Question

What is the energy of an ultraviolet photon that has a wavelength of \(1.18 \times 10^{-8} \mathrm{m} ?\)

Short answer

The energy of the ultraviolet photon with a wavelength of \(1.18 \times 10^{-8} \mathrm{m}\) is approximately \(1.686 \times 10^{-17} \mathrm{J}\).

Step-by-step solution

Write the equation for energy of a photon

Recall the equation to find the energy of a photon: Energy (E) = Planck's constant (h) × the speed of light (c) / wavelength (λ)

Plug in the given values into the equation

Now, we'll plug in the values for Planck's constant (h), the speed of light (c), and the wavelength (λ) into the equation: E = \(\frac{6.63 \times 10^{-34} \mathrm{Js} \times 3.00 \times 10^8 \mathrm{m/s}}{1.18 \times 10^{-8} \mathrm{m}}\)

Calculate the energy of the photon

Now, we'll perform the calculation to find the energy of the photon: E = \(\frac{1.989 \times 10^{-25} \mathrm{Jm}}{1.18 \times 10^{-8} \mathrm{m}}\) E = \(1.686 \times 10^{-17} \mathrm{J}\)

State the final answer

The energy of the ultraviolet photon with a wavelength of \(1.18 \times 10^{-8} \mathrm{m}\) is approximately \(1.686 \times 10^{-17} \mathrm{J}\).

Key concepts

Planck's Constant

Planck's constant is a fundamental constant in quantum mechanics, symbolized as 'h'. It relates the energy of a photon with its frequency, serving as a proportional constant in the equation E=hf, where E is energy and f is frequency. This small number, with a value of approximately 6.63 x 10^-34 Joule seconds (Js), may seem minuscule, yet it represents a cornerstone of quantum physics, demonstrating the quantization of energy in the microscopic world.

Understanding Planck's constant helps us grasp how and why energy is quantized and not continuous, as well as why we can precisely calculate the energy of light particles, or photons, emitted by atoms. In the exercise, using Planck's constant is crucial for calculating the energy of an ultraviolet photon. This challenges the classical physics perspective and indicates a profound shift to a quantized view of energy.

Speed of Light

The speed of light, denoted as 'c', is a constant that represents the speed at which light travels in a vacuum. It's a fundamental constant of nature with a value of approximately 3 x 10^8 meters per second (m/s).

The prominence of the speed of light isn't solely tied to its role in physics as the maximum speed at which all energy, matter, and information in the universe can travel. It also plays a pivotal role when understanding the relationship between energy and electromagnetic waves, such as ultraviolet light. When we calculate the energy of a photon in exercises like the one provided, the speed of light is used alongside Planck's constant and the wavelength to paint a full picture of a photon's energy profile.

Wavelength

Wavelength, represented by the Greek letter lambda (λ), describes the distance over which a wave's shape repeats. It is a critical concept in understanding waves, including those of the electromagnetic spectrum which range from radio waves to gamma rays. Wavelength is inversely proportional to a photon's frequency and, by extension, its energy—an equation that becomes clearer when using Planck's constant and the speed of light.

In the context of our ultraviolet photon problem, the given wavelength helps to determine just how much energy the photon carries. The shorter the wavelength, the higher the frequency, and thus the greater the energy, which is why ultraviolet photons, with their relatively short wavelengths, are more energetic than visible or infrared light.

Ultraviolet Photon

An ultraviolet photon is a quantum of ultraviolet light, which is part of the electromagnetic spectrum. This type of light has shorter wavelengths than visible light, hence carrying more energy than visible light photons. Ultraviolet photons are noteworthy for their roles in various processes, like producing vitamin D in the skin or causing fluorescence.

The exercise highlights how to calculate the energy contained in these energized particles. Knowing the energy of ultraviolet photons is not only crucial in scientific calculations but also in understanding their potential effects on biological systems and materials, which can range from beneficial to harmful impacts, such as the degradation of materials or risks of skin cancer due to prolonged exposure to ultraviolet light.