Hydrogen-like atom – ground-state binding energy In its lowest (ground) state, the electron bound to a proton (hydrogen atom) has a binding/ionization energy equal to which of the following?

Introduction:The Bohr model and quantum mechanics both predict discrete energy levels for the hydrogen atom. The ground-state energy determines the photon energy required to ionize hydrogen from n = 1 to the continuum, a benchmark value in spectroscopy and astrophysics.

Given Data / Assumptions:

• Hydrogen atom (one electron, one proton) at rest.
• Energy levels depend on principal quantum number n.
• Binding energy is the energy needed to remove the electron to infinity (zero reference).

Concept / Approach:

In the Bohr model, allowed energies are E_n = −13.6 eV / n^2. For the ground state n = 1, E_1 = −13.6 eV. The ionization energy is the magnitude required to raise the electron to zero energy, hence 13.6 eV. This value matches precise spectroscopic measurements and underpins the Rydberg constant.

Step-by-Step Solution:

Verification / Alternative check:

The Lyman series limit corresponds to 13.6 eV photons (≈ 91.2 nm), consistent with this binding energy.

Why Other Options Are Wrong:

51.2 eV equals 4 × 12.8 eV, not a hydrogen ground-state value; 100.5 eV and 1.6 or 0.2 eV are inconsistent with the known spectrum.

Common Pitfalls:

Confusing excitation energies (e.g., Balmer lines) with ionization energy; forgetting sign convention for bound-state energies.

Final Answer:

13.6 eV