Q.In the spectrum of hydrogen atom which transition will yield longest wavelength? (A) n = 2 to n = 1 (B) n = 5 to n = 4 (C) n = 7 to n = 6 (D) n = 8 to n = 7

[!TLDR] Longest wavelength means smallest photon energy, and adjacent-level energy gaps shrink as n increases -- among the given options, n=8 to n=7 has by far the smallest energy gap (a …

Wavelength and photon energy are inversely related ($E=hc/\lambda$), so the LONGEST wavelength corresponds to the SMALLEST energy difference between the two levels. Using $E_n=-13.6/n^2$ eV, compute the energy gap $\Delta E=E_m-E_n$ for each option's transition ($m\rightarrow n$, m the higher level): for n=2 to n=1 (option A), $\Delta E=13.6(1-1/4)=10.2$ eV; for n=5 to n=4 (option B), $\Delta E=13.6(1/16-1/25)\approx0.306$ eV; for n=7 to n=6 (option C), $\Delta E=13.6(1/36-1/49)\approx0.100$ eV; for n=8 to n=7 (option D), $\Delta E=13.6(1/49-1/64)\approx0.065$ eV. Option D has by far the smallest energy gap of the four, and hence yie …

Compute the transition energy $\Delta E=13.6\left(\frac{1}{n^2}-\frac{1}{m^2}\right)$ eV for each option and pick the smalles …

Assuming the transition between the two LOWEST-numbered levels (option A, n=2 to n=1) gives the longest wavelength, when in fact it is the OPPOSITE -- the largest energy gap and h …