Difference between a hydrogen ion and a proton

I've run into a bit of a problem on this weeks coursework.

A proton and an electron initially at rest combine to form hydrogen. Find the wavelength of the emitted photon?

So, as far as I can see, there are two ways to do this: Find the mass of an electron + a proton, find the mass of hydrogen, work out Δm and use E=mc^2 and work out wavelength from E

Secondly, I'd assume, since I thought ionised hydrogen is the same as a proton, just use E=13.6 eV then work out wavelength from that.

The problem is they give very different answers.

Now all my friends are telling me to go with the first, so I reckon that's probably the right one. But why? Is a proton different to a H+ ion?

• Your two approaches are exactly the same. 13.6 eV is the binding energy of hydrogen. What binding energy means is the difference between the mass of the bound state and the mass of the constituent parts. – Matt Reece Nov 23 '14 at 21:29
• In terms of why these 2 approaches are giving you different answers, you are looking at a difference in energy $\sim 10eV$. the mass of a proton $\sim 10^9 eV$. In order to get back that energy difference of $13.6eV$ to 3sf you will need the mass of the proton and the hydrogen atom to 10sf, and a similar level of precision with any unit conversion factors. I'm betting that is where your error occurred. Going back to $E=Mc^2$ is not the best approach to this calculation. – By Symmetry Nov 23 '14 at 21:56
• Sorry, I should add here that the answer is two orders of magnitude out - 2800 eV compared to 13.6. I used 9 sig figs: (9.10938291*10^-31)+(1.67262178*10^-27)-1.67352766*10^-27 = 5.05*10^-33 = Δm (5.05*10^-33)*(3*10^8)^2 = 4.55*10^-16 joules, or 2800 MeV – Jack Maggs Nov 23 '14 at 23:31

Finally $H^+$ is equivalent to a proton - unless you are thinking of a hydrogen nucleus of mass 2 with one proton and one neutron, but we normally write that as $D^+$.
• @Sofia - yes this is true! It would be nice to be able to build a mass spectrometer that could tell the difference between say $He^+$ and $He^{+*}$ - ground state Helium ion and excited Helium ion, but I don't think we have enough mass resolution for that at the moment. More about this topic in answer to question at physics.stackexchange.com/questions/145972/… – tom Nov 23 '14 at 23:09