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Chapter 6: Problem 32

A stellar object is emitting radiation at \(3.0 \mathrm{~mm} .\) (a) What type of electromagnetic spectrum is this radiation? (b) If a detector is capturing \(3.0 \times 10^{8}\) photons per second at this wavelength, what is the total energy of the photons detected in 1 day?

Short Answer

Expert verified
(a) The radiation with a wavelength of \(3.0 \, \mathrm{mm}\) belongs to the microwave region of the electromagnetic spectrum. (b) The total energy of the photons detected in 1 day is approximately \(5.653 \times 10^{-15} \, \mathrm{J}\).

Step by step solution

01

Identify the type of electromagnetic spectrum

To identify the type of electromagnetic spectrum, we can refer to a table or a chart listing the different types and their respective wavelength ranges. Let's check the wavelength of the emitted radiation, which is \(3.0 \, \mathrm{mm}\), and compare it to the ranges of various types of electromagnetic spectrum.
02

Calculate the energy of a single photon

In order to calculate the total energy of the detected photons, we need to find the energy of a single photon first. We can use the formula for the energy of a photon: \(E = \dfrac{hc}{\lambda}\), where \(E\) is the energy of a photon, \(h \approx 6.626 \times 10^{-34} \, \mathrm{Js}\) is the Planck's constant, \(c \approx 3.0 \times 10^8 \, \mathrm{m/s}\) is the speed of light, and \(\lambda = 3.0 \, \mathrm{mm} = 3.0 \times 10^{-3}\, \mathrm{m}\) is the wavelength of the radiation.
03

Calculate the total energy of the photons detected in 1 day

Now that we have the energy of a single photon, we can calculate the total energy of the photons detected in 1 day. To do this, we need to multiply the energy of a single photon by the number of photons captured per second and by the total number of seconds in 1 day (86400 seconds): \(E_{total} = E \times N \times t\), where \(E_{total}\) is the total energy of the photons detected in 1 day, \(N = 3.0 \times 10^8 \, \mathrm{photons/s}\) is the number of photons captured per second, and \(t = 86400 \, \mathrm{s}\) is the number of seconds in 1 day. Now, let's use the given values and calculate the energy of a single photon and the total energy of photons detected in 1 day.

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Most popular questions from this chapter

A certain orbital of the hydrogen atom has \(n=4\) and \(l=3\). (a) What are the possible values of \(m_{l}\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

List the following types of electromagnetic radiation in order of descending wavelength: (a) UV lights used in tanning salons \((300-400 \mathrm{nm}) ;\) (b) radiation from an FM radio station at \(93.1 \mathrm{MHz}\) on the dial; (c) radiation from mobile phones \((450-2100 \mathrm{MHz}) ;\) (d) the yellow light from sodium vapor streetlights; (e) the red light of a light-emitting diode, such as in an appliance's display.

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 \(\mathrm{Li}^{2+}\) ions but not to neutral Li atoms? (b) The ground-state energies of \(\mathrm{B}^{4+}, \mathrm{C}^{5+},\) and \(\mathrm{N}^{6+}\) are tabulated as follows: By examining these numbers, propose a relationship between the ground-state energy of hydrogen-like systems and the nuclear charge, \(Z\). (Hint: Divide by the ground-state energy of hydrogen $\left.-2.18 \times 10^{-18} \mathrm{~J}\right)$ (c) Use the relationship you derive in part (b) to predict the ground-state energy of the \(\mathrm{Be}^{3+}\) ion.

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.

Classify each of the following statements as either true or false: (a) A hydrogen atom in the \(n=3\) state can emit light at only two specific wavelengths, \((\mathbf{b})\) a hydrogen atom in the \(n=2\) state is at a lower energy than one in the \(n=1\) state, and (c) the energy of an emitted photon equals the energy difference of the two states involved in the emission.
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