It takes #7.21 * 10^-19# #J# of energy to remove an electron from an iron atom. What is the maximum wavelength of light that can do this?

The idea here is that the energy of a photon is proportional to its frequency.

Simply put, photons that have a high frequency will also have a high energy.

Mathematically, this relationship between the energy of a photon and its frequency is described by the Planck - Einstein equation

#color(blue)(|bar(ul(color(white)(a/a)E = h * nucolor(white)(a/a)|)))" "#. where

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

Now, notice that the problem is asking you for the wavelength of the photon. As you know, frequency and wavelength have an Inverse relationship described by the equation

#color(blue)(|bar(ul(color(white)(a/a)nu * lamda = c color(white)(a/a)|)))" "#, where

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

So, this relationship tells you that long wavelengths are associated with low frequencies, and short wavelengths are associated with high frequencies.

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This means that a maximum wavelength of light implies a minimum frequency, which in turn implies a minimum energy associated with the photon.

You will have

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

Plug this into the Planck - Einstein equation to get

#E = h * c/(lamda)#

Rearrange to solve for #lamda# and plug in your values to find

#lamda = (h * c)/E#

#lamda = (6.626 * 10^(-34)color(red)(cancel(color(black)("J")))color(red)(cancel(color(black)("s"))) * 3 * 10^8"m" color(red)(cancel(color(black)("s"^(-1)))))/(7.21 * 10^(-19)color(red)(cancel(color(black)("J")))) = 2.76 * 10^(-7)"m"#

If you want, you can express this value in nanometers by using the conversion factor

#"1 m" = 10^9"nm"#

The answer will thus be

#lamda = color(green)(|bar(ul(color(white)(a/a)"276 nm"color(white)(a/a)|)))#

This wavelength places you in the ultraviolet - shortwave UV region of the EM spectrum.

![http://www.watertreatmentguide.com/ultraviolet_systems.htm]()