Assigning the Lines in a Hydrogen Spectrum Report
In a practical class, students record the emission spectrum of a low-pressure hydrogen discharge tube using a grating spectrograph whose manual quotes a usable range of 200–800 nm and a resolution of 0.5 nm. One student logs four lines at 656.3, 486.1, 434.0 and 410.2 nm. Because the laboratory also owns a vacuum-UV detector with a MgF2 window, the student records one further line at 121.6 nm and labels all five entries as Balmer lines with n₁ = 2. In the discussion section the student writes that the series limit calculated from these data gives 'the ionisation energy of hydrogen, about 328 kJ mol⁻¹'. A demonstrator points out that the tube was not flushed after a helium discharge experiment the previous day, so traces of helium may be present. The report must decide which entries are genuine hydrogen lines, assign quantum numbers, extract the Rydberg constant, and state what the series limit actually measures. The data sheet gives R_H = 1.097 × 10⁷ m⁻¹, hc = 1240 eV nm and 1 eV per particle = 96.5 kJ mol⁻¹.
1. Show that the four visible lines belong to the Balmer series, assign n₂ for each, and calculate R_H from the 656.3 nm line. Is the agreement with the data-sheet value acceptable at the stated resolution?
Read the worked answer
Use 1/λ = R_H(1/2² − 1/n₂²).
For 656.3 nm, 1/λ = 1/(656.3 × 10⁻⁹) = 1.524 × 10⁶ m⁻¹. Since the bracket is (1/4 − 1/n₂²), solving gives n₂ = 3.
R_H = (1.524 × 10⁶) × 36/5 = 1.097 × 10⁷ m⁻¹.
Assignment by the same route:
- 486.1 nm → n₂ = 4
- 434.0 nm → n₂ = 5
- 410.2 nm → n₂ = 6 (Check 410.2 nm: 1/λ = 2.438 × 10⁶ m⁻¹; 1/4 − 2.438 × 10⁶/1.097 × 10⁷ = 0.2500 − 0.2222 = 0.02778 = 1/36, so n₂ = 6. Consistent.)
The Rydberg constant recovered differs from the data-sheet value by about 0.01%, i.e. far better than the 0.5 nm resolution would suggest, because the line positions are centroids of strong, sharp lines. The conclusion is that the four lines are Balmer lines of hydrogen, and the spacings shrink towards the series limit as n₂ increases, exactly as the formula demands.
2. The 121.6 nm entry is also labelled as Balmer. Test that claim quantitatively, identify the line correctly, and evaluate the student's statement that the Balmer series limit is the ionisation energy of hydrogen.
Read the worked answer
For 121.6 nm, 1/λ = 8.224 × 10⁶ m⁻¹. Dividing by R_H gives 0.7497. The Balmer bracket (1/4 − 1/n₂²) can never exceed 0.25, so no Balmer assignment is possible - the claim fails by a factor of three.
Using the Lyman form (1 − 1/n₂²): 0.7497 corresponds to 1/n₂² = 0.2503, i.e. n₂ = 2. The line is Lyman α, the n = 2 → 1 transition, which needs a vacuum-UV detector - consistent with the instrument used.
Series limit: 1/λ = R_H/4, so λ = 364.6 nm. Energy = 1240/364.6 = 3.40 eV = 328 kJ mol⁻¹. This is the energy to take an electron from n = 2 to n = ∞, not from the ground state. The true ionisation energy is the n = 1 → ∞ limit: 1/λ = R_H, λ = 91.2 nm, 13.6 eV = 1312 kJ mol⁻¹. The student's number is arithmetically right but describes the wrong process; the wording, not the arithmetic, must be corrected.
3. The demonstrator warns that traces of helium may contaminate the tube. Using the resolution quoted, decide whether any of the four visible lines should be rejected as helium, and state what a helium impurity would actually do to the recorded spectrum.
Read the worked answer
The resolution is 0.5 nm, so a helium line would have to fall within about ±0.5 nm of a recorded position to be mistaken for it.
Strong neutral helium lines in the visible are at 587.6, 501.6, 492.2 and 447.1 nm. The nearest approach to any recorded hydrogen line is 492.2 vs 486.1 nm - a gap of 6.1 nm, twelve times the resolution. The others are further away. None can be confused.
A second check is internal consistency: the four wavelengths fit one Rydberg formula with a single constant to 0.01%. An impurity line would not obey that formula, so it would appear as an extra line rather than shift a hydrogen line.
Therefore no recorded line should be rejected; the helium contamination would add faint extra lines (most obviously around 447.1 and 587.6 nm) that a careful student would report as unassigned. The 121.6 nm entry is likewise hydrogen, not helium: the He I resonance line lies at 58.4 nm, far below it.