# How many photons are produced in a laser pulse of 0.210 J at 535 nm?

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Your strategy here will be to use the **Planck - Einstein relation** to calculate the energy of a *single photon* of wavelength

The Planck - Einstein relation looks like this

#color(blue)(ul(color(black)(E = h * nu)))#

Here

#E# - the energy of the photon#h# - Planck's constant, equal to#6.626 * 10^(-34)"J s"# #nu# - the frequency of the photon

Notice that this equation relates the energy of the photon to its *frequency*. More specifically, it shows that the energy of the photon is **directly proportional** to its frequency.

In simple terms, the **higher** the frequency, the **more energetic** the photon.

Now, the frequency of the photon is related to its wavelength by the equation

#color(blue)(ul(color(black)(lamda * nu = c)))#

Here

#lamda# - the wavelength of the photon#c# - the speed of light in a vacuum, usually given as#3 * 10^8"m s"^(-1)#

Rearrange to solve for

#lamda * nu = c implies nu = c/(lamda)#

Plug in your value to find -- **do not** forget to convert the wavelength from *nanometers* to *meters*

#nu = (3 * 10^8color(red)(cancel(color(black)("m")))"s"^(-1))/(535 * 10^(-9)color(red)(cancel(color(black)("m")))) = 5.6075 * 10^(14)"s"^(-1)#

Plug this value into the Planck - Einstein equation to find the energy of a single photon

#E = 6.626 * 10^(-34)"J" color(red)(cancel(color(black)("s"))) * 5.6075 * 10^(14)color(red)(cancel(color(black)("s"^(-1))))#

#E = 3.716 * 10^(-19)"J"#

Now use the total energy of the laser pulse to find how many photons were needed for this output

#0.210 color(red)(cancel(color(black)("J"))) * "1 photon"/(3.716 * 10^(-19)color(red)(cancel(color(black)("J")))) = color(darkgreen)(ul(color(black)(5.65 * 10^(17)"photons")))#

The answer is rounded to three **sig figs**.

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