Question

The electromagnetic spectrum that lies between \(0.40\) and \(0.76 \mu \mathrm{m}\) is what we call visible light. Within this spectrum, the color violet has the shortest wavelength while the color red has the longest wavelength. Determine which of these colors, violet \((\lambda=0.40 \mu \mathrm{m})\) or red \((\lambda=0.76 \mu \mathrm{m})\), propagates more photon energy.

Short answer

Answer: Violet light carries more photon energy than red light.

Step-by-step solution

Write down the given values and constants

We are given the following values: - Wavelength of violet light: \(\lambda_v=0.40 \mu \mathrm{m}\) - Wavelength of red light: \(\lambda_r=0.76 \mu \mathrm{m}\) And the following constants: - Planck's constant (h): \(6.63\times 10^{-34} \mathrm{Js}\) - Speed of light (c): \(3\times 10^{8} \mathrm{m/s}\)

Convert wavelengths to meters

We need to convert the given wavelengths from micrometers to meters: - Wavelength of violet light: \(\lambda_v = 0.40 \times 10^{-6} \mathrm{m}\) - Wavelength of red light: \(\lambda_r = 0.76 \times 10^{-6} \mathrm{m}\)

Calculate the photon energy of violet light

Using the formula \(E = \frac{hc}{\lambda}\), we can calculate the energy of violet light: \(E_v = \frac{6.63\times 10^{-34} \mathrm{Js} \times 3\times 10^{8} \mathrm{m/s}}{0.40 \times 10^{-6} \mathrm{m}} = 4.97\times 10^{-19} \mathrm{J}\)

Calculate the photon energy of red light

Using the same formula, we can calculate the energy of red light: \(E_r = \frac{6.63\times 10^{-34} \mathrm{Js} \times 3\times 10^{8} \mathrm{m/s}}{0.76 \times 10^{-6} \mathrm{m}} = 2.61\times 10^{-19} \mathrm{J}\)

Compare the photon energies

Now we can compare the photon energies of violet and red light: - Violet light energy: \(4.97\times 10^{-19} \mathrm{J}\) - Red light energy: \(2.61\times 10^{-19} \mathrm{J}\) Since the energy of violet light is higher than the energy of red light, violet light propagates more photon energy.

Key concepts

Photon Energy

Photon energy is a fundamental concept in understanding how light behaves. It refers to the energy carried by a single photon, the smallest unit of light. This energy is crucial because it relates directly to the frequency and wavelength of light. It can be determined using the formula:
\[ E = \frac{hc}{\lambda} \]
where:

• \( E \) is the photon energy in joules (J)
• \( h \) is Planck's constant \((6.63 \times 10^{-34} \text{ Js})\)
• \( c \) is the speed of light \((3 \times 10^8 \text{ m/s})\)
• \( \lambda \) is the wavelength of the light in meters (m)

This equation shows that the energy of a photon is inversely proportional to the wavelength. Hence, shorter wavelengths mean higher photon energy. This is why violet light, with its shorter wavelength, carries more energy than red light in the visible spectrum.

Visible Light

Visible light is a small portion of the electromagnetic spectrum that can be perceived by the human eye. It ranges from about \(0.40 \mu m\) to \(0.76 \mu m\). Within this range, different wavelengths correspond to different colors, each with unique properties.

• Violet light has the shortest wavelength around \(0.40 \mu m\) and appears at one end of the visible spectrum.
• Red light has the longest wavelength around \(0.76 \mu m\) and appears at the other end of the visible spectrum.

The sequence of colors visible to the human eye, often remembered as the acronym ROYGBIV (Red, Orange, Yellow, Green, Blue, Indigo, Violet), represents the order of light from longest to shortest wavelength. Each color corresponds to a specific frequency and therefore a specific energy level, affecting how we perceive color and intensity.

Wavelength

Wavelength is a key parameter in the study of light and electromagnetic waves. It defines the distance between consecutive peaks of a wave and plays an integral role in determining the characteristics of light, such as color and energy.
Wavelength is inversely related to frequency, as expressed in the formula:
\[ c = \lambda u \]
where:

• \( c \) is the speed of light in a vacuum
• \( \lambda \) is the wavelength
• \( u \) is the frequency

Shorter wavelengths correlate with higher frequencies and vice versa. Consequently, shorter-wavelength light (like violet) contributes to higher photon energy, while longer-wavelength light (like red) contributes to lower photon energy. Understanding wavelength helps in determining how different colors of light behave and interact with matter.