Question

A) Electrons accelerated by a potential difference of 13.16 V pass through a gas of hydrogen atoms at room temperature. Calculate the wavelength of light emitted with the longest possible wavelength.

B) Calculate the wavelength of light emitted with the shortest possible wavelength.

Answer 1

(B)

Since the electron has been accelerated through a potential difference of 13.16 V, its kinetic energy is 13.16 eV.

The energy levels for hydrogen are given by the standard formula:

En = -13.6/n² eV

The n=1 level has energy E1 = -13.6/1² = -13.6eV (ground state)

We need to find the highest energy level to which the 13.16eV electron can raise a ground state electron.

The highest possible resulting electron energy would be -13.6+13.16 = -0.44eV.

The n=2 level has energy E2 = -13.6/2² = -3.4eV

The n=3 level has energy E3 = -13.6/3² = -1.5eV

The n=4 level has energy E4 = -13.6/4² = -0.85eV

The n=5 level has energy E5 = -13.6/5² = -0.544eV

The n=5 level has energy E6 = -13.6/6² = -0.378eV

So the 13.06eV electron can raise a ground state electron to the n=5 level but not to the n=6 level.

(To raise a ground state electron to the n=6 level, the incident electron would need an energy of -0.378-(-13.6) = 13.22eV.)

After the excitation, the highest energy photon which can be emitted (i.e. the shortest wavelength) will be one produced when the electron drops back from the n=5 to the n=1 level in a single transition.

Photon energy, E = E5-E1

= -0.544-(-13.6)

= 13.056eV

= 13.056 x 1.6E-19 J

= 2.09E-18J

E=hc/?

? = hc/E

= 6.63E-34 x 3E8 / 2.08E-18

= 9.52E-8 m

(A)

For longest wavelength:

Photon energy, E = E5-E4

= -0.544-(-0.85)

= 0.306 eV

= 0.306 x 1.6E-19 J

= 4.896E-20J

E=hc/?

? = hc/E

= 6.63E-34 x 3E8 / 4.896E-20

= 4.06E-6 m