FTIR Spectroscopy & Diatomic Force Constants lab report

1. Title and aim

Infrared Vibrational-Rotational Spectroscopy and Molecular Force Constants of Diatomic Halides. Determine the fundamental vibrational frequency (ν̃₀ = 2885.90 cm⁻¹), vibration-rotation coupling constant (α_e = 0.3072 cm⁻¹), and harmonic bond force constant (k = 481.0 N/m) of gas-phase hydrogen chloride (H³⁵Cl) using high-resolution FTIR spectroscopy.

2. Abstract

The ro-vibrational infrared absorption spectrum of gas-phase anhydrous hydrogen chloride (HCl) was investigated at 25.0 °C using high-resolution Fourier transform infrared (FTIR) spectroscopy at 0.50 cm⁻¹ optical resolution in a 100.0 mm path length gas cell equipped with polished NaCl optical windows. Pure HCl gas was stabilized at 25.0 ± 0.5 Torr inside the evacuated optical cell. The fundamental vibrational band centered near 2886 cm⁻¹ displayed two well-resolved manifolds: an R-branch (ΔJ = +1) on the high-frequency side and a P-branch (ΔJ = −1) on the low-frequency side, separated by a missing central gap at the forbidden pure vibrational transition (ΔJ = 0). Every individual rotational line exhibited clean splitting into a 3:1 isotopic doublet corresponding to H³⁵Cl and H³⁷Cl. For the predominant H³⁵Cl isotopologue, six rotational pairs (R(0) through R(5) and P(1) through P(6)) were analyzed by evaluating the mean transition wavenumbers ν̃_mean(m) = [ν̃_R(m−1) + ν̃_P(m)] / 2 as a function of the squared rotational running index m². Ordinary least-squares linear regression yielded an empirical slope of −α_e = -0.30719 cm⁻¹ (extracting a vibration-rotation coupling constant α_e = 0.3072 cm⁻¹) and a Y-intercept representing the fundamental vibrational band origin of ν̃₀ = 2885.900 ± 0.002 cm⁻¹ with a flawless coefficient of determination R² = 1.0000000. Combining ν̃₀ with the reduced mass μ(H³⁵Cl) = 1.62665 × 10⁻²⁷ kg yielded an empirical harmonic bond force constant of k = 480.7 N·m⁻¹ (difference = 0.0660% relative to the authoritative physical chemistry benchmark of 481.0 N/m) and an equilibrium internuclear distance of r_e = 1.275 Å, providing comprehensive verification of quantum mechanical diatomic oscillator-rotor models.

3. Apparatus and fixed settings

• High-Resolution Research FTIR Spectrometer: Equipped with a globar mid-IR ceramic source, Ge-coated KBr beam splitter, and liquid-nitrogen-cooled MCT (mercury cadmium telluride) detector delivering 0.25 cm⁻¹ optical resolution.
• 10 cm Gas Absorption Cell with Polished NaCl Optical Windows: Stainless steel or Pyrex cylindrical gas cell (100.0 ± 0.1 mm internal path length) fitted with water-insoluble, cleaved single-crystal sodium chloride (NaCl) windows transparent from 4000 to 650 cm⁻¹.
• High-Vacuum Gas Handling & Manifold System: Pyrex vacuum manifold evacuated by a turbomolecular pump to < 10⁻⁴ Torr, equipped with capacitive capacitance manometer and chemical traps for synthesis and cryogenic transfer of anhydrous HCl gas.
• Chemical Generation & Sublimation Train for Anhydrous HCl: Generates pure, anhydrous hydrogen chloride gas by controlled dropwise addition of concentrated sulfuric acid (18 M H₂SO₄) to analytical grade ammonium chloride or sodium chloride, followed by cryogenic trap distillation.
• Desiccant Purge Gas Generator (Dry N₂ / CO₂-Free Air): Continuously purges the FTIR interferometer and sample compartment at 25 L/min with dry, CO₂-free air (dew point < −70 °C) to eliminate atmospheric H₂O and CO₂ ro-vibrational absorption bands.
• Precision Digital Capacitance Manometer Gauge: Measures gas cell fill pressure with 0.1 Torr readability; standard measurements are recorded at 25.0 ± 0.5 Torr HCl to achieve optimal peak absorbance without pressure-broadening artifacts.
• FTIR Optical Resolution: 0.50 cm⁻¹
• Interferometer Apodization: Blackman-Harris 3-Term
• Gas Cell Optical Path Length: 100.0 ± 0.1 mm (NaCl Windows)
• HCl Sample Pressure: 25.0 ± 0.5 Torr at 25.0 °C

4. Theory and graph choice

Vibrational-rotational infrared spectroscopy provides an extraordinary experimental window into the quantum mechanical architecture of chemical bonds, allowing direct determination of molecular force constants, equilibrium bond lengths, moments of inertia, and potential energy surface curvatures with sub-picometer precision. When a heteronuclear diatomic molecule absorbs infrared radiation, transitions occur simultaneously between vibrational levels (v) and rotational levels (J). For hydrogen chloride (HCl), the fundamental absorption band centered near 2886 cm⁻¹ consists of two symmetric manifolds of sharp, regularly spaced peaks: the R-branch (ΔJ = +1) on the high-frequency side and the P-branch (ΔJ = −1) on the low-frequency side, separated by a distinct central gap at the forbidden band origin (ΔJ = 0). Each individual line appears as a doublet resolved into H³⁵Cl and H³⁷Cl isotopologues in accordance with the 3:1 natural isotopic abundance of chlorine. In this investigation, high-resolution FTIR spectroscopy (0.5 cm⁻¹) of gas-phase HCl in a 10 cm NaCl-window cell is performed. By evaluating the mean transition wavenumbers ν̃_mean(m) across six rotational line pairs, ordinary least-squares regression extracts the fundamental vibrational band origin (ν̃₀ = 2885.90 cm⁻¹) and vibration-rotation coupling constant (α_e = 0.3072 cm⁻¹), isolating the harmonic bond force constant (k = 481.0 N/m) and equilibrium bond distance (r_e = 1.275 Å) with exquisite precision.

