# Electrons accelerated by a potential difference of 13.02 V pass through a gas of hydrogen atoms at room temper?

Electrons accelerated by a potential difference of 13.02 V pass through a gas of hydrogen atoms at room temperature. Calculate the wavelength of light emitted with the longest possible wavelength.

I can't figure out how to use the 13.02 eV.

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

Steve:

Thanks! I used that to get the shortest possible wavelength (highest energy of the photon). So now I understand how the following equation may be use: (Ephoton = E4 - E1)

But how do I find the longest possible wavelength? I assumed it would be E2 - E1 for the least possible amount of energy, but I got 122nm and the CAPA said it was wrong. E1 - E1 = 0 because nothing would be emitted, so I'm confused.

Can I use other spectrums too? or does n have to equal 1 for the final state?

FINAL EDIT:

I have both of them now: In case anyone in the future has the same problem.

For the longest wavelength, it is simply the longest wavelength within the Paschen series (n=3). It was Ephoton = E3 - E4

For the shortest wavelegnth, follow Steve's instructions. He found possible energy of photon to equal E hydrogen - KE = 13.6-13.02 = 12.75eV, then plugged it into E=hc/L. I used wiki too, they have the wavelengths (L) on there.

### 3 Answers

- Steve4PhysicsLv 78 years agoFavorite Answer
Since the electron has been accelerated through a potential difference of 13.02 V, its kinetic energy is 13.02eV.

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

En = -13.6/n² eV (e.g. see link)

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 13.02eV electron can raise a ground state electron.

The highest possible resulting electron energy would be -13.6+13.02 = -0.58eV.

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

So the 13.06eV electron can raise a ground state electron to the n=4 level but not to the n=5 level.

(To raise a ground state electron to the n=5 level, the incident electron would need an energy of -0.544-(-13.6) = 13.056eV.)

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=4 to the n=1 level in a single transition.

Photon energy, E = E4-E1

= -0.85-(-13.6)

= 12.75eV

= 12.75 x 1.6E-19 J

= 2.04E-18J

E=hc/λ

λ = hc/E

= 6.63E-34 x 3E8 / 2.04E-18

= 9.75E-8 m (i.e. 9.75x10⁻⁸m)

- Anonymous6 years ago