What is the wavelength?

Question

(a) What is the wavelength (in nm) of radiation that has an energy content of 

8.01 × 10³ kJ/mol?

(b) In which region of the electromagnetic spectrum is this radiation found?

Answer 1

A) There are 2 important formulas to know in order to answer this question:

1. c = λ*ν
2. E = h*ν

We don't have a direct formula to figure out the wavelength (λ) based on the energy (E) of the radiation. However we can do some algebra on our first formula to obtain:

1. c = λ*ν → c/λ = v

With this we can then substitute for frequency (v) in our second formula:

1. E = h*ν → E = h*(c/λ)

Now we have a formula that has one variable we know, energy (E), two constants we know, plank's constant (h) and speed of light (c), and our variable we want to find, wavelength (λ). We perform algebra one more time to isolate the wavelength variable (λ):

1. E = h*(c/λ) → λ = (h*c)/E

We are almost ready to plugin for our known values into our final equation: E = 8.01*10³ kJ/mol, c = 3.00*10⁸ m/s, and h = 6.626 x 10^–34 (J*s)/photon. Notice that our E units are in kJ/mol. In order to find the wavelength of a single photon, we must convert our E units into J/photon.

E = 8.01*10³ kJ/mol * (10³ J/ 1kJ) * (1 mol photons/ 6.022*10²³ photons) = 1.33*10^-17 J/photon

E = 1.33*10^-17 J/photon

Finally, we can plugin the values into our equation and find the wavelength of the radiation energy measured:

λ = (h*c)/E = [ (6.262*10^-34 (J*s)/photon) * (3.00 * m/s) ] / 1.33*10^-17 J/photon = 1.41*10^-8 m

Now, we change our answer into nano-meters (nm):

λ = 1.41*10^-8 m * (10⁹ nm/ 1 m) = 1.41*10¹ nm

λ = 14.1 nm

B) Comparing the wavelength of our radiation to the electromagnetic radiation spectrum we find that 14.1 nm corresponds to X-Rays.