Which electronic transition in a hydrogen atom will require the largest amount of energy ? (1) n = 1 to n = 2 (2) n = 2 to n = 4 (3) n = 5 to n = 1 (4) n = 3 to n = 2?

1 Answer
Apr 5, 2018

Transition #(1)#, from #n=1# to #n=2#

Explanation:

Start by eliminating choices #(3)# and #4#: the closer to the nucleus (positively-charged) an electron (negatively-charged) lies, the lower its potential energy. Transitions lowering this energy tend to be spontaneous and releases energy (in the form of electromagnetic radiations when it comes to atomic electrons).

Bohr modeled the potential energy of an electron at the #n#-th principal energy level of a hydrogen atom with the equation

#E(n)=-k/n^2#

where #k=13.6*"eV"#, which is numerically equal to the energy required to move an electron from the principal energy level #n=1# of a hydrogen atom to an infinite separation from the nucleus.

Energy changes involved in the two transitions, #Delta E_1# and #Delta E_2#, would thus be proportional to the respective differences in the reciprocals of the principal energy levels- squared. That is:

#Delta E_1=E(2)-E(1) prop (- 1/2^2)-(-1/1^2)=3/4#
#Delta E_2=E(4)-E(2) prop (- 1/4^2)-(-1/2^2)=3/16#

#Delta E_1>Delta E_2# and therefore the first transition would require the most significant amount of energy input.