Final answer:
The number of possible electron shell transitions for a pure sample of atomic gas with atoms having six principal quantum numbers is 15.
Step-by-step explanation:
In the quantum-mechanical model of an atom, the principal quantum number determines the energy level of electrons. A pure sample of atomic gas with atoms that have six principal quantum numbers can yield several emission lines. Each emission line represents a different electron shell transition. The number of possible electron shell transitions can be determined by calculating the number of combinations of transitions between the six quantum number levels.
The formula for calculating the number of combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of quantum number levels and r is the number of levels for a shell transition. Plugging in the values, we get C(6,2) + C(6,3) + C(6,4) + C(6,5) + C(6,6), which equals 15. Therefore, the number of possible electron shell transitions for which energy is radiated is 15.