Emission Spectra & Rydberg Constant lab report

1. Title and aim

A Study of Spectra: Optical Emission Spectroscopy, the Balmer Series and the Rydberg Constant. To observe and map discrete atomic line emission spectra, measure the wavelengths of the visible Balmer series of hydrogen (H_α, H_β, H_γ, H_δ, H_ε, H_ζ) using a calibrated grating spectrometer, test the Bohr-Balmer wavelength relationship, and determine the Rydberg constant R_H.

2. Abstract

Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of Rydberg constant of hydrogen of 1.0970e+07 m⁻¹, differing by 0.018% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.

3. Apparatus and fixed settings

• High-Resolution Transmission Grating Spectrometer: Provides an optical bench with a precision bilateral entrance slit, collimating telescope with achromatic objective, central rotatable circular spectrometer table with replica diffraction grating (600 lines/mm), and viewing telescope with calibrated micrometer reticle graduated in tenths of a nanometer.
• Capillary Hydrogen Geissler Gas Discharge Tube: Emits intense atomic hydrogen emission radiation produced by electron-impact dissociation and collisional excitation of molecular H₂ gas in a constricted capillary bore under low pressure (~1 mmHg).
• High-Voltage Regulated Induction Power Supply: Delivers a stable 5,000 V AC/DC excitation potential to strike and sustain a steady glow discharge across the gas tube without filament thermal sputtering.
• Sodium and Mercury Calibration Reference Lamps: Provide certified spectroscopic wavelength benchmark lines (sodium yellow doublet at 589.0 nm and 589.6 nm, mercury green line at 546.1 nm) for precise spectrometer reticle zero-offset and dispersion calibration.
• Bunsen Flame Chemical Excitation Attachment: Permits comparative study of flame line emission spectra of alkali and alkaline-earth metal chlorides (NaCl, LiCl, SrCl₂, CaCl₂) introduced into a non-luminous gas flame on platinum/nichrome loops.
• Digital Angle and Wavelength Micrometer Readout: Measures angular diffraction deviation θ and directly reads optical wavelength λ to 0.1 nm resolution using anti-parallax vernier drum optics.
• Replica Transmission Grating Constant: d = 1.6667 μm (600 lines / mm)
• Spectral Range: 380.0 nm to 700.0 nm (visible electromagnetic window)
• Spectrometer Slit Width: 0.05 mm (achieving resolution λ / Δλ > 1500)
• Reference Rydberg Constant: R_H = 1.0967758 × 10⁷ m⁻¹ (10.9678 μm⁻¹)
• Lower Balmer Energy Level: n₁ = 2 (E₂ = −3.401 eV)

4. Theory and graph choice

Optical emission spectroscopy is the premier analytical technique that unlocked the internal quantum architecture of the atom. In 1885, Johann Jakob Balmer showed that the visible spectral lines of atomic hydrogen follow a simple mathematical relationship, later generalized by Johannes Rydberg and theoretically explained by Niels Bohr's quantized planetary atomic model in 1913. When hydrogen gas in a low-pressure discharge tube is excited by a high-voltage electric field, energetic electron collisions dissociate molecular H₂ and promote atomic electrons into excited quantum states (n = 3, 4, 5, ...). De-excitation back down to the n = 2 state releases discrete photons constituting the Balmer series: ruby-red H_α (n = 3 → 2 at 656.3 nm), cyan H_β (n = 4 → 2 at 486.1 nm), blue-violet H_γ (n = 5 → 2 at 434.0 nm), deep violet H_δ (n = 6 → 2 at 410.2 nm), and near-UV transitions H_ε (397.0 nm) and H_ζ (388.9 nm). In this experiment, students measure these six discrete emission wavelengths using a calibrated grating spectrometer, compute the optical wavenumbers ν̃ = 1/λ, perform linear regression of ν̃ against (1/4 − 1/n²), extract the Rydberg constant R_H (≈ 1.0968 × 10⁷ m⁻¹), and explore the correspondence between discrete atomic emission and dark solar Fraunhofer absorption lines.

The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units m⁻¹.

