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In a Franck-Hertz experiment in which mercury vapor has been replaced by atomic hydrogen, it is observed pronounced maximum in the current that circulates through the galvanometer for potential values of 10.2 V and 12.09 V; and if it analyzes spectroscopically the emergent radiation three spectral lines are observed. Explain the results.

I'm not sure about the three spectral lines if there are only two potential values; i.e for me there are only two wavelengths using the equation $ E= h\frac{c}{\lambda}$. And the energy is obtained by multiplying the potential value with the charge of the electron $E= eV$.

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In the Franck-Hertz experiment, most of the vapor is in the ground state (quantum number n=1) to begin with. The 10.2 V signal corresponds to the 10.2 eV transition from n=1 to n=2 in hydrogen. The 12.09 V signal corresponds to the 12.09 eV transition from n=1 to n=3.

The 3 spectroscopic lines in question are then for the 3 possible decays from these excited states: 3 to 2, 3 to 1 and 2 to 1.

To find the level structure yourself, check out the NIST Atomic Spectra Database Levels Form and entering "H I" (for neutral hydrogen) and changing the units to eV.

For hydrogren, it's even easier to just use the Rydberg formula, and the lines you are asking about are the Lyman series.

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