Which level of the single ionized carbon has the same energy as the ground state energy of hydrogen atom?

Solution

Physics sir · 3 steps

Energy goes as Z²/n², so match that ratio to hydrogen's ground state, where Z = 1 and n = 1.

1. Which ion. The Bohr energy formula works only for a nucleus with one electron. The question means the carbon ion left with a single electron, C⁵⁺, so take Z = 6.

   Eₙ = -13.6 Z²/n² eV

2. Match the energies. Hydrogen's ground state is -13.6 eV, with Z = 1 and n = 1. For carbon you need 6²/n² = 1, so n² = 36.
3. So. n = 6. The 6th level of hydrogen-like carbon has energy -13.6 × 36/36 = -13.6 eV, the same as hydrogen's ground state. The answer is (b).

The idea to keep. For any hydrogen-like ion, the level n = Z has exactly hydrogen's ground-state energy, -13.6 eV, because Z²/n² = 1 there.

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