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Chapter 6: Problem 25

(a) Calculate the energy of a photon of electromagnetic radiation whose frequency is \(2.94 \times 10^{14} \mathrm{~s}^{-1}\). (b) Calculate the energy of a photon of radiation whose wavelength is 413 \(\mathrm{nm} .\) (c) What wavelength of radiation has photons of energy $6.06 \times 10^{-19} \mathrm{~J} ?$

Short Answer

Expert verified
The energy of a photon with a frequency of \(2.94 \times 10^{14} s^{-1}\) is \(1.95 \times 10^{-19} J\). The energy of a photon with a wavelength of \(413 nm\) is \(4.81 \times 10^{-19} J\). The wavelength of radiation with photon energy of \(6.06 \times 10^{-19} J\) is \(328 nm\).

Step by step solution

01

Identify the given parameters

In this case, the given frequency (v) is \(2.94 \times 10^{14} s^{-1}\).
02

Calculate the energy

Use the formula for energy: \(E = h * v\), with Planck's constant (h) equal to \(6.63 \times 10^{-34} J∙s\). Then, the energy is \(E = (6.63 \times 10^{-34} J∙s)(2.94 \times 10^{14} s^{-1}) = 1.95 \times 10^{-19} J\). #b: Calculate the energy of a photon given the wavelength#
03

Identify the given parameters

In this case, the given wavelength (λ) is \(413 nm\). But before using this value, it should be converted to meters: \(λ = 413 × 10^{−9} m\).
04

Calculate the frequency

Use the relationship between speed of light, frequency, and wavelength: \(c = λv\). Rearrange the equation for frequency: \(v = \frac{c}{λ}\). Plug in the speed of light (c) = \(3 \times 10^{8} m/s\) and the wavelength (λ): \(v = \frac{3 \times 10^{8} m/s}{413 × 10^{-9} m} = 7.26 \times 10^{14} s^{-1}\).
05

Calculate the energy

Use the formula for energy: \(E = h \cdot v\). Then, the energy is \(E = (6.63 \times 10^{-34} J∙s)(7.26 × 10^{14} s^{-1}) = 4.81 \times 10^{-19} J\). #c: Calculate the wavelength of radiation given the photon's energy#
06

Identify the given parameters

In this case, the given energy (E) is \(6.06 \times 10^{-19} J\).
07

Calculate the frequency

Use the formula for energy: \(E = h \cdot v\). Rearrange the equation for frequency: \(v = \frac{E}{h}\). Plug in the Planck's constant (h) and the photon's energy (E): \(v = \frac{6.06 \times 10^{-19} J}{6.63 × 10^{-34} J∙s} = 9.14 \times 10^{14} s^{-1}\).
08

Calculate the wavelength

Use the relationship between the speed of light, frequency, and wavelength: \(c = λv\). Rearrange the equation for wavelength: \(λ = \frac{c}{v}\). Plug in the speed of light (c) and the frequency (v): \(λ = \frac{3 \times 10^{8} m/s}{9.14 × 10^{14} s^{-1}} = 3.28 × 10^{-7} m\), or \(328 nm\).

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Most popular questions from this chapter

One of the emission lines of the hydrogen atom has a wavelength of $94.974 \mathrm{nm}$. (a) In what region of the electromagnetic spectrum is this emission found? (b) Determine the initial and final values of \(n\) associated with this emission.

(a) What is the frequency of radiation that has a wavelength of $10 \mu \mathrm{m}\(, about the size of a bacterium? \)(\mathbf{b})$ What is the wavelength of radiation that has a frequency of $5.50 \times 10^{14} \mathrm{~s}^{-1} ?$ (c) Would the radiations in part (a) or part \((b)\) be visible to the human eye? (d) What distance does electromagnetic radiation travel in \(50.0 \mu \mathrm{s} ?\)

The electron microscope has been widely used to obtain highly magnified images of biological and other types of materials. When an electron is accelerated through a particular potential field, it attains a speed of $9.47 \times 10^{6} \mathrm{~m} / \mathrm{s}$ What is the characteristic wavelength of this electron? Is the wavelength comparable to the size of atoms?

Is energy emitted or absorbed when the following electronic transitions occur in hydrogen? (a) from \(n=3\) to \(n=2\), (b) from an orbit of radius \(0.846 \mathrm{nm}\) to one of radius 0.212 \(\mathrm{nm},(\mathbf{c})\) an electron adds to the \(\mathrm{H}^{+}\) ion and ends up in the \(n=2\) shell?

In the experiment shown schematically below, a beam of neutral atoms is passed through a magnetic field. Atoms that have unpaired electrons are deflected in different directions in the magnetic field depending on the value of the electron spin quantum number. In the experiment illustrated, we envision that a beam of hydrogen atoms splits into two beams. (a) What is the significance of the observation that the single beam splits into two beams? (b) What do you think would happen if the strength of the magnet were increased? (c) What do you think would happen if the beam of hydrogen atoms were replaced with a beam of helium atoms? Why? (d) The relevant experiment was first performed by Otto Stern and Walter Gerlach in \(1921 .\) They used a beam of \(\mathrm{Ag}\) atoms in the experiment. By considering the electron configuration of a silver atom, explain why the single beam splits into two beams.
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