Answer:
1a. The energy difference between the upper and lower energy levels is Eu - Ep = 2.21 x 10-19 J - 2.72 x 10-19 J = -5.1 x 10-20 J. Since this energy is emitted as a photon of light during stimulated emission, we can use the energy of the photon, E = hc/λ, where h is Planck's constant and c is the speed of light, to calculate the wavelength of the emitted light:
-5.1 x 10-20 J = hc/λ
λ = hc/(-5.1 x 10-20 J) ≈ 3.88 x 10-7 m or 388 nm
b. The pumping energy per electron is the energy difference between the ground state and the pumped state, which is EL - EG - Ep = (0.30 x 109 J - 0 - 2.72 x 10-19 J) = 0.30 x 109 J. However, this energy is per mole of electrons, so we need to convert it to energy per electron:
0.30 x 109 J/mol / (6.02 x 1023 electrons/mol) ≈ 5.0 x 10-14 J/electron
c. The maximum possible efficiency of the system is given by the ratio of the energy of the emitted photon to the pumping energy per electron:
Efficiency = E/(EL - EG - Ep) ≈ 1.3 x 10-10 or 0.000000013%
2 a. In a laser cavity, the standing wave pattern is established due to the constructive and destructive interference of the waves travelling in opposite directions. The condition for constructive interference is that the path difference between the two waves must be an integer multiple of the wavelength, i.e., 2L = nλ, where L is the cavity length, λ is the wavelength, and n is a whole number. Solving for λ, we get:
λ = 2L/n
But the wave must also have a node at the 100% reflecting mirror, which means that the distance from the mirror to the node must be an integer multiple of half the wavelength. Therefore, we have:
L = mλ/2
where m is a whole number. Substituting this expression for L in the previous equation, we get:
λ = 2mL/n
But we know that m = n/2 for a node at the other mirror, so we can substitute this expression for m to get:
λ = 4L/n
Substituting the given values, we get:
λ = 4(0.2050 mm)/n = 0.8200/n μm
b. To determine whether a stationary wave of wavelength 821.0 nm can exist in the cavity, we need to check whether it satisfies the condition derived in part (a). Substituting λ = 821.0 nm = 0.8210 μm, we get:
n = 4(0.2050 mm)/0.8210 μm = 976.5
Since n must be a whole number, we conclude that a stationary wave of wavelength 821.0 nm cannot exist in the cavity.
c. To find the next highest wavelength of stationary wave above 820 nm that can exist in the cavity, we can simply plug in the next highest value of n. Since n must be a whole number, we take n = 30 to get:
λ = 4(0.2050 mm)/30 = 0.0273 mm = 27.3 μm
3 .Sure! Here is an energy level diagram for the given gas laser with the energies labeled in both electron volts (eV) and joules (J):
| Laser Emission (500 nm)
E3 = 0 eV | 1.6 x 10^-18 J (Pumped state)
|
E2 = 4.8 eV| 7.7 x 10^-19 J (Collisional decay)
|
E1 = 6.4 eV| 1.03 x 10^-18 J (Ground state)
The ground state E1 is the lowest energy level, at 6.4 eV or 1.03 x 10^-18 J. The pumped state E3 is 1.6 x 10^-18 J above the ground state, and rapidly decays by collision and the loss of 4.8 eV (E3 to E2). This collisional decay releases energy in the form of heat. When an atom in the pumped state decays to the lower energy level E2, it emits energy in the form of a photon with a wavelength of 500 nm, which is in the visible range of the electromagnetic spectrum.
I hope this helps!