5. Method

1. Illuminate the entrance slit with an incandescent lamp. Focus the viewing telescope crosshairs sharply on the slit image. Rotate the collimator tube until the slit image is strictly parallel to the vertical reticle hair, and adjust the collimator lens to produce a perfectly parallel incident beam (Schuster’s method). Set the diffraction grating perpendicular to the optical axis on the central turntable.
2. Place the sodium vapor lamp in front of the entrance slit. Locate the intense first-order yellow emission lines. Align the reticle crosshair on the resolved sodium doublet (D₁ = 589.6 nm, D₂ = 589.0 nm) and calibrate the spectrometer drum index. Verify the calibration against the mercury 546.1 nm green line to confirm linear dispersion across the focal plane.
3. Mount the hydrogen Geissler tube directly in front of the entrance slit within a protective shielded housing. Energize the high-voltage supply to initiate the pink-purple glow discharge. Observe the spectrum through the telescope and identify the four brilliant visible Balmer lines against the faint molecular background band.
4. Rotate the telescope slowly toward the red region and bisect the vivid ruby-red H_α line (n = 3 → 2) with the crosshairs. Read and record its wavelength λ = 656.3 nm. Progressively measure the cyan H_β line (n = 4 → 2 at 486.1 nm), the blue-violet H_γ line (n = 5 → 2 at 434.0 nm), the deep violet H_δ line (n = 6 → 2 at 410.2 nm), and the near-ultraviolet limits H_ε (n = 7 → 2 at 397.0 nm) and H_ζ (n = 8 → 2 at 388.9 nm).
5. Introduce platinum wires wetted with sodium chloride (yellow 589 nm), lithium chloride (crimson 671 nm), and strontium chloride (red 680 nm, blue 461 nm) into a non-luminous Bunsen flame. Record the characteristic flame line positions. Direct solar light through the slit using an exterior heliostat mirror to observe the dark Fraunhofer absorption lines (C, D, F, G) matching the hydrogen and sodium emission wavelengths.
6. For each spectral line n, compute the Bohr quantum geometric factor x = 1/4 − 1/n² and the optical wavenumber y = ν̃ = 1 / λ (in m⁻¹). Plot ν̃ on the vertical axis against (1/4 − 1/n²) on the horizontal axis. Compute ordinary least-squares linear regression to determine the slope S = R_H (in m⁻¹) and the vertical intercept b. Compare the measured Rydberg constant against the standard CODATA value (1.0967758 × 10⁷ m⁻¹).

6. Observations and calculation table

All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Dots show the sample observations. The line is the ordinary least-squares fit.

9. Uncertainty and comparison

Percentage difference = |estimate − model value| / |model value| × 100 = 0.0176%. Percentage difference is not experimental uncertainty.

For n = 5 observations, Σresidual² = 51991.80373; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 1418.80265. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 292.82218.

Propagating only the relevant fit coefficient gives a fit-only standard error of 1418.80265 m⁻¹. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.

Spectrometer circular vernier scale reading resolution (±1 arcminute, producing ±0.2 nm wavelength uncertainty), diffraction grating ruling spacing tolerance and surface thermal expansion (d = 1.6667 μm for 600 lines/mm grating, with thermal expansion coefficient ~1.0 × 10⁻⁵ K⁻¹ altering groove spacing by ~0.03% across ambient room temperature swings), collimator slit width and diffraction broadening (a wide slit broadens spectral line profiles and causes peak centroid asymmetry, minimized by narrowing the bilateral slit until the line profile is sharp and distinct), crosshair visual coincidence and optical parallax in the telescope eyepiece reticle (mitigated by focusing the eyepiece on the crosshair before focusing the objective on the collimator slit image at infinity), small non-normal incident angle alignment error on the grating face (minimized by leveling the spectrometer prism table with the Schuster method and verifying symmetric left-right order diffraction angles θ_L and θ_R), and unmodelled fine-structure relativistic spin-orbit splitting (~0.005 nm in hydrogen Balmer lines, completely unresolved by a 600 lines/mm grating) govern the uncertainty envelope. Ordinary least-squares regression of wavenumber 1/λ against the Balmer factor (1/4 − 1/n²) averages out random vernier scale alignment and reading fluctuations, with the slope isolating the pure Rydberg constant R_H with sub-0.02% error and the vertical intercept isolating any baseline optical alignment offset.

10. Result and discussion

The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.

11. Precautions and limitations

• Allow the gas discharge tube 5 minutes to warm up and stabilize its emission intensity before recording precision reticle settings.
• Always approach the spectral line from the same rotational direction with the micrometer drum to eliminate mechanical gear backlash.
• Keep the entrance slit narrowed to the minimum width (~0.05 mm) that provides adequate brightness to maximize wavelength resolution and prevent line broadening.
• Calibrate the zero-point reading against the certified 589.0/589.6 nm sodium doublet before and after taking measurements on the hydrogen tube.
• Never touch the optical optical surfaces of the replica diffraction grating with fingers or solvents, as moisture and oil permanently destroy holographic ruling microstructures.

Calculations assume an ideal transmission grating obeying d sin θ = m λ with zero prism curvature, negligible Doppler thermal line broadening, and negligible ambient refractive index variations in laboratory air. Real spectrometers exhibit minor chromatic aberration in collimator lenses, mechanical backlash in the micrometer vernier drum, and a refractive index of air n_air ≈ 1.00028 that causes air wavelengths to be approximately 0.18 nm longer than vacuum wavelengths.

12. Conclusion

The worked example illustrates how to obtain Rydberg constant of hydrogen from measurements, graph variables and a fitted coefficient. The estimate is 1.0970e+07 m⁻¹. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.