Final answer:
The energy difference for a transition in the Paschen series from n=7 to n=3 is calculated using the formula for the energy levels of a hydrogen atom, resulting in an energy difference of 1.973 x 10^-19 Joules.
Step-by-step explanation:
To calculate the energy difference for a transition in the Paschen series from a higher energy shell n=7 to the next lower level of n=3, we use the formula for the energy levels of a hydrogen atom, typically represented as E_n = -R_H * (1/n^2), where R_H is the Rydberg constant. The energy difference (\(\Delta E\)) for the transition would be given by \(\Delta E = E_{initial} - E_{final} = -R_H * (1/n_{initial}^2 - 1/n_{final}^2)\).
Using the Rydberg constant \(R_H = 2.179 \times 10^{-18} J\), the energy difference for the transition from n=7 to n=3 is calculated as follows:
\(\Delta E = -2.179 \times 10^{-18} J * (1/7^2 - 1/3^2) = -2.179 \times 10^{-18} J * (1/49 - 1/9)\)
\(\Delta E = -2.179 \times 10^{-18} J * (-40/441)\)
\(\Delta E = 2.179 \times 10^{-18} J * \frac{40}{441}\)
\(\Delta E = 1.973 \times 10^{-19} J\)
Therefore, the energy difference for a transition in the Paschen series from n=7 to n=3 is \(1.973 \times 10^{-19} J\).