If Rydberg's constant is R, the longest wavelength of radiation in Paschen series will be α/(7 R), where α= ________.

Solution

Physics sir · 3 steps

Longest wavelength means smallest energy jump, so in Paschen you only drop from n = 4 to n = 3.

1. Pick the transition. Every Paschen line ends on n = 3. Wavelength is longest when the energy gap is smallest, and the smallest gap is from the nearest level above, n = 4.

   1/λ = R(1/3²-1/4²) = R((16-9)/144) = 7R/144

2. Flip it. Inverting gives λₘₐₓ = 144/7R. Compare with α/7R to read off α.
3. So. α = 144.

The common slip: 63. Taking the jump from n = ∞ gives the shortest wavelength, 9/R = 63/7R, so α = 63. That is the series limit, not the longest line.

The idea to keep. In any series the lower level is fixed. The longest wavelength comes from the next level up, and the shortest comes from n = ∞.

Answer from the NTA final answer key. Solution by PrepWiser.