A beam of monochromatic light is used to excite the electron in Li + + from the first orbit to the third orbit. The wavelength of monochromatic light is found to be x × 10⁻¹⁰ m. The value of x is ___________.
[Given hc = 1242 eV nm]

Solution

Physics sir · 4 steps

Lithium with two electrons gone is hydrogen-like with Z = 3, so every energy level is 9 times hydrogen's.

1. Hydrogen-like levels. Li²⁺ has one electron and Z = 3, so Bohr's formula applies. Its levels are Eₙ = -(13.6 Z²)/n² eV = -122.4/n² eV.
2. Energy absorbed. Going from n = 1 to n = 3, the electron absorbs Δ E = 122.4(1 - 1/9) = 122.4 × 8/9 = 108.8 eV.

   λ = hc/(Δ E) = (1242 eV nm)/(108.8 eV) = 11.415 nm

3. Convert units. 1 nm = 10× 10⁻¹⁰ m, so λ = 114.15 × 10⁻¹⁰ m.
4. So. x ≈ 114.

The common slip: 1027. Leaving out the Z² factor uses hydrogen's 12.09 eV gap and gives λ = 102.7 nm, that is x ≈ 1027, not 114.

The idea to keep. For any one-electron ion, take hydrogen's 13.6 eV and multiply by Z². Then λ = hc/Δ E, and with hc = 1242 eV nm the wavelength comes out directly in nm.

Answer from the NTA final answer key. Solution by PrepWiser.