Answer: 3.9 x 1016 hertz
First, we have to change 7.66 nm to units we understand! As said above “nm” stands for nanometer, which is one billionth of a meter, so 7.66 nm = 7.66 x 10-9 meters. f = c/λ = 3 x 108 m/s / 7.66 x 10-9 m = 3.9 x 1016 /sec = 3.9 x 1016 hertz
The light for methane is really beating fast! Look back at the EMS graphic above and you’ll see that a frequency of 1016 cycles/sec is near the middle of the ultraviolet (UV). To detect the 7.66 μm CH4 spectral line requires observing in the ultraviolet, a part of the spectrum that is blocked by the Earth’s atmosphere so space telescopes are required.
Radio waves, which have the lowest frequency, oscillate "only" 1 million times each second, and at the high frequency end of the EMS, gamma rays vibrate up and down 1020 times a second. These rates are very different from sound waves, which are typically at hundreds to thousands of vibrations per second. The physics of electromagnetism is something we can calculate but the rates and sizes of waves are beyond our normal experiences. Except that we do see light, and we detect UV with our skin – too much UV and we get sunburned.
Answer: 1.4 x 109 hertz
f = c/λ = 3 x 108 m/s / 2.1 x 10-1 m = 1.4 x 109 hertz
Discovery of Infrared Radiation (IR)
The first indication that there was more than just visible light came in 1800 when the astronomer William Herschel was experimenting with the temperature of different frequencies of light. He laid out a series of thermometers behind a prism that broke the sun’s light into a rainbow of colors. He positioned the thermometers so that the different colors of light from the prism would fall on them. He then recorded the temperature of each thermometer. One thermometer was outside the range of the spectrum cast by the prism. It sat just beyond the edge of the red part of the spectrum. Herschel wanted to use this thermometer as a control data point showing what the temperature would read if no light fell on the thermometer. To his great surprise, that thermometer recorded a temperature higher than any of the others! There must be invisible light beyond the red part of the spectrum. Herschel had discovered infrared radiation.
Image: Herschel discovering infrared radiation - from CalTech.