Energy Levels

The energy of the $n$th level of hydrogen is given in NCERT as $E_n=-\frac{13.6}{n^2}$ eV, a set of discrete negative values reflecting the electron's binding to the nucleus. When the electron falls from a higher to a lower level, the photon carries away exactly the difference $E_{photon}=E_{high}-E_{low}$. Here $E_3=-\frac{13.6}{9}=-1.51$ eV and $E_2=-\frac{13.6}{4}=-3.40$ eV, so the emitted photon energy is $-1.51-(-3.40)=1.89$ eV. The value 10.2 eV is wrong because that is the $n=2\to n=1$ transition, the first Lyman line, not a Balmer transition. The value 12.09 eV is wrong because it corresponds to the $n=3\to n=1$ jump. The value 3.40 eV is wrong because it is merely the magnitude of $E_2$, not a difference between levels, so it confuses a single level's energy with a transition. As stated in NCERT Class 12, Chapter 12 (Atoms), transitions ending at $n=2$ form the visible Balmer series, and 1.89 eV lies in the red (H-alpha) region of the spectrum. Because the levels crowd together as $n$ grows, jumps between higher levels release less energy than jumps down to the ground state, which is why Balmer lines are visible while Lyman lines are ultraviolet. A magnitude check confirms the photon energy is smaller than the $10.2$ eV Lyman-alpha photon, as expected for a transition between the more closely spaced higher levels.