Question

Consider a hydrogen atom following the Bohr model.

A) What are the four lowest energy levels?

B) How many different types of photons are observed as transitions between these levels?

C) What are the energies of the photons that would be part of an absorption spectrum assuming that the hydrogen is in its ground state?

Please show work! Thanks!

Answer 1

Le tme begin with a little theoric mark of the bohr model:

• Bohr began with the assumption that electrons were orbiting the nucleus, much like the earth orbits the sun.
• From classical physics, a charge traveling in a circular path should lose energy by emitting electromagnetic radiation
• If the "orbiting" electron loses energy, it should end up spiraling into the nucleus (which it does not). Therefore, classical physical laws either don't apply or are inadequate to explain the inner workings of the atom
• Bohr borrowed the idea of quantized energy from Planck
• He proposed that only orbits of certain radii, corresponding to defined energies, are "permitted"
• An electron orbiting in one of these "allowed" orbits:
  • Has a defined energy state
  • Will not radiate energy
  • Will not spiral into the nucleus

According to this, the equation he use to determine the Energy of an atom of Hydrogen is:

En = -Rh x (1/n²) where "n" is the quantum number and corresponds to the different allowed orbits for the electron. Rh is the rydberg constant, and the value for that is -2.18x10^-18 J or if you don't want to use this value in J, you can do the conversion by simply multiplying that value for 6.241509 x 10¹⁸ eV. If you do that:

Rh = -2.18x10^-18 x 6.241509x10¹⁸ = -13.6 eV

Now, the four lowest energy levels are (Use the equation 1):

E1 = -13.6 x (1/1) = -13.6 eV

E2 = -13.6 x (1/2²) = -3.4 eV

E3 = -13.6 x (1/3²) = -1.511 eV

E4 = -13.6 x (1/4²) = -0.85 eV

Now, the energies of the photons are calculated for this equation (equation 2)

Eph = Em - En where "m" is an energy level lower than "n".

So for this case, we can actually say that if Hydrogen is in it's ground state then:

Eph1 = E1 - E4 = -13.6 - (-0.85) = -12.75 eV

Eph = E1 - E3 = -13.6 - (-1.511) = -12.089 eV

Eph3 = E1 - E2 = -13.6 - (-3.4) = -10.2 eV

If there is something else you need, or something to add, or to correct, tell me in the comments and I'll add or edit my answer for you.