Resolving a Line-Spectrum Puzzle in a Discharge Tube
Students view the emission from a low-pressure hydrogen discharge tube through a diffraction grating of 600 lines/mm and observe a violet line and a red line. They also see a faint yellow line that does not match the expected hydrogen series and a continuous background glow. A second group, using a tube labelled 'deuterium', reports the same lines shifted very slightly toward shorter wavelength but within their reading error. One student claims the yellow line proves a new element is present; another says it is a grating artefact. The grating is 1.0 m from the scale, and the first-order red line falls at 39.0 cm from the central maximum. The accepted red wavelength is 656 nm.
1. Use the red-line position to check the grating calibration and compute the grating spacing, then judge whether the yellow line can be a grating artefact.
Read the worked answer
Grating spacing. 600 lines/mm means d = 1/600 mm = 1.667×10^-6 m = 1667 nm.
Angle from geometry. tan θ = 0.390/1.000, so θ = arctan(0.390) ≈ 21.3°. Then sin θ ≈ 0.363.
Predicted wavelength. λ = d sin θ = 1667 × 0.363 ≈ 605 nm. That is well short of 656 nm, a discrepancy of about 8%. Something is off: either the grating is not truly 600 lines/mm, the distance is not 1.0 m, or the scale reading is to a different order.
Calibration check. Solving for d using the known 656 nm: d = λ/sin θ = 656/0.363 ≈ 1807 nm, i.e. about 553 lines/mm. So the nominal 600 lines/mm label is inconsistent with the geometry by ~8%; the grating should be recalibrated against a known line before any unknown is identified.
Artefact test. A grating artefact (ghost or second-order overlap) would appear at a position fixed by the grating, not by the source. The yellow line's position should be checked in first and second order: a genuine line obeys sin θ ∝ λ and appears at a predictable angle in each order, whereas a ghost often does not scale correctly. Given the 8% calibration uncertainty, the yellow line cannot be confidently assigned until the grating is calibrated.
2. Explain physically why hydrogen and deuterium spectra differ only slightly, and state whether the students' data could distinguish them.
Read the worked answer
Origin of the shift. The Rydberg formula depends on the reduced mass of the electron–nucleus system: 1/λ = R∞ (1 + m_e/M) (1/n1^2 − 1/n2^2). Replacing the proton (M ≈ 1836 m_e) with a deuteron (M ≈ 3670 m_e) changes the reduced-mass correction from about 1/1836 to 1/3670, a fractional change of roughly 0.027%.
Magnitude. For the red line at 656 nm, the shift is about 0.027% × 656 nm ≈ 0.18 nm, toward shorter wavelength for deuterium (larger mass, slightly larger effective Rydberg). This is far below the ~8% calibration uncertainty and below what a hand-held scale reading at 1 m could resolve.
Verdict. The students' observation that the shift is 'within reading error' is correct and expected. Their data cannot distinguish hydrogen from deuterium; a diffraction grating at this scale is inadequate. Distinguishing them would need a high-resolution spectrometer (resolution better than 0.1 nm) or a Fabry–Pérot etalon, plus careful calibration. The claim of 'same lines' is consistent with both isotopes and proves nothing either way.
3. Recommend how to establish the origin of the yellow line and what evidence would settle the question.
Read the worked answer
Step 1 - Calibrate the grating. Use a known source (e.g. a sodium lamp with its 589.0/589.6 nm doublet, or a mercury lamp with known lines) to determine the true line spacing and the true grating-to-scale distance. This removes the 8% ambiguity.
Step 2 - Test order scaling. Measure the yellow line in first and second order. A genuine spectral line satisfies d sin θ = nλ with the same λ in each order; a ghost or artefact will fail this test or appear at a position that does not scale with order.
Step 3 - Vary conditions. Change tube current and pressure. Impurity lines (often from residual gases or electrode material) change intensity differently from the hydrogen series as conditions vary; a genuine new element would persist with a characteristic pattern of multiple lines, not a single yellow line.
Step 4 - Compare with reference spectra. Match the calibrated wavelength against tabulated impurity lines (e.g. sodium from glass, or mercury) before invoking a new element.
Settling evidence. The yellow line is accepted as a real spectral feature only if it survives calibration, obeys the grating equation in multiple orders, and matches a known atomic transition. A single uncalibrated line is not sufficient to claim a new element.