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Chapter 6: Problem 21

(a) A red laser pointer emits light with a wavelength of \(650 \mathrm{~nm}\). What is the frequency of this light? (b) What is the energy of 1 mole of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state they lose the excess energy in the form of \(650 \mathrm{~nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Short Answer

Expert verified
The frequency of the laser light is \(4.62 \times 10^{14} \, Hz\). The energy of 1 photon is \(3.06 \times 10^{-19} \, J\), and the energy of 1 mole of photons is \(1.84 \times 10^5 \, J/mol\). The energy gap between the ground state and the excited state in the laser material is \(3.06 \times 10^{-19} \, J\).

Step by step solution

01

Calculate the frequency of the laser light

To find the frequency of light, we use the speed of light (c) formula: c = 位谓, where 位 is the wavelength in meters and 谓 is the frequency. We are given the wavelength as 650 nm (nanometers), so we need to convert it to meters first: \(650 \, nm * 1 \times 10^{-9} = 6.50 \times 10^{-7} m\) Now we can use the speed of light formula: \(谓 = \frac{c}{位}\) \(谓 = \frac{3.00 \times 10^8 \, m/s}{6.50 \times 10^{-7} \, m} \)
02

Find the frequency

Calculate the frequency: \(谓 = 4.62 \times 10^{14} \, Hz\) The frequency of the laser light is \(4.62 \times 10^{14} \, Hz\).
03

Calculate the energy of 1 photon

To find the energy of 1 photon, we use the Planck's constant (h) and the frequency: E = h谓, where E represents the energy and h is Planck's constant, \(6.626 \times 10^{-34} \, Js\). Calculate the energy of 1 photon: \(E = (6.626 \times 10^{-34} \, Js)(4.62 \times 10^{14} \, Hz)\)
04

Find the energy of 1 photon

Calculate the energy: \(E = 3.06 \times 10^{-19} \, J\) The energy of 1 photon is \(3.06 \times 10^{-19} \, J\).
05

Calculate the energy of 1 mole of photons

To find the energy of 1 mole of photons, we multiply the energy of 1 photon by Avogadro's number (\(6.022 \times 10^{23} \, mol^{-1}\)): Energy of 1 mole of photons = Energy of 1 photon * Number of particles in 1 mole Energy of 1 mole of photons = \(3.06 \times 10^{-19} \, J \times 6.022 \times 10^{23} \, mol^{-1}\)
06

Find the energy of 1 mole of photons

Calculate the energy of 1 mole of photons: Energy of 1 mole of photons = \(1.84 \times 10^5 \, J/mol\) The energy of 1 mole of photons is \(1.84 \times 10^5 \, J/mol\).
07

Calculate the energy gap

The energy gap between the ground state and the excited state in the laser material is equal to the energy of 1 photon: Energy gap = Energy of 1 photon Energy gap = \(3.06 \times 10^{-19} \, J\) The energy gap between the ground state and the excited state in the laser material is \(3.06 \times 10^{-19} \, J\).

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Most popular questions from this chapter

Sodium metal requires a photon with a minimum energy of \(4.41 \times 10^{-19} \mathrm{~J}\) to emit electrons. (a) What is the minimum frequency of light necessary to emit electrons from sodium via the photoelectric effect? (b) What is the wavelength of this light? (c) If sodium is irradiated with light of \(439 \mathrm{~nm}\), what is the maximum possible kinetic energy of the emitted electrons? (d) What is the maximum number of electrons that can be freed by a burst of light whose total energy is \(1.00 \mu \mathrm{J} ?\)

(a) What is the relationship between the wavelength and the frequency of radiant energy? (b) Ozone in the upper atmosphere absorbs energy in the \(210-230-\mathrm{nm}\) range of the spectrum. In what region of the electromagnetic spectrum does this radiation occur?

Arrange the following kinds of electromagnetic radiation in order of increasing wavelength: infrared, green light, red light, radio waves, X-rays, ultraviolet light.

For each of the following electronic transitions in the hydrogen atom, calculate the energy, frequency, and wavelength of the associated radiation, and determine whether the radiation is emitted or absorbed during the transition: (a) from \(n=4\) to \(n=1\), (b) from \(n=5\) to \(n=2,(\mathrm{c})\) from \(n=3\) to \(n=6\). Does any of these transitions emit or absorb visible light?

Calculate the uncertainty in the position of (a) an electron moving at a speed of \((3.00 \pm 0.01) \times 10^{5} \mathrm{~m} / \mathrm{s},(\mathrm{b})\) a neutron moving at this same speed. (The masses of an electron and a neutron are given in the table of fundamental constants in the inside cover of the text.) (c) What are the implications of these calculations to our model of the atom?
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