calculate the energy difference for a transition in the paschen series for a transition from the higher energy shell n=5?

Rydberg constant, R = 1.097 × 10⁷ /m
Speed of light, c = 3.00 × 10⁸ m/s
Planck constant, h = 6.626 × 10⁻³⁴ J s

The transition from n₂ = 5 to n₁ = 3

Energy difference for a transition, E = hc/λ
also, Rydberg formula: 1/λ = R [(1/n₁²) - (1/n₂²)]

Energy difference for a transition, E
= hcR [(1/n₁²) - (1/n₂²)]
= (6.626 × 10⁻³⁴) × (3.00 × 10⁸) × (1.097 × 10⁷) × [(1/3²) - (1/5²)] J
= 1.55 × 10⁻¹⁹ J

The Paschen series involves electrons falling to n=3 from higher levels. You can use the Rydberg equation to calculate the wavelength of the photon emitted in this transition and then calculate the energy change. For a hydrogen atom:

1/wavelength = 1.09678X10^7 m^-1 (1/n^2f - 1/n^2i)

For the transition from n=5 to n = 3:
1/wavelength = 1.09678X10^7 m^-1 (1/9 - 1/25) = 7.800X10^5 m^-1
wavelength = 1.282X10^-6 m

E = hc/wavelength
E = 6.626X10^-34 Js (2.998X10^8 m/s) / 1.282X10^-6 m
E = 1.549X10^-19 J