Energy levels of hydrogen atom

For hydrogen, the allowed energy levels are given by $E_n = -\frac{13.6}{n^2}\,\text{eV}, \qquad n=1,2,3,\dots$

Ground state and excited states

• Ground state: $n=1$ $E_1 = -13.6\,\text{eV}$
• Excited states: $n=2,3,4,\dots$

Check the options

Option A: $-6.8$ eV

$-\frac{13.6}{n^2} = -6.8$ $\frac{13.6}{n^2} = 6.8$ $n^2 = 2$ This gives $n=\sqrt{2}$, which is not allowed. So this is not a possible energy level.

Option B: $-3.4$ eV

$E_2 = -\frac{13.6}{2^2} = -\frac{13.6}{4} = -3.4\,\text{eV}$ This is a valid hydrogen energy level and corresponds to the first excited state.

Option C: $+6.8$ eV

Bound-state energies of hydrogen are always negative. Positive energy means the electron is free (ionized), not in a bound excited state. So this is not a possible excited state energy.

Option D: $+13.6$ eV

Again, positive energy corresponds to an unbound electron, not a bound excited state. So this is not a possible excited state energy.

Conclusion

The only possible excited-state energy of hydrogen among the given options is $\boxed{-3.4\,\text{eV}}$ which is Option B.