# Police often monitor traffic with "K-band" radar guns, which operate in the microwave region at 22.235 GHz (1 GHz=10^9 Hz). Find the wavelength in nm of this radiation.

ncchemist | Certified Educator

We need to convert 22.235 GHz into nm.  We know that energy is equal to the wavenumber of radiation (v) times Plank's constant (h):

E = hv

We also know that energy is equal to Plank's constant (h) times the speed of light (c) divided by the wavelength (lambda) of the radiation:

E = hc/lambda

Now we can equate the two sides and cancel the common term of h:

hv = hc/lambda

v = c/lambda

lambda = c/v

In other words, the wavelength of radiation is equal to the speed of light divided by the wavenumber (or frequency) of the radiation.  We are given that the frequency of the radiation is 22.235 GHz.  This is equal to 22.235 x 10^9 Hz, which is the same as 22.235 x 10^9 s^-1 (hertz is the equivalent of cycles per second).  Now divide the two numbers:

(2.99 x 10^8 m/s)/(2.235 x 10^9 s^-1) = 0.134 meters

We now convert meters to nm:

0.134 m * (1 nm/10^-9 m) = 1.34 x 10^8 nm