In the first excited state of a hydrogen atom, the electron energy is -3.4 eV. Estimate its orbital radius. Take 1 eV = 1.6 × 10⁻¹⁹ J, electron charge magnitude e = 1.6 × 10⁻¹⁹ C and Coulomb constant k = 9 × 10⁹ N·m²/C².
Options

Question

In the first excited state of a hydrogen atom, the electron energy is -3.4 eV. Estimate its orbital radius. Take 1 eV = 1.6 × 10⁻¹⁹ J, electron charge magnitude e = 1.6 × 10⁻¹⁹ C and Coulomb constant k = 9 × 10⁹ N·m²/C².

Options

G1 Physics · Accepted Answer

Step 1: In a circular Coulomb orbit, K = ke²/(2r) and U = −ke²/r. Hence total energy E = −ke²/(2r). Step 2: |E| = 3.4 × 1.6 × 10⁻¹⁹ = 5.44 × 10⁻¹⁹ J. Step 3: Rearrange: r = ke²/(2|E|). Step 4: r = [9 × 10⁹ × (1.6 × 10⁻¹⁹)²]/[2 × 5.44 × 10⁻¹⁹] = 2.12 × 10⁻¹⁰ m. Final answer: Approximately 2.1 × 10⁻¹⁰ m (Option B).

In the first excited state of a hydrogen atom, the electron energy is −3.4 eV. Estimate its orbital radius. Take 1 eV = 1.6 × 10⁻¹⁹ J, electron charge magnitude e = 1.6 × 10⁻¹⁹ C and Coulomb constant k = 9 × 10⁹ N·m²/C².