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Chapter 6: Problem 24

(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 one 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
\(a)\) The frequency of the red laser light is approximately \(4.62 \times 10^{14} \ Hz\). \(b)\) The energy of one photon of the red laser light is approximately \(3.06 \times 10^{-19} \ J\). \(c)\) The energy gap between the ground state and the excited state in the laser material is approximately \(3.06 \times 10^{-19} \ J\).

Step by step solution

01

Write the given wavelength in meters.

The given wavelength is 650 nm, which is equivalent to \(6.50 \times 10^{-7} \ m\) in meters.
02

Use the speed of light formula to find the frequency.

We can find the frequency using the formula, \(c = \lambda \nu\). Since the speed of light (c) is approximately \(3.0 \times 10^8 \ m/s\), we get: \(\nu = \frac{c}{\lambda} = \frac{3.0 \times 10^8 \ m/s}{6.50 \times 10^{-7} \ m} = 4.62 \times 10^{14} \ Hz\) The frequency of the red laser light is approximately \(4.62 \times 10^{14} \ Hz\). #b. Calculating the energy of one photon#
03

Use Planck's constant to find the energy of a single photon.

The energy of a photon can be found using the formula, \(E = h \nu\). Planck's constant (h) is approximately \(6.63 \times 10^{-34} \ Js\). By substituting the values, we get: \(E = (6.63 \times 10^{-34} \ Js)(4.62 \times 10^{14} \ Hz) = 3.06 \times 10^{-19} \ J\) The energy of one photon of the red laser light is approximately \(3.06 \times 10^{-19} \ J\). #c. Finding the energy gap between the ground state and excited state#
04

Analyze the energy conservation.

The energy gap between the ground state and excited state is equal to the energy of a single photon. Thus, the energy gap is also \(3.06 \times 10^{-19} \ J\). The energy gap between the ground state and the excited state in the laser material is approximately \(3.06 \times 10^{-19} \ J\).

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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 \(405 \mathrm{nm},\) 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 \(1.00 \mu \mathrm{J}\) ?

What is the maximum number of electrons in an atom that can have the following quantum numbers: (a) \(n=2\), \(m_{s}=-\frac{1}{2},\) (b) \(n=5, l=3 ;\) (c) \(n=4, l=3, m_{l}=-3\) (d) \(n=4, l=0, m_{l}=0 ?\)

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{f}=1 .\) (a) Determine the region of the electromagnetic spectrum in which the lines of the Lyman series are observed. (b) Calculate the wavelengths of the first three lines in the Lyman series- those for which \(n_{i}=2,3,\) and 4 .

Bohr's model can be used for hydrogen-like ions -ions that have only one electron, such as \(\mathrm{He}^{+}\) and \(\mathrm{Li}^{2+}\). (a) Why is the Bohr model applicable to \(\mathrm{He}^{+}\) ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+},\) and \(\mathrm{Li}^{2+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{C}^{5+}\) ion.

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