Question

(a) A red laser pointer emits light with a wavelength of \(650 \mathrm{~nm}\). What is the frequency of this light? (b) What is the energy of 1 mole of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state they lose the excess energy in the form of \(650 \mathrm{~nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Short answer

The frequency of the laser light is \(4.62 \times 10^{14} \, Hz\). The energy of 1 photon is \(3.06 \times 10^{-19} \, J\), and the energy of 1 mole of photons is \(1.84 \times 10^5 \, J/mol\). The energy gap between the ground state and the excited state in the laser material is \(3.06 \times 10^{-19} \, J\).

Step-by-step solution

Calculate the frequency of the laser light

To find the frequency of light, we use the speed of light (c) formula: c = λν, where λ is the wavelength in meters and ν is the frequency. We are given the wavelength as 650 nm (nanometers), so we need to convert it to meters first: \(650 \, nm * 1 \times 10^{-9} = 6.50 \times 10^{-7} m\) Now we can use the speed of light formula: \(ν = \frac{c}{λ}\) \(ν = \frac{3.00 \times 10^8 \, m/s}{6.50 \times 10^{-7} \, m} \)

Find the frequency

Calculate the frequency: \(ν = 4.62 \times 10^{14} \, Hz\) The frequency of the laser light is \(4.62 \times 10^{14} \, Hz\).

Calculate the energy of 1 photon

To find the energy of 1 photon, we use the Planck's constant (h) and the frequency: E = hν, where E represents the energy and h is Planck's constant, \(6.626 \times 10^{-34} \, Js\). Calculate the energy of 1 photon: \(E = (6.626 \times 10^{-34} \, Js)(4.62 \times 10^{14} \, Hz)\)

Find the energy of 1 photon

Calculate the energy: \(E = 3.06 \times 10^{-19} \, J\) The energy of 1 photon is \(3.06 \times 10^{-19} \, J\).

Calculate the energy of 1 mole of photons

To find the energy of 1 mole of photons, we multiply the energy of 1 photon by Avogadro's number (\(6.022 \times 10^{23} \, mol^{-1}\)): Energy of 1 mole of photons = Energy of 1 photon * Number of particles in 1 mole Energy of 1 mole of photons = \(3.06 \times 10^{-19} \, J \times 6.022 \times 10^{23} \, mol^{-1}\)

Find the energy of 1 mole of photons

Calculate the energy of 1 mole of photons: Energy of 1 mole of photons = \(1.84 \times 10^5 \, J/mol\) The energy of 1 mole of photons is \(1.84 \times 10^5 \, J/mol\).

Calculate the energy gap

The energy gap between the ground state and the excited state in the laser material is equal to the energy of 1 photon: Energy gap = Energy of 1 photon Energy gap = \(3.06 \times 10^{-19} \, J\) The energy gap between the ground state and the excited state in the laser material is \(3.06 \times 10^{-19} \, J\).