Final answer:
The wavelength of microwave radiation with an energy of 1.91 x 10^-23 J is approximately 1042 cm, calculated using the Planck-Einstein relation and converting the result to centimeters.
Step-by-step explanation:
To calculate the wavelength of radiation from its energy we can use the Planck-Einstein relation which connects the energy of a photon (E) to its frequency (ν) and wavelength (λ) through the equation E = hν, with h being Planck's constant (6.626 x 10-34 J·s).
The speed of light, c, gives us a relation between the wavelength and frequency (ν = c/λ), which can be rearranged to λ = c/ν. Substituting the expression for frequency we have λ = hc/E.
Using the given energy (1.91 x 10-23 J), Planck's constant, and the speed of light (which is approximately 3.00 x 108 m/s), we can calculate the wavelength in meters, and then convert it to centimeters:
λ = (6.626 x 10-34 J·s)(3.00 x 108 m/s) / (1.91 x 10-23 J) = 1.04 x 10-2 m = 1.04 cm.
Therefore, the wavelength of microwave radiation with an energy of 1.91 x 10-23 J is approximately 1042 cm.