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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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The work function of a certain material is \(5.8 \mathrm{eV}\). What is the photoelectric threshold for this material?

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