1. Frequency: [tex]3.23\cdot 10^{20} Hz[/tex]
The energy given is the energy per mole of particles:
[tex]E=1.29\cdot 10^{11} J/mol[/tex]
1 mole contains a number of Avogadro of particles, [tex]N_A[/tex], equal to
[tex]N_A=6.022\cdot 10^{23}[/tex] particles
So, by setting the following proportion, we can calculate the energy of a single photon:
[tex]1.29 \cdot 10^{11} J/mol : 6.022 \cdot 10^{23} ph/mol = E_1 : 1 ph\\E_1 = \frac{(1.29\cdot 10^{11} J/mol)(1 ph)}{6.022\cdot 10^{23} ph/mol}=2.14\cdot 10^{-13} J[/tex]
This is the energy of a single photon; now we can calculate its frequency by using the formula:
[tex]E_1 = hf[/tex]
where
[tex]h=6.63\cdot 10^{-34} Js[/tex] is the Planck's constant
f is the photon frequency
Solving for f, we find
[tex]f=\frac{E_1}{h}=\frac{2.14\cdot 10^{-13} J}{6.63\cdot 10^{-34} Js}=3.23\cdot 10^{20} Hz[/tex]
2. Wavelength: [tex]9.29\cdot 10^{-13} m[/tex]
The wavelength of the photon is given by the equation:
[tex]\lambda=\frac{c}{f}[/tex]
where
[tex]c=3\cdot 10^8 m/s[/tex]
is the speed of the photon (the speed of light). Substituting,
[tex]\lambda=\frac{3 \cdot 10^8 m/s}{3.23\cdot 10^{20} Hz}=9.29\cdot 10^{-13} m[/tex]
