As we move to higher energy levels (larger 'n' values) in the Bohr model of a hydrogen atom, the energy difference between successive levels:

Correct — B. In Bohr's model, the energy of level n is Eₙ = −13.6⁄n² eV.

Gap n=1→2: 13.6 × (1 − 1⁄4) = 10.2 eV
Gap n=2→3: 13.6 × (1⁄4 − 1⁄9) ≈ 1.89 eV
Gap n=3→4: 13.6 × (1⁄9 − 1⁄16) ≈ 0.66 eV

Each gap is smaller than the one before: the levels crowd together as they approach E = 0 → option (b).

Why the others are wrong

• (a)
  Increases continuously — The opposite of what happens. The orbit radius grows with n, as n², but the energy gaps shrink: 10.2 eV, then 1.89 eV, then 0.66 eV.
• (c)
  Remains constant — Equal gaps would need energy to change in equal steps with n. Bohr energy goes as 1⁄n², so equal steps in n give ever-smaller steps in energy.
• (d)
  First increases, then decreases — There is no rise at the start. The largest gap is the very first one, n = 1 to 2 (10.2 eV), and every later gap is smaller than the one before it.

Bohr's model gives the hydrogen electron fixed orbits with energy Eₙ = −13.6⁄n² eV: −13.6 eV at n = 1, −3.4 eV at n = 2, about −1.51 eV at n = 3.

The gap between neighbours is 13.6 × (2n + 1) ⁄ [n²(n + 1)²] eV, which falls at every step.

As n → ∞ the energy reaches 0 and the electron is free, so 13.6 eV is hydrogen's ionisation energy from the ground state.

Key facts

• Bohr energy of hydrogen: Eₙ = −13.6⁄n² eV.
• Ground-state energy is −13.6 eV, so hydrogen's ionisation energy is 13.6 eV.
• Orbit radius grows as n²: rₙ = 0.529 n² Å.
• Successive gaps: 10.2 eV (n = 1→2), about 1.89 eV (2→3), about 0.66 eV (3→4).

Common traps

• Confusing the energy of a level, which rises towards zero with n, with the gap between levels, which shrinks.
• Carrying the n² growth of the orbit radius over to the energy gaps.

No directly related past PYQ was found.