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Chapter 4: Problem 154

What is the wavelength in nanometers of electromagnetic waves that have an energy of \(1.00 \times 10^{-15} \mathrm{~J} ?\)

Short Answer

Expert verified
The wavelength of the electromagnetic waves with an energy of \(1.00 \times 10^{-15} \mathrm{~J}\) is approximately 198.9 nm.

Step by step solution

01

Identify the given values

First, we need to identify the given values: Energy, E = \(1.00 \times 10^{-15} \mathrm{~J}\) Next, we also need some constant values: Planck's constant, h = \(6.626 \times 10^{-34} \mathrm{~Js}\) Speed of light, c = \(3.0 \times 10^{8} \mathrm{~m/s}\)
02

Use the energy of a photon formula to find frequency

Now we write down the energy of a photon formula and solve for the frequency (f): E = hf We need to find f, so rearrange the formula: f = E / h Now, plug in the given values for E and h: f = \((1.00 \times 10^{-15} \mathrm{~J}) / (6.626 \times 10^{-34} \mathrm{~Js})\) Calculate the frequency: f ≈ \(1.509 \times 10^{18} \mathrm{~Hz}\)
03

Use the speed of light formula to find the wavelength

Now we use the speed of light formula and solve for the wavelength (λ): c = λf We need to find λ, so rearrange the formula: λ = c / f Now, plug in the calculated value for frequency and the constant speed of light: λ = \((3.0 \times 10^{8} \mathrm{~m/s}) / (1.509 \times 10^{18} \mathrm{~Hz})\) Calculate the wavelength: λ ≈ \(1.989 \times 10^{-10} \mathrm{~m}\)
04

Convert the wavelength to nanometers

We have found the wavelength in meters; now we just need to convert it to nanometers (nm): 1 m = \(1 \times 10^{9} \mathrm{~nm}\) So, multiply the wavelength in meters by \(1\times 10^9\) to convert to nanometers: Wavelength = \((1.989 \times 10^{-10} \mathrm{~m}) \times (1 \times 10^{9} \mathrm{~nm/m})\) Wavelength ≈ 198.9 nm The wavelength of the electromagnetic waves with an energy of \(1.00 \times 10^{-15} \mathrm{~J}\) is approximately 198.9 nm.

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