The n = 5 to n = 3 electron transition is part of the Paschen series.
To calculate the energy that corresponds to this specific electron transition you'll have to use the Rydberg equation
#E = R_E * (1/n_"final"^2 - 1/n_"initial"^2)#, where
#R_E# - the Rydberg constant, equal to #2.78 * 10^(-18)"J"#;
#n_"final"# - the final energy level of the transition;
#n_"initial"# - the initial energy level of the transition.
Plug your values into the above equation and solve for #E#
#E = 2.178 * 10^(-18)"J" * (1/3^2 - 1/5^2) = color(green)(1.55 * 10^(-19)"J")#
To determine the frequency of the light emitted in this transition, use the relationship that exists between energy, frequency, and Planck's constant
#E = h * nu#, where
#h# - Planck's constant, equal to #6.626 * 10^(-34)"J s"#
#nu# - the frequency of the emitted photon.
Solving for #nu# will give you
#nu = E/h = (1.55 * 10^(-19)cancel("J"))/(6.626 * 10^(-34)cancel("J") "s") = color(green)(2.34 * 10^(15)"s"^(-1)#
