Using Bohr's postulates, derive an expression for the radius of the nth orbit of the hydrogen atom and show that rₙ ∝ n². Find the radius of the third orbit if a₀ = 0.53 Å.
How to lay out the answer. Follow the frame, then write it in your own words.
1 · State the two postulates used. The electron moves in a stationary circular orbit, and its angular momentum is mvr = nh/2π.
2 · Draw the orbit. [Diagram: nucleus +e, electron −e on a circle of radius rₙ, v along the tangent, F towards the centre]
3 · Write the force balance. mvₙ²/rₙ = (1/4πε₀)(e²/rₙ²), so mvₙ²rₙ = e²/(4πε₀) ... (i)
4 · Write the quantum condition. vₙ = nh/(2πmrₙ) ... (ii)
5 · Substitute (ii) in (i) and simplify. n²h²/(4π²mrₙ) = e²/(4πε₀), so rₙ = n²h²ε₀/(πme²).
6 · Show the dependence and solve. All other quantities are constants, so rₙ ∝ n². r₃ = 9 × 0.53 Å = 4.77 Å.
7 · Close in one line. So, the allowed orbits have radii in the ratio 1 : 4 : 9, and the first is the Bohr radius, 0.53 Å.
Scoring tip: Number the equations (i) and (ii) and write 'substituting (ii) in (i)'. Examiners give step marks for each line.
