The orbital radius of an electron in a hydrogen-like atom is 4.5 a0 (where a0 is the Bohr radius) and its orbital angular momentum is 3h/(2*pi). Given R is the Rydberg constant, which wavelength would NOT be emitted when this atom de-excites?

Question

Accepted Answer

9/(16R). From L=n*hbar=3*hbar => n=3. From r=n²*a0/Z=4.5*a0 => 9/Z=4.5 => Z=2 (He+). Allowed emission transitions from n=3: (3->1), (3->2), (2->1). Using 1/lambda=Z²*R*(1/n1²-1/n2²)=4R*(...): - 3->1: 4R*(1-1/9)=32R/9 => lambda=9/(32R). Emitted. - 3->2: 4R*(1/4-1/9)=5R/9... wait: 4R*(9-4)/36=4R*5/36=5R/9 => lambda=9/(5R). Emitted. - 2->1: 4R*(1-1/4)=3R => lambda=1/(3R). Emitted. Wavelength 9/(16R) corresponds to 1/lambda=16R/9 => 4*(1/n1²-1/n2²)=16/9 => no integer solution. NOT emitted.

The orbital radius of an electron in a hydrogen-like atom is 4.5 a0 (where a0 is the Bohr radius) and its orbital angular momentum is 3h/(2*pi). Given R is the Rydberg constant, which wavelength would NOT be emitted when this atom de-excites?

Solution

Related JEE Advanced Physics questions