as meters, seconds, and Joules, for these constants is necessary to ensure the final calculated energy is correctly expressed in Joules.
Step-by-Step Calculation Using the Formula
The calculation of a photon’s energy (\(E\)) from its wavelength (\(\lambda\)) is accomplished by combining the relationship between energy and frequency with the relationship between frequency, wavelength, and the speed of light. This combination yields the formula \(E = hc/\lambda\). In this equation, \(h\) is Planck’s constant, \(c\) is the speed of light, and \(\lambda\) is the given wavelength.
Before substituting values into the formula, the wavelength must be in meters for the units to cancel correctly, resulting in energy measured in Joules. Wavelengths are frequently provided in nanometers (nm), which requires a conversion step where 1 nanometer equals \(10^{-9}\) meters. For instance, if a photon has a wavelength of 500 nm, this converts to \(5.00 \times 10^{-7}\) m.
Using the example of a 500 nm photon, the calculation involves multiplying Planck’s constant and the speed of light. This step uses the values \((6.626 \times 10^{-34} \text{ J}\cdot\text{s}) \times (3.00 \times 10^8 \text{ m/s})\), which results in a numerator of \(1.9878 \times 10^{-25} \text{ J}\cdot\text{m}\). The seconds unit in Planck’s constant cancels with the seconds unit in the speed of light, leaving Joules times meters.
The final step is to divide this result by the wavelength, \(5.00 \times 10^{-7} \text{ m}\). Dividing \(1.9878 \times 10^{-25} \text{ J}\cdot\text{m}\) by \(5.00 \times 10^{-7} \text{ m}\) yields the photon energy. The meters unit in the denominator cancels with the meters unit in the numerator, leaving the final energy in Joules. The calculated energy for the 500 nm photon is approximately \(3.976 \times 10^{-19}\) Joules.
Interpreting and Converting Energy Units
The result of the calculation is expressed in Joules (J), the standard SI unit for energy. However, in contexts dealing with individual atoms, molecules, or photons, the Joule represents a relatively large amount of energy. For this reason, scientists often convert the energy value into electron volts (eV), which is a more practical unit for atomic-scale measurements.
An electron volt is defined as the amount of energy gained by a single electron accelerated through an electric potential difference of one volt. The conversion factor is defined by the elementary charge, where 1 eV is equivalent to approximately \(1.602 \times 10^{-19}\) Joules. To convert the calculated energy from Joules to electron volts, one simply divides the Joule value by this conversion factor.
Taking the previous result of \(3.976 \times 10^{-19} \text{ J}\), dividing it by \(1.602 \times 10^{-19} \text{ J/eV}\) yields the energy in electron volts. The resulting energy for the 500 nm photon is about 2.48 eV. This unit conversion is valuable in fields like photochemistry, where the energy of a photon must be directly compared to the ionization energy or bond energy of an electron in an atom.