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Chapter 36: Problem 58

An Einstein ( \(E\) ) is a unit of measurement equal to Avogadro's number \(\left(6.02 \cdot 10^{23}\right)\) of photons. How much en ergy is contained in 1 Einstein of violet light \((\lambda=400 . \mathrm{nm}) ?\)

Short Answer

Expert verified
Answer: The energy contained in 1 Einstein of violet light with a wavelength of 400 nm is approximately \(2.99 \cdot 10^5\, \text{J}\).

Step by step solution

01

Calculate the frequency of violet light

In order to find the frequency of violet light, we can use the formula \(f = \dfrac{c}{\lambda}\). Given that the wavelength of violet light is 400 nm, we can convert this to meters: \(\lambda = 400 \cdot 10^{-9} \, \text{m}\) Now, we can substitute the values of \(\lambda\) and \(c = 3 \cdot 10^{8} \, \text{m/s}\) into the formula for the frequency: \(f = \dfrac{3 \cdot 10^{8}\, \text{m/s}}{400 \cdot 10^{-9}\, \text{m}} = 7.5 \cdot 10^{14}\, \text{Hz}\)
02

Calculate the energy of a single photon

To find the energy of a photon, we can use the equation \(E = h \cdot f\) where \(h\) is Planck's constant, approximately equal to \(6.63 \cdot 10^{-34}\, \text{Js}\). We now have the frequency \(f = 7.5 \cdot 10^{14}\, \text{Hz}\), so we can compute the energy of a single photon: \(E = (6.63 \cdot 10^{-34}\, \text{Js}) \cdot (7.5 \cdot 10^{14}\, \text{Hz}) = 4.97 \cdot 10^{-19}\, \text{J}\)
03

Calculate the total energy in 1 Einstein of photons

Now, since 1 Einstein is equal to Avogadro's number of photons, we can calculate the total energy in 1 Einstein of photons by multiplying the energy of a single photon by Avogadro's number, approximately equal to \(6.02 \cdot 10^{23}\) photons: \(E_{\text{total}} = (4.97 \cdot 10^{-19}\, \text{J}) \cdot (6.02 \cdot 10^{23}\, \text{photons}) = 2.99 \cdot 10^5\, \text{J}\) Therefore, the en energia contained in 1 Einstein of violet light is approximately \(2.99 \cdot 10^5\, \text{J}\).

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Most popular questions from this chapter

The Solar Constant measured by Earth satellites is roughly \(1400 . \mathrm{W} / \mathrm{m}^{2}\). Though the Sun emits light of different wavelengths, the peak of the wavelength spectrum is at \(500 . \mathrm{nm}\) a) Find the corresponding photon frequency. b) Find the corresponding photon energy. c) Find the number flux of photons arriving at Earth, assuming that all light emitted by the Sun has the same peak wavelength.

Compton used photons of wavelength \(0.0711 \mathrm{nm} .\) a) What is the wavelength of the photons scattered at \(\theta=180 .\) ? b) What is energy of these photons? c) If the target were a proton and not an electron, how would your answer in (a) change?

Calculate the wavelength of a) a \(2.00 \mathrm{eV}\) photon, and b) an electron with kinetic energy \(2.00 \mathrm{eV}\).

A photovoltaic device uses monochromatic light of wavelength 700 . \(\mathrm{nm}\) that is incident normally on a surface of area \(10.0 \mathrm{~cm}^{2}\). Calculate the photon flux rate if the light intensity is \(0.300 \mathrm{~W} / \mathrm{cm}^{2}\).

After you told him about de Broglie's hypothesis that particles of momentum \(p\) have wave characteristics with wavelength \(\lambda=h / p\), your 60.0 -kg roommate starts thinking of his fate as a wave and asks you if he could be diffracted when passing through the 90.0 -cm-wide doorway of your dorm room. a) What is the maximum speed at which your roommate can pass through the doorway in order to be significantly diffracted? b) If it takes one step to pass through the doorstep, how long should it take your roommate to make that step (assume the length of his step is \(0.75 \mathrm{~m}\) ) in order for him to be diffracted? c) What is the answer to your roommate's question? Hint: Assume that significant diffraction occurs when the width of the diffraction aperture is less that 10.0 times the wavelength of the wave being diffracted.
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