Simulation · Physics · Class 12
Bohr orbits: how r, v and E scale with n and Z
From the lesson Bohr's postulates in Atoms. Change the values and watch what happens.
Bohr orbits: how r, v and E scale with n and ZPhysics · Class 12
The idea behind it
NCERT §12.4
- Niels Bohr spent several months in Rutherford's laboratory in 1912 and accepted the nuclear model. In 1913 he concluded that classical electromagnetism, so successful on a large scale, cannot be applied to processes inside the atom, and he combined classical and early quantum ideas in three postulates.
- First postulate: an electron can revolve in certain stable orbits without radiating, contrary to classical theory. Each such stationary state of the atom has a definite total energy.
- Second postulate: the allowed orbits are those whose angular momentum is a whole multiple of h/2π, L = nh/2π (Eq. 12.5), where h is Planck's constant (NCERT quotes 6.6 × 10⁻³⁴ J s here). Angular momentum is quantised.
- Third postulate: an electron may jump from one stationary orbit to another of lower energy, emitting a photon whose energy equals the difference: hν = Ei − Ef (Eq. 12.6), with Ei > Ef.
- Combining L = nh/2π with the force balance gives the radius of the nth orbit, rn = (n²/m)(h/2π)²(4πε₀/e²) (Eq. 12.7), so rn ∝ n². For n = 1 this is the Bohr radius a₀ = 5.3 × 10⁻¹¹ m.
- Putting rn into E = −e²/(8πε₀r) gives En = −me⁴/(8n²ε₀²h²) (Eq. 12.8), which is −2.18 × 10⁻¹⁸ J/n² (Eq. 12.9), or En = −13.6 eV/n² (Eq. 12.10).
- Since v = nh/(2πmrn) and rn ∝ n², the speed goes as vn ∝ 1/n: 2.2 × 10⁶ m/s in the first orbit, half that in the second.
- The negative En means the electron is bound; energy must be supplied to take it infinitely far from the proton.