Calculate the energy of one mole of photon of radiation of:a) Frequency 4.6 GHzb) Wave numbers 37,000 cm-1c) What is the energy of one mole of these photons

To calculate the energy of one mole of photons, we can use the equation E = hν, where E is the energy, h is Planck's constant (6.626 x 10^-34 J·s), and ν is the frequency of the radiation.

a) Frequency 4.6 GHz: First, we need to convert the frequency from gigahertz (GHz) to hertz (Hz). Since 1 GHz = 10^9 Hz, the frequency is 4.6 x 10^9 Hz. Now, we can calculate the energy using the equation E = hν: E = (6.626 x 10^-34 J·s) x (4.6 x 10^9 Hz) E ≈ 3.05 x 10^-24 J

b) Wave numbers 37,000 cm^-1: To calculate the energy using wave numbers, we need to convert wave numbers to frequency. The relationship between wave number (ν̄) and frequency (ν) is given by the equation ν̄ = ν/c, where c is the speed of light (3 x 10^8 m/s). First, we need to convert wave numbers from cm^-1 to m^-1. Since 1 cm^-1 = 100 m^-1, the wave number is 37,000 x 100 m^-1 = 3.7 x 10^6 m^-1. Now, we can calculate the frequency using the equation ν̄ = ν/c: ν = (3.7 x 10^6 m^-1) x (3 x 10^8 m/s) ν ≈ 1.11 x 10^15 Hz Finally, we can calculate the energy using the equation E = hν: E = (6.626 x 10^-34 J·s) x (1.11 x 10^15 Hz) E ≈ 7.34 x 10^-19 J

c) Energy of one mole of these photons: To calculate the energy of one mole of photons, we need to multiply the energy calculated in parts a) and b) by Avogadro's number (6.022 x 10^23 mol^-1). For part a): Energy of one mole = (3.05 x 10^-24 J) x (6.022 x 10^23 mol^-1) Energy of one mole ≈ 1.84 x 10^0 J/mol

For part b): Energy of one mole = (7.34 x 10^-19 J) x (6.022 x 10^23 mol^-1) Energy of one mole ≈ 4.42 x 10^5 J/mol