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Chapter 7: Problem 20

A particular form of electromagnetic radiation has a frequency of \(8.11 \times 10^{14} \mathrm{~Hz}\). (a) What is its wavelength in nanometers? In meters? (b) To what region of the electromagnetic spectrum would you assign it? (c) What is the energy (in joules) of one quantum of this radiation?

Short Answer

Expert verified
The wavelength of this radiation in meters is approximately 0.37 meters, and in nanometers, it is approximately 370 nm. This places it in the ultraviolet category of the electromagnetic spectrum. The energy of one quantum of this radiation is approximately \(5.36 \times 10^{-19} Joules\).

Step by step solution

01

Calculate Wavelength in Meters

First, solve for wavelength in meters using the formula \(c = \lambda \times v\). Rearrange it to get \(\lambda = c/v\). Substitute \(v = 8.11 \times 10^{14} Hz\) and \(c = 3.0 \times 10^{8} m/s\) into the equation and solve to find the wavelength in meters.
02

Convert Wavelength to Nanometers

The wavelength in meters can be converted to nanometers by multiplying by \(1 \times 10^{9}\), since there are one billion (or \(1 \times 10^{9}\)) nanometers in a meter. So, \(\lambda (in nm) = \lambda (in m) \times 1 \times 10^{9}\).
03

Categorize into Electromagnetic Spectrum

With the wavelength calculated, it can be compared to the known ranges of the electromagnetic spectrum to determine which category the radiation belongs to. The electromagnetic spectrum ranges from radio waves with longest wavelengths to gamma rays with shortest wavelengths.
04

Calculate Energy

The Energy of one quantum of this radiation can be calculated using the formula \(E = h \times v\), where \(h = 6.63 \times 10^{-34} Js\) (Planck's constant) and \(v = 8.11 \times 10^{14} Hz\) (given frequency). Multiply these values to get the energy in joules.

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