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Chapter 7: Problem 42

One type of electromagnetic radiation has a frequency of \(107.1\) 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 \(3.97 \times 10^{-19} \mathrm{~J} /\) photon. Identify each type of electromagnetic radiation and place them in order of increasing photon energy and increasing frequency.

Short Answer

Expert verified
The order of increasing photon energy and increasing frequency for the given electromagnetic radiations is Radio Waves, Visible Light, and Gamma Rays.

Step by step solution

01

Find the missing values for each type of radiation

1. Radiation with frequency \(f_1 = 107.1 \mathrm{~MHz}\): We need to find the wavelength \(\lambda_1\) and energy \(E_1\): \(\lambda_1 = \cfrac{c}{f_1}\) \(E_1 = h \cdot f_1\) 2. Radiation with wavelength \(\lambda_2 = 2.12 \times 10^{-10} \mathrm{~m}\): We need to find the frequency \(f_2\) and energy \(E_2\): \(f_2 = \cfrac{c}{\lambda_2}\) \(E_2 = h \cdot f_2\) 3. Radiation with photon energy \(E_3 = 3.97 \times 10^{-19} \mathrm{~J}\): We need to find the frequency \(f_3\) and wavelength \(\lambda_3\): \(f_3 = \cfrac{E_3}{h}\) \(\lambda_3 = \cfrac{c}{f_3}\)
02

Calculate the missing values

1. Radiation with frequency \(f_1 = 107.1 \mathrm{~MHz}\): \(f_1 = 107.1 \times 10^6 \mathrm{~Hz} \) \(\lambda_1 = \cfrac{3 \times 10^8 \mathrm{~m/s}}{107.1 \times 10^6 \mathrm{~Hz}} = 2.8 \times 10^{-3} \mathrm{~m} \) \(E_1 = 6.63 \times 10^{-34} \mathrm{~Js} \cdot 107.1 \times 10^6 \mathrm{~Hz} = 7.1 \times 10^{-28} \mathrm{~J} \) 2. Radiation with wavelength \(\lambda_2 = 2.12 \times 10^{-10} \mathrm{~m}\): \(f_2 = \cfrac{3 \times 10^8 \mathrm{~m/s}}{2.12 \times 10^{-10} \mathrm{~m}} = 1.42 \times 10^{18} \mathrm{~Hz} \) \(E_2 = 6.63 \times 10^{-34} \mathrm{~Js} \cdot 1.42 \times 10^{18} \mathrm{~Hz} = 9.42 \times 10^{-16} \mathrm{~J} \) 3. Radiation with photon energy \(E_3 = 3.97 \times 10^{-19} \mathrm{~J}\): \(f_3 = \cfrac{3.97 \times 10^{-19} \mathrm{~J}}{6.63 \times 10^{-34} \mathrm{~Js}} = 5.99 \times 10^{14} \mathrm{~Hz} \) \(\lambda_3 = \cfrac{3 \times 10^8 \mathrm{~m/s}}{5.99 \times 10^{14} \mathrm{~Hz}} = 5.01 \times 10^{-7} \mathrm{~m} \)
03

Identify each type of electromagnetic radiation

Now, let's compare the values with the electromagnetic spectrum to identify each type of radiation: 1. Radiation 1: \(f_1 = 107.1 \times 10^6 \mathrm{~Hz} \) and \(\lambda_1 = 2.8 \times 10^{-3} \mathrm{~m} \) - This belongs to the "Radio Waves" category. 2. Radiation 2: \(f_2 = 1.42 \times 10^{18} \mathrm{~Hz} \) and \(\lambda_2 = 2.12 \times 10^{-10} \mathrm{~m} \) - This belongs to the "Gamma Rays" category. 3. Radiation 3: \(f_3 = 5.99 \times 10^{14} \mathrm{~Hz} \) and \(\lambda_3 = 5.01 \times 10^{-7} \mathrm{~m} \) - This belongs to the "Visible Light" category.
04

Place them in order of increasing photon energy and increasing frequency

1. Increasing photon energy: \(E_1\) (Radio Waves) < \(E_3\) (Visible Light) < \(E_2\) (Gamma Rays) 2. Increasing frequency: \(f_1\) (Radio Waves) < \(f_3\) (Visible Light) < \(f_2\) (Gamma Rays) In conclusion, the order of increasing photon energy and increasing frequency is Radio Waves, Visible Light, and Gamma Rays.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Radio Waves
Radio waves are a fascinating part of the electromagnetic spectrum. They have the longest wavelengths, usually ranging from 1 millimeter to 100 kilometers, and the lowest frequencies, from about 3 kHz to 300 GHz. This means that radio waves have less energy compared to other types of electromagnetic radiation.

