Chemistry · Structure Of Atom · NEET
Start from Bohr's rule that angular momentum is quantised: mvr = nh/2π. de Broglie said every moving particle has a wavelength λ = h/mv, so mv = h/λ. Put mv = h/λ into Bohr's rule: (h/λ)·r = nh/2π. Cancel h from both sides: r/λ = n/2π, which rearranges to 2πr = nλ. So Bohr's angular-momentum rule and de Broglie's wave idea are the SAME statement.
2πr is the length once around the circular orbit (the circumference). nλ is n complete de Broglie waves laid end to end. The equation says the orbit length must hold a whole number of waves. If it held, say, 3.5 waves, the wave crest meeting a trough would cancel itself out — that orbit cannot exist. Only orbits that fit 1, 2, 3... whole waves are allowed.
Exactly n waves. In the 1st orbit (n=1) there is 1 whole wave, in the 2nd orbit (n=2) there are 2 whole waves, in the 3rd orbit 3 waves, and so on. The number of waves equals the orbit number n. This is a common one-line NEET fact.
Use λ = 2πr/n. First find the orbit radius with rₙ = a₀·n²/Z (a₀ = 52.9 pm = 0.529 Å). Then divide the circumference 2πrₙ by n. Example for n=2 in hydrogen (Z=1): r₂ = 52.9 × 4 = 211.6 pm, so λ = 2π(211.6)/2 = 211.6π pm.
It gets BIGGER. For hydrogen, rₙ grows as n², so 2πrₙ grows as n². Dividing by n leaves λ ∝ n. So the wavelength grows in proportion to n: λ(n=2) is twice λ(n=1), λ(n=3) is three times, and so on. Radius grows faster (n²) but wavelength grows only as n.
Yes — they are two forms of one idea. mvr = nh/2π is Bohr's postulate (came first, was assumed). 2πr = nλ is de Broglie's explanation (came later, explains WHY the postulate is true). One turns into the other the moment you substitute λ = h/mv. NEET may ask either form.
In a hydrogen atom, the de Broglie wavelength of an electron in the second Bohr orbit is: [Bohr radius a₀ = 52.9 pm]
Which one is the wrong statement?
Try the real previous-year questions from this chapter — each with the answer and a full solution.
2πr = nλ, where r is the orbit radius, n is the orbit number (1, 2, 3...), and λ = h/mv is the de Broglie wavelength of the electron. It says the orbit circumference holds exactly n whole electron waves.
A wave going once around must join back to itself smoothly. If a fraction of a wave were left over, the wave would meet itself out of step and cancel by destructive interference. Only whole numbers of waves survive, which is why n = 1, 2, 3... and orbits are quantised.
Bohr just assumed mvr = nh/2π without a reason. de Broglie's wave idea gives the reason: the electron is a standing wave around the orbit. So 2πr = nλ turns an assumption into a physical picture, which is a common NEET conceptual question.
The relation itself holds for any single-electron (hydrogen-like) species. Only the radius changes: rₙ = 0.529 × n²/Z Å. Put that radius in and use λ = 2πrₙ/n.
