One type of electromagnetic radiation has a frequency of 107.1 \(\mathrm{MHz}\) , another type has a wavelength of \(2.12 \times 10^{-10} \mathrm{m},\) and another type of electromagnetic radiation has photons with energy equal to \(3.97 \times 10^{-19} \mathrm{J} / \mathrm{photon}\) . Identify each type of electromagnetic radiation and place them in order of increasing photon energy and increasing frequency.

Short Answer

Expert verified
The electromagnetic radiations can be ordered as follows: Order of increasing photon energy: 1. Type 1 (\( E_1 \approx 4.76 \times 10^{-24} \mathrm{J} \)) 2. Type 3 (\( E_3 = 3.97 \times 10^{-19} \mathrm{J/photon} \)) 3. Type 2 (\( E_2 \approx 9.94 \times 10^{-16} \mathrm{J} \)) Order of increasing frequency: 1. Type 1 (\( f_1 = 107.1 \times 10^6 \mathrm{Hz} \)) 2. Type 3 (\( f_3 \approx 5.99 \times 10^{14} \mathrm{Hz} \)) 3. Type 2 (\( f_2 \approx 1.42 \times 10^{18} \mathrm{Hz} \))

Step by step solution

01

Type 1: Frequency Given

We are given the frequency of type 1 electromagnetic radiation: \( f_1 = 107.1 \mathrm{MHz} \), we need to convert this into Hz by multiplying by \( 10^6 \). So, \( f_1 = 107.1 \times 10^6 \mathrm{Hz} \). Now we can find the wavelength using the equation: \( f \times \lambda = c \). \( \lambda_1 = \frac{c}{f_1} = \frac{3 \times 10^8 \mathrm{m/s} }{107.1 \times 10^6 \mathrm{Hz}} \) And finally, we can find the photon energy using the equation: \( E = h \times f \). \( E_1 = (6.626 \times 10^{-34} \mathrm{Js} ) \times (107.1 \times 10^6 \mathrm{Hz}) \)
02

Type 2: Wavelength Given

We are given the wavelength of type 2 electromagnetic radiation: \( \lambda_2 = 2.12 \times 10^{-10} \mathrm{m} \). Now we can find the frequency using the equation: \( f \times \lambda = c \). \( f_2 = \frac{c}{\lambda_2} = \frac{3 \times 10^8 \mathrm{m/s}}{2.12 \times 10^{-10} \mathrm{m}} \) And finally, we can find the photon energy using the equation: \( E = h \times f \). \( E_2 = (6.626 \times 10^{-34} \mathrm{Js}) \times (\frac{3 \times 10^8 \mathrm{m/s}}{2.12 \times 10^{-10} \mathrm{m}}) \)
03

Type 3: Photon Energy Given

We are given the photon energy of type 3 electromagnetic radiation: \( E_3 = 3.97 \times 10^{-19} \mathrm{J/photon} \). Now we can find the frequency using the equation: \( E = h \times f \). \( f_3 = \frac{E_3}{h} = \frac{3.97 \times 10^{-19} \mathrm{J}}{6.626 \times 10^{-34} \mathrm{Js}} \) And finally, we can find the wavelength using the equation: \( f \times \lambda = c \). \( \lambda_3 = \frac{c}{f_3} = \frac{3 \times 10^8 \mathrm{m/s}}{(\frac{3.97 \times 10^{-19} \mathrm{J}}{6.626 \times 10^{-34} \mathrm{Js}})} \) Now that we have all the information, let's compare the energies and frequencies.
04

Comparing Energies and Frequencies

We need to calculate and compare the energies \( E_1, E_2, E_3 \) and frequencies \( f_1, f_2, f_3 \) to find the order. Using a calculator, we find: - \( E_1 \approx 4.76 \times 10^{-24} \mathrm{J} \) - \( E_2 \approx 9.94 \times 10^{-16} \mathrm{J} \) - \( f_1 = 107.1 \times 10^6 \mathrm{Hz} \) - \( f_2 \approx 1.42 \times 10^{18} \mathrm{Hz} \) Order of increasing photon energy: 1. Type 1 (\( E_1 \)) 2. Type 3 (\( E_3 \)) 3. Type 2 (\( E_2 \)) Order of increasing frequency: 1. Type 1 (\( f_1 \)) 2. Type 3 (\( f_3 \)) 3. Type 2 (\( f_2 \))

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