Math

QuestionHow many photons are in a laser pulse with energy 7.53 mJ7.53 \mathrm{~mJ} at a wavelength of 601 nm601 \mathrm{~nm}? Use 3 sig figs.

Studdy Solution

STEP 1

Assumptions1. The wavelength of the laser pulse is 601 nm601 \mathrm{~nm} . The energy of the laser pulse is 7.53 mJ7.53 \mathrm{~mJ}
3. We use the Planck-Einstein relation to calculate the energy of a single photon =hc/λ = h \cdot c / \lambda, where $$ is the energy of a photon, $h$ is Planck's constant ($6.626 \times10^{-34} \mathrm{~J \cdot s}$), $c$ is the speed of light ($3.00 \times10^{8} \mathrm{~m/s}$), and $\lambda$ is the wavelength.
4. The number of photons is calculated by dividing the total energy by the energy of a single photon.

STEP 2

First, we need to calculate the energy of a single photon using the Planck-Einstein relation.=hcλ = \frac{h \cdot c}{\lambda}

STEP 3

Now, plug in the given values for Planck's constant, the speed of light, and the wavelength to calculate the energy of a single photon. Note that we need to convert the wavelength from nanometers to meters.
=6.626×1034 Js3.00×108 m/s601×109 m = \frac{6.626 \times10^{-34} \mathrm{~J \cdot s} \cdot3.00 \times10^{8} \mathrm{~m/s}}{601 \times10^{-9} \mathrm{~m}}

STEP 4

Calculate the energy of a single photon.
=6.626×1034 Js3.00×108 m/s601×109 m=3.31×1019 J = \frac{6.626 \times10^{-34} \mathrm{~J \cdot s} \cdot3.00 \times10^{8} \mathrm{~m/s}}{601 \times10^{-9} \mathrm{~m}} =3.31 \times10^{-19} \mathrm{~J}

STEP 5

Now that we have the energy of a single photon, we can find the number of photons in the laser pulse by dividing the total energy by the energy of a single photon.
Numberofphotons=TotalenergyEnergyofasinglephotonNumber\, of\, photons = \frac{Total\, energy}{Energy\, of\, a\, single\, photon}

STEP 6

Plug in the values for the total energy and the energy of a single photon to calculate the number of photons. Note that we need to convert the total energy from millijoules to joules.
Numberofphotons=.53×103 J3.31×1019 JNumber\, of\, photons = \frac{.53 \times10^{-3} \mathrm{~J}}{3.31 \times10^{-19} \mathrm{~J}}

STEP 7

Calculate the number of photons in the laser pulse.
Numberofphotons=7.53×103 J3.31×1019 J=2.27×1016Number\, of\, photons = \frac{7.53 \times10^{-3} \mathrm{~J}}{3.31 \times10^{-19} \mathrm{~J}} =2.27 \times10^{16}The laser pulse contains 2.27×10162.27 \times10^{16} photons.

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