Question

1. light with a wavelength of 525 nm is green. calculate the energy in joules for a green light photon. 2. ultraviolet radiation has a frequency of 6.8×10^{18} hz. calculate the energy, in joules, of the photon. 3. find the energy, in joules per photon, of microwave radiation with a frequency of 7.91×10^{10} hz. 4. a sodium vapor lamp emits light photons with a wavelength of 5.89×10^{-7} m. what is the energy of these photons? 5. what is the wavelength of uv light that has an energy of 1.03×10^{-18}?

Explanation:

Step1: Recall the formula for photon energy

The energy of a photon is given by \(E = h
u\), where \(h = 6.626\times10^{-34}\space J\cdot s\) (Planck's constant) and \(
u\) is the frequency of the photon. Also, \(c=\lambda
u\) (where \(c = 3\times10^{8}\space m/s\) is the speed of light and \(\lambda\) is the wavelength), so \(
u=\frac{c}{\lambda}\). Then \(E=\frac{hc}{\lambda}\).

Step2: Calculate the energy for problem 1

Given \(\lambda = 525\space nm=525\times10^{-9}\space m\).
Substitute into \(E=\frac{hc}{\lambda}\):
\(E=\frac{6.626\times 10^{-34}\times3\times 10^{8}}{525\times 10^{-9}}\)
\(E=\frac{19.878\times10^{-26}}{525\times10^{-9}}\)
\(E = 3.786\times10^{-19}\space J\)

Step3: Calculate the energy for problem 2

Given \(
u=6.8\times 10^{18}\space Hz\).
Substitute into \(E = h
u\):
\(E=6.626\times10^{-34}\times6.8\times10^{18}\)
\(E=(6.626\times6.8)\times10^{-34 + 18}\)
\(E = 45.0568\times10^{-16}=4.506\times10^{-15}\space J\)

Step4: Calculate the energy for problem 3

Given \(
u = 7.91\times10^{10}\space Hz\).
Substitute into \(E=h
u\):
\(E=6.626\times10^{-34}\times7.91\times10^{10}\)
\(E=(6.626\times7.91)\times10^{-34+10}\)
\(E = 52.41166\times10^{-24}=5.241\times10^{-23}\space J\)

Step5: Calculate the energy for problem 4

Given \(\lambda=5.89\times 10^{-7}\space m\).
Substitute into \(E=\frac{hc}{\lambda}\):
\(E=\frac{6.626\times 10^{-34}\times3\times 10^{8}}{5.89\times 10^{-7}}\)
\(E=\frac{19.878\times10^{-26}}{5.89\times10^{-7}}\)
\(E=3.375\times10^{-19}\space J\)

Step6: Calculate the wavelength for problem 5

Given \(E = 1.03\times10^{-18}\space J\).
From \(E=\frac{hc}{\lambda}\), we can solve for \(\lambda\): \(\lambda=\frac{hc}{E}\)
Substitute \(h = 6.626\times10^{-34}\space J\cdot s\), \(c = 3\times10^{8}\space m/s\) and \(E = 1.03\times10^{-18}\space J\)
\(\lambda=\frac{6.626\times 10^{-34}\times3\times 10^{8}}{1.03\times 10^{-18}}\)
\(\lambda=\frac{19.878\times10^{-26}}{1.03\times10^{-18}}\)
\(\lambda = 1.93\times10^{-7}\space m = 193\space nm\)

Answer:

1. \(3.79\times10^{-19}\space J\)
2. \(4.51\times10^{-15}\space J\)
3. \(5.24\times10^{-23}\space J\)
4. \(3.38\times10^{-19}\space J\)
5. \(193\space nm\)