Radio waves are used extensively in communication systems. Think about radios, television broadcasts, and even cell phones. These applications take advantage of radio waves' ability to travel long distances and penetrate through buildings and walls. This is why when you're driving that's far away from a radio station, you may still receive a clear radio signal.
  • Radio wavelengths are typically longer than 0.1 meters.
  • Frequencies are lower than 300 GHz.
  • They are used in various communication technologies.
Despite being the least energetic waves on the electromagnetic spectrum, radio waves play a crucial role in everyday technology. Understanding their properties helps us appreciate how they carry our favorite tunes or urgent news bulletins right to our devices.
Gamma Rays
Gamma rays are at the other end of the electromagnetic spectrum compared to radio waves. They have the shortest wavelengths, less than picometers, and the highest frequencies which exceed 10 exahertz (Hz). As a result, gamma rays carry the most energy among electromagnetic waves.

Due to their high energy, gamma rays are used in applications where penetrating power is needed. They're common in medical treatments like cancer radiotherapy, where they target and destroy harmful cells. They're also utilized in various scientific fields for detecting and analyzing cosmic phenomena. Astronomers, for instance, use gamma-ray observations to study distant galaxies or neutron stars.
  • Gamma rays have wavelengths shorter than 10 picometers.
  • Their frequencies are above 10 exahertz.
The energy and penetrating ability of gamma rays make them invaluable for both scientific research and practical applications. However, due to their high energy, they must be used carefully to avoid harmful biological effects.
Visible Light
Visible light is unique in the electromagnetic spectrum because it is the range of wavelengths that the human eye can detect. This range spans approximately 400 to 700 nanometers, which corresponds to a frequency range of around 430 to 790 terahertz (THz).

This bandwidth is responsible for all the colors we perceive in the natural world. When light hits an object, the object absorbs certain wavelengths and reflects others. The colors we see are the wavelengths that aren't absorbed by the object's material. For example, a leaf appears green because it reflects green light while absorbing other colors.
  • Visible wavelengths lie between 400 and 700 nanometers.
  • Frequencies range from 430 to 790 terahertz.
  • This spectrum allows us to perceive colors.
Visible light plays a vital role in our daily lives, influencing everything from how we perceive art and nature to how we communicate and use technology. Understanding visible light, therefore, enriches our interaction with the world around us.

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Most popular questions from this chapter

Calculate the velocities of electrons with de Broglie wavelengths of \(1.0 \times 10^{2} \mathrm{~nm}\) and \(1.0 \mathrm{~nm}\), respectively.

Are the following statements true for the hydrogen atom only, true for all atoms, or not true for any atoms? a. The principal quantum number completely determines the energy of a given electron. b. The angular momentum quantum number, \(\ell\), determines the shapes of the atomic orbitals. c. The magnetic quantum number, \(m_{\ell}\), determines the direction that the atomic orbitals point in space.

The electron affinities of the elements from aluminum to chlorine are \(-44,-120,-74,-200.4\), and \(-384.7 \mathrm{~kJ} / \mathrm{mol}\), respectively. Rationalize the trend in these values.

Defend and criticize Bohr's model. Why was it reasonable that such a model was proposed, and what evidence was there that it "works"? Why do we no longer "believe" in it?

Assume that we are in another universe with different physical laws. Electrons in this universe are described by four quantum numbers with meanings similar to those we use. We will call these quantum numbers \(p, q, r\), and \(s .\) The rules for these quantum numbers are as follows: \(p=1,2,3,4,5, \ldots\) \(q\) takes on positive odd integers and \(q \leq p\) \(r\) takes on all even integer values from \(-q\) to \(+q\). (Zero is considered an even number.) \(s=+\frac{1}{2}\) or \(-\frac{1}{2}\) a. Sketch what the first four periods of the periodic table will look like in this universe. b. Wh?t are the atomic numbers of the first four elements you would expect to be least reactive? c. Give an example, using elements in the forst four rows, of ionic compounds with the formulas \(\mathrm{XY}, \mathrm{XY}_{2}, \mathrm{X}_{2} \mathrm{Y}, \mathrm{XY}_{3}\), and \(\mathrm{X}_{2} \mathrm{Y}_{3}\) d. How many electrons can have \(p=4, q=3 ?\) e. How many electrons can have \(p=3, q=0, r=0 ?\) f. How many electrons can have \(p=6\) ?
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