Which Energy Levels in Hydrogen Produce the Red Line?

[catkin]

Homework Statement

This is from Advanced Physics by Adams and Allday, part 8 Modern Physics, Practice Exam Question 15.

• The spectrum of atomic hydrogen contains a prominent red line having a wavelength of 6.60 x 10^-7 m. Calculate the energy of a photon with this wavelength.
• The ionisation energy of hydrogen is 2.18 x 10^-18 J. The next allowed energy level above the ground state in hydrogen has an energy -5.40 x 10^-19 J. Show by calculation that the lowest energy level cannot be involved in the production of the prominent red line in a.

Homework Equations

E = h f
v = f λ

The Attempt at a Solution

f = v / λ
E = h v / λ
= 6.63E-34 * 3.00E+8 / 6.60E-7
= 3.0E-19 J ct2sf (Book gives same answer. Calculated 3.013636364e-19)

A drop from the next allowed energy level above the ground state to the ground state would release 5.40E-19 J. This does not match energy calculated in a so this drop cannot be the one that produces the red line having a wavelength of 6.60E-7 m.

A free electron dropping to the next allowed energy level above the ground state would release 2.18E-18 - 5.40E-19 = 1.64E-18 J. This is more than the energy that produces the red line having a wavelength of 6.60E-7 m so I cannot, on the available data, show that some drop down to the next allowed energy level above the ground state does not produce the red line having a wavelength of 6.60E-7.

Hmm ...

 

[Hootenanny]

I believe by lowest energy level the question means the ground state, since this is the lowest allowed energy level. If this is the case, then you can show that the observed emission line cannot possibly be cause by a transition to the ground state.

 

[catkin]

Ah! Thank you Hootenanny

Then an answer to b is:

The ground state is the lowest energy level. When an excited electron drops from the next allowed energy level to the ground state the energy given off is 2.18E-18 - 5.40E-19 = 1.64E-18 J. This is more than the energy calculated in a so this drop cannot be the one that produces the red line in a. Drops from other allowed energy levels to ground state give off more energy so also cannot produce the red line in a. Thus the lowest energy level cannot be involved in the production of the prominent red line in a.

 

[Hootenanny]

Ah! Thank you Hootenanny

Then an answer to b is:

The ground state is the lowest energy level. When an excited electron drops from the next allowed energy level to the ground state the energy given off is 2.18E-18 - 5.40E-19 = 1.64E-18 J. This is more than the energy calculated in a so this drop cannot be the one that produces the red line in a. Drops from other allowed energy levels to ground state give off more energy so also cannot produce the red line in a. Thus the lowest energy level cannot be involved in the production of the prominent red line in a.

Sounds spot on to me

 

[catkin]

Thanks

 

[Hootenanny]

Thanks

A pleasure