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Chapter 7: Problem 16

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

Expert verified
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

01

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.
02

Convert Units

Since the question asks for the wavelength in nanometers (nm), convert the calculated wavelength from meters to nanometers using the conversion factor \(1 m = 1 \times 10^9 nm\).
03

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.

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