Question

The blue color of the sky results from the scattering of sunlight by air molecules. The blue light has a frequency of about \(7.5 \times 10^{14} \mathrm{~Hz}\). (a) Calculate the wavelength, in \(\mathrm{nm}\), associated with this radiation, and (b) calculate the energy, in joules, of a single photon associated with this frequency.

Short answer

The wavelength of the blue light is approximately 400 nm, and the energy of a single photon associated with this frequency is about 5.0 x 10^-19 Joules.

Step-by-step solution

Calculate the Wavelength

Use the formula for the speed of light, \(c = \lambda f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency. As the speed of light is a constant (\(c = 3.0 \times 10^8 m/s\)), you can rearrange the equation to solve for the wavelength: \(\lambda = c/f\). Substituting the given frequency value (\(f = 7.5 \times 10^{14} Hz\)) into the formula will give the wavelength value in meters.

Convert Units

Since the question asks for the wavelength in nanometers (nm), convert the calculated wavelength from meters to nanometers using the conversion factor \(1 m = 1 \times 10^9 nm\).

Calculate Photon Energy

Utilize the Planck's equation, \(E = hf\), where \(E\) is the photon energy, \(h\) is the Planck's constant (\(6.63 \times 10^{-34} Js\)), and \(f\) is the frequency. By replacing \(f\) with the given frequency, the photon energy in Joules can be determined.

Key concepts

Wavelength Calculation

To calculate the wavelength of light when given the frequency, we can use a fundamental relationship in physics involving the speed of light. The speed of light is a constant, denoted by \(c\), and is approximately \(3.0 \times 10^{8} \, \text{m/s}\). The connection between this constant speed, wavelength \(\lambda\), and frequency \(f\) is described by the equation:

• \( c = \lambda f \)
• To find wavelength, rearrange the equation as: \( \lambda = \frac{c}{f} \)

Substitute the given frequency \(f = 7.5 \times 10^{14} \, \text{Hz}\) into the equation to find your wavelength in meters.
After calculating, you may need to convert the units to nanometers, because the size of wavelengths in the visible spectrum is typically expressed in this unit. Since \(1 \text{ meter} = 1 \times 10^{9} \text{ nm}\), simply multiply your result by \(1 \times 10^{9}\) to switch from meters to nanometers. With this process, you will find that the wavelength for blue light is around the range typical for this color in the visible spectrum.

Photon Energy

Photon energy, the energy carried by a single photon, can be calculated using Planck's equation. This equation beautifully ties together the frequency of the photon and its energy with the Planck's constant \(h\), an essential constant in quantum mechanics that is approximately \(6.63 \times 10^{-34} \, \text{Js}\).
The formula is:

• \( E = hf \)
• Here, \(E\) is the energy, \(h\) is Planck's constant, and \(f\) is the frequency.

By inserting the given frequency of the blue light \(7.5 \times 10^{14} \, \text{Hz}\), into this formula, you will be able to calculate the energy in joules for each photon.
This calculation indicates that even seemingly tiny wavelengths of light comprise photons with a notable amount of energy when considered at the quantum level.

Frequency and Wavelength Relationship

Exploring the relationship between frequency and wavelength is vital for understanding how light behaves. Both of these properties are inherently linked by the formula \(c = \lambda f\), emphasizing that light wave speed is a product of its wavelength and frequency.
Here’s some insight:

• As the frequency of light increases, the wavelength becomes shorter. This is because the speed of light \(c\) stays constant.
• In the context of the visible spectrum, higher frequency light (like blue light) will have a shorter wavelength than lower frequency light (like red light).
• Therefore, different colors of light correspond to varying frequencies and wavelengths.

Understanding this relationship helps explain why light scatters differently in various mediums, such as why the sky appears blue. Shorter wavelengths of blue light scatter much more efficiently than the longer wavelengths of red light as they pass through the atmosphere, hence the blue appearance of the sky.
This intricate dance between frequency and wavelength is fundamental to the nature of electromagnetic waves, including visible light.