What was the 2018 JEE Main question on de Broglie wavelength and emitted photon wavelength really asking?
The 2018 JEE Main question asked which linear relation holds between Λ_n and λ_n for large n, specifically Λ_n ≈ A + B/λ_n² with constants A and B.
An electron in a hydrogen atom drops from various excited states n to ground state emitting a photon. Here λ_n is the de Broglie wavelength of the orbiting electron in the nth state, λ_g is the value for the ground state, and Λ_n is the wavelength of the emitted photon for that transition.
The core demand is to combine the Rydberg formula with the de Broglie relation and keep the first-order correction in 1/n² to eliminate n in favour of λ_n.
How do you derive Λ_n ≈ A + B/λ_n² using the official 2018 method?
The official derivation gives Λ_n ≈ A + B/λ_n². Start from Bohr energies E_n = -13.6/n² eV. The photon energy for the n to ground transition is 13.6(1 - 1/n²) eV, which for large n carries a correction term proportional to 13.6/n².
The Rydberg formula is more direct:
For large but finite n this becomes
Invert to first order with (1 - x)^{-1} ≈ 1 + x where x = R/n² divided by R:
This is Λ_n ≈ A + B/n². In the Bohr model electron velocity v_n ∝ 1/n, so de Broglie λ_n = h/(m v_n) gives λ_n ∝ n and therefore λ_n² ∝ n². Substitute n² ∝ λ_n² to replace the second term and obtain Λ_n ≈ A + B/λ_n².
What algebraic method error produces a wrong option in this question?
Treating photon energy as exactly -13.6/n² instead of expanding (1 - 1/n²) in the Rydberg formula leads to the wrong relation. This produces 1/Λ_n ≈ constant/n², which inverts to Λ_n ∝ n².
The error then forgets to relate the correction term 1/n² back to λ_n² through the proportionality λ_n ∝ n. The result is an expression such as Λ_n ∝ λ_n² or Λ_n ∝ 1/λ_n instead of the offset form.
The flaw arises from stopping at zeroth-order approximation rather than retaining the first-order term required for large but finite n. Write the binomial expansion and replace every power of n with the matching power of λ_n before you invert.
Why is this 2018 JEE question tagged under both Rutherford and Bohr models?
The question is tagged under both because the Rutherford model alone is insufficient. Rutherford gives a nuclear atom with orbiting electrons but cannot explain discrete spectral lines or orbit stability.
Bohr adds mvr = nh/2π, justified later by de Broglie standing waves. This produces the E_n ∝ 1/n² and v ∝ 1/n used in the solution. The de Broglie wavelength λ_n in the question tests the wave-particle link that bridges the two models.
What two similar Atoms and Nuclei questions use the same large-n and proportionality techniques?
The same large-n expansion and λ_n ∝ n substitution apply directly to these problems.
Question 1
Find the ratio of de Broglie wavelengths of electron in n=2 and n=1 Bohr orbits.
Velocity v ∝ 1/n, so λ = h/(m v) gives λ ∝ n. The ratio is therefore 2:1. This is the same proportionality used in the 2018 paper to replace n with λ_n.
Question 2
Find the wavelength of photon for transition from n=∞ to n=1 in hydrogen.
Use the series limit of the Lyman formula: 1/Λ = R(1 - 0) so Λ = 1/R. In the large-n approximation of the 2018 question this limit is exactly the constant A in Λ_n ≈ A + B/λ_n². The same first-order expansion and substitution λ_n² ∝ n² converts any correction term into the required 1/λ_n² dependence.
You can search every JEE Main paper from 2002 and every Advanced paper from 2007 by chapter, each with a worked solution.
What must you memorise for large-n Bohr questions before JEE Main 2025?
- v_n = 2.18×10^6 / n m/s so λ_n ∝ n and λ_n² ∝ n².
- 1/Λ_n = R(1 - 1/n²) and the binomial expansion on the inverse to keep the 1/n² correction.
- Angular momentum quantization mvr = nh/2π and its de Broglie origin λ_n = 2πr/n.
- Expected solving time for this 2018 question was 300 s; target under 180 s by directly writing λ_n² ∝ n² on the first line and substituting before inverting.
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Frequently asked questions
What relation holds between Λ_n and λ_n for large n in 2018 JEE question?
For large n, Λ_n ≈ A + B/λ_n² where A and B are constants. This is obtained by using the Rydberg formula, binomial approximation for (1 - 1/n²)^-1 and replacing n² with a term proportional to λ_n² since λ_n ∝ n.
Why is the 2018 JEE question tagged under Rutherford and Bohr models?
The Rutherford model proposed the nuclear structure but failed to explain stable orbits and line spectra. Bohr introduced quantized orbits and angular momentum condition mvr = nh/2π. The de Broglie wavelength in the question connects to the standing wave concept that justifies Bohr's model.
What is the common mistake in solving the 2018 Bohr model wavelength question?
The common error is using the photon energy as exactly 13.6/n² without expanding the (1 - 1/n²) term properly. This leads to incorrect proportionality like Λ_n ∝ n² or Λ_n ∝ λ_n² without the constant offset A. Always retain the first order term in 1/n².
How do you derive Λ_n ≈ A + B/λ_n² in the 2018 JEE question?
Start from 1/Λ_n = R(1 - 1/n²). For large n this is ≈ R - R/n². Invert using (1 - x)^-1 ≈ 1 + x to get Λ_n ≈ 1/R + (1/R)/n². Since λ_n ∝ n from Bohr model, n² ∝ λ_n². Substitute to replace 1/n² with 1/λ_n² yielding Λ_n ≈ A + B/λ_n².
What is the de Broglie wavelength ratio for n=2 to n=1 in Bohr model?
Since velocity v_n is proportional to 1/n, the de Broglie wavelength λ_n = h/(mv_n) is proportional to n. Therefore the ratio λ_2 : λ_1 = 2 : 1. This same proportionality is used in the 2018 question to replace n² by λ_n².