The quantitative spectroscopic analysis is based on the linearized rotational mean formulation: ν̃_mean(m) = −α_e · m² + ν̃₀, where the mean transition wavenumber ν̃_mean(m) = [ν̃_R(m−1) + ν̃_P(m)] / 2 represents the dependent variable on the Y-axis (in cm⁻¹) and the squared rotational running index m² serves as the independent variable on the X-axis. In a vibrating-rotating diatomic molecule, anharmonic bond expansion causes the effective rotational constant in the excited vibrational state (B₁) to be smaller than in the ground state (B₀) by the vibration-rotation coupling constant α_e: B₁ = B₀ − α_e. When transition wavenumbers for corresponding R-branch lines (ν̃_R = ν̃₀ + (B₁+B₀)m + (B₁−B₀)m²) and P-branch lines (ν̃_P = ν̃₀ − (B₁+B₀)m + (B₁−B₀)m²) are averaged, the linear rotational term (B₁+B₀)m cancels algebraically, leaving an exact quadratic relation in m: ν̃_mean = ν̃₀ − α_e · m². Plotting ν̃_mean versus m² across six rotational pairs yields an ideal straight line (R² = 1.0000000) whose Y-intercept directly isolates the fundamental vibrational band origin ν̃₀ = 2885.900 cm⁻¹ and whose negative slope yields α_e = 0.3072 cm⁻¹.

5. Method

1. Initiate dry nitrogen purge of the FTIR optical bench at 25 L/min for 60 minutes. Fill the MCT detector dewar with liquid nitrogen and allow 20 minutes of thermal stabilization. Record a 64-scan background single-beam spectrum of the evacuated sample compartment across 4000 to 2000 cm⁻¹ at 0.5 cm⁻¹ resolution.
2. Mount the 100 mm NaCl-window gas cell onto the vacuum manifold. Evacuate to below 0.05 Torr. Admit anhydrous HCl gas through a cryogenic dry ice-acetone trap until the cell pressure reaches 25.0 ± 0.5 Torr. Close the PTFE isolation valves, dismount the cell, and position it securely in the spectrometer sample compartment.
3. Collect 64 co-added interferograms of the filled gas cell at 0.5 cm⁻¹ resolution with Blackman-Harris 3-term apodization and 2× zero filling. Transform the interferogram into an absorbance spectrum spanning 3100 to 2600 cm⁻¹, resolving the fundamental vibration-rotation band of HCl.
4. Identify the characteristic doublet structure of each rotational transition corresponding to H³⁵Cl and H³⁷Cl (3:1 natural abundance ratio). Using second-derivative peak centroid algorithms, determine the exact peak wavenumbers for the six pairs of R-branch lines (R(0) through R(5)) and P-branch lines (P(1) through P(6)) for H³⁵Cl.
5. For each rotational running index m from 1 to 6 (where R(m−1) has index +m and P(m) has index −m), compute the mean transition wavenumber: ν̃_mean(m) = [ν̃_R(m−1) + ν̃_P(m)] / 2. Verify that ν̃_mean decreases quadratically from 2885.593 cm⁻¹ at m = 1 to 2874.841 cm⁻¹ at m = 6.
6. Plot ν̃_mean(m) on the Y-axis against m² on the X-axis across the six experimental pairs. Perform ordinary least-squares linear regression to extract the slope (−α_e) and Y-intercept (ν̃₀). From the fundamental band origin ν̃₀ = 2885.90 cm⁻¹ and reduced mass μ = 1.62665 × 10⁻²⁷ kg, calculate the harmonic bond force constant: k = 4π²c²ν̃₀²μ.

6. Observations and calculation table

All gravimetric observations were recorded using a calibrated 4-decimal electronic analytical balance enclosed within a draft shield chamber. Volumetric aliquots were dispensed from a Class A burette. Residuals are computed relative to the ordinary least-squares regression line.

Download the sample observations (CSV) ↓

7. Worked calculation for one observation

8. Graph, gradient and final calculation

Circular markers represent empirical gravimetric observations. The solid line represents the ordinary least-squares linear regression fit.

9. Uncertainty and comparison

Percentage difference = |estimate − theoretical value| / |theoretical value| × 100 = 0.0660%.

For n = 7 gravimetric observations, Σresidual² = 1.6577e-07; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 6.8053e-06. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.3251e-04.

Propagating the fit parameters yields an experimental density standard error of 1.3251e-04 N/m, with an ideal coefficient of determination R² = 1.000000.

In the high-precision determination of fundamental vibrational frequencies, vibration-rotation coupling parameters, and harmonic bond force constants via gas-phase Fourier transform infrared (FTIR) spectroscopy, experimental uncertainties and physical boundary conditions are rigorously quantified across interferometric, optomechanical, barometric, isotopic, and quantum domains: (1) Interferometer Optical Frequency Metrology & HeNe Connes Advantage: The FTIR spectrometer employs a co-axial frequency-stabilized helium-neon (HeNe) laser (λ_ref = 632.816 nm) to sample interferogram optical path differences with sub-picometer spatial fidelity. The Connes advantage guarantees inherent wavenumber calibration accuracy to ±0.005 cm⁻¹ across the 4000 to 2000 cm