How to Calculate Photon Energy from its Wavelength: A Comprehensive Guide

The energy of a photon is directly related to its wavelength, and understanding this relationship is crucial for various applications in physics, chemistry, and engineering. This comprehensive guide will walk you through the step-by-step process of calculating the energy of a photon from its wavelength, providing you with the necessary theoretical background, formulas, examples, and problem-solving techniques.

Planck’s Relation: The Fundamental Equation

The relationship between the energy of a photon and its wavelength is governed by Planck’s relation, which is given by the equation:

E = hc/λ

Where:
– E is the energy of the photon (in Joules)
– h is Planck’s constant, which has a value of approximately 6.626 × 10^-34 J·s
– c is the speed of light, which is approximately 3 × 10^8 m/s
– λ is the wavelength of the photon (in meters)

This equation establishes the fundamental connection between the energy, frequency, and wavelength of a photon, and it is the basis for all calculations related to photon energy.

Converting Wavelength Units

Before using the Planck’s relation equation, it is essential to ensure that the wavelength is expressed in the appropriate unit. Photon wavelengths are typically measured in nanometers (nm), angstroms (Å), or micrometers (μm). To use the equation, the wavelength must be converted to meters (m).

Here are the conversion factors:
– 1 nm = 10^-9 m
– 1 Å = 10^-10 m
– 1 μm = 10^-6 m

For example, if the given wavelength is 520 nm, you would convert it to meters as follows:

520 nm = 520 × 10^-9 m = 5.2 × 10^-7 m

Calculating Photon Energy: Step-by-Step Example

Let’s calculate the energy of a photon with a wavelength of 520 nm.

1. Convert the wavelength to meters:
   520 nm = 520 × 10^-9 m = 5.2 × 10^-7 m

2. Plug the values into the Planck’s relation equation:
   E = (6.626 × 10^-34 J·s) × (3 × 10^8 m/s) / (5.2 × 10^-7 m)

3. Calculate the energy:
   E ≈ 2.38 × 10^-19 Joules

4. Convert the energy to electronvolts (eV):
   1 eV = 1.602 × 10^-19 J
Energy in eV = (2.38 × 10^-19 J) / (1.602 × 10^-19 J/eV) ≈ 2.38 eV

Therefore, the energy of a photon with a wavelength of 520 nm is approximately 2.38 × 10^-19 Joules or 2.38 electronvolts (eV).

Inverse Relationship between Wavelength and Energy

The relationship between the energy and wavelength of a photon is inversely proportional, meaning that shorter wavelengths correspond to higher frequencies and, consequently, higher energy levels. This principle is crucial for understanding the behavior of different types of electromagnetic radiation and their applications in various fields, such as:

• Medical imaging (e.g., X-rays, CT scans)
• Wireless communications (e.g., radio waves, microwaves)
• Quantum mechanics (e.g., the photoelectric effect)
• Spectroscopy (e.g., UV-Vis, IR, Raman)

As the wavelength of a photon decreases, its energy increases. This inverse relationship can be expressed mathematically as:

E = hc/λ

where a smaller value of λ results in a larger value of E.

Photon Energy Calculations: Additional Examples

To further solidify your understanding, let’s explore a few more examples of calculating photon energy from its wavelength.

1. Ultraviolet (UV) Photon
2. Wavelength: 250 nm
3. Calculation:
   λ = 250 nm = 250 × 10^-9 m
E = (6.626 × 10^-34 J·s) × (3 × 10^8 m/s) / (250 × 10^-9 m)
E ≈ 7.96 × 10^-19 J ≈ 4.96 eV

4. The energy of a UV photon with a wavelength of 250 nm is approximately 7.96 × 10^-19 Joules or 4.96 electronvolts (eV).

5. Infrared (IR) Photon

6. Wavelength: 1000 nm
7. Calculation:
   λ = 1000 nm = 1 × 10^-6 m
E = (6.626 × 10^-34 J·s) × (3 × 10^8 m/s) / (1 × 10^-6 m)
E ≈ 1.99 × 10^-19 J ≈ 1.24 eV

8. The energy of an IR photon with a wavelength of 1000 nm is approximately 1.99 × 10^-19 Joules or 1.24 electronvolts (eV).

9. Visible Light Photon

10. Wavelength: 650 nm (red light)
11. Calculation:
    λ = 650 nm = 650 × 10^-9 m
E = (6.626 × 10^-34 J·s) × (3 × 10^8 m/s) / (650 × 10^-9 m)
E ≈ 1.92 × 10^-19 J ≈ 1.19 eV
12. The energy of a red light photon with a wavelength of 650 nm is approximately 1.92 × 10^-19 Joules or 1.19 electronvolts (eV).

These examples demonstrate the versatility of the Planck’s relation equation in calculating the energy of photons across different regions of the electromagnetic spectrum, from ultraviolet to infrared and visible light.

Photon Energy Calculator

For more complex calculations or to explore the relationship between the wavelength, frequency, and energy of a photon, you can use online calculators like the one provided by Omnicalculator:

Photon Energy Calculator

This interactive tool allows you to input the wavelength or frequency of a photon and instantly calculate its energy in various units, including Joules, electronvolts, and more. It’s a valuable resource for quickly and accurately determining the energy of a photon based on its wavelength.

Conclusion

In this comprehensive guide, we have explored the fundamental relationship between the energy of a photon and its wavelength, as described by Planck’s relation. We have provided step-by-step examples, formulas, and conversion factors to help you calculate the energy of a photon from its wavelength, as well as discussed the inverse relationship between these two quantities.

Understanding this concept is crucial for various applications in physics, chemistry, and engineering, ranging from medical imaging and wireless communications to quantum mechanics and spectroscopy. By mastering the techniques presented in this guide, you will be well-equipped to tackle photon energy calculations and deepen your understanding of the behavior of electromagnetic radiation.

Reference Links:

1. How is the energy carried per photon of light related to the wavelength of the light?
2. Photon Energy Calculator
3. How to calculate the energy of a photon? – BYJU’S