The emission of which atom's first excited state releases light of the smallest energy? #"Na"#, #"Mg"#, #"K"#, #"Ca"#, #"Rb"#?

Rubidium.

#E = hnu = (hc)/lambda -> E prop 1/lambda#

Large wavelength #-># small frequency #-># low energy

Since light emission is caused by electron relaxation towards the ground-state energy level from the next-up energy level, we are looking at that energy gap. Let's start with one valence electron excited, and then relaxing back down to the ground-state.

To do that, let's consider the electron configurations of each one. We have in the ground state:

Na: #[Ne]3s^1#
Mg: #[Ne]3s^2#
K: #[Ar]4s^1#
Ca: #[Ar]4s^2#
Rb: #[Kr]5s^1#

and excited state based on the selection rules:

Selection rules:

#DeltaL = 0, pm1#
#DeltaS = 0#

Na: #[Ne]3p^1#
Mg: #[Ne]3s^1 3p^1#
K: #[Ar]4p^1#
Ca: #[Ar]4s^1 4p^1#
Rb: #[Kr]5p^1#

There is furthermore a trend where the lower energy levels converge towards the higher ones as we move down the periodic table, so the lowest energy difference would be in the Rb transition (780.2 nm). Let's see what the relaxation transitions would be:

Na: #3p^1 -> 3s^1# (~589 nm)
Mg: #3s^1 3p^1 -> 3s^2# (~280 nm)
K: #4p^1 -> 4s^1# (766~769 nm)
Ca: #4s^1 4p^1 -> 4s^2# (~395 nm)
Rb: #5p^1 -> 5s^1# (~780.2 nm)

http://mike.depalatis.net/ionptable/calcium/
http://mike.depalatis.net/ionptable/magnesium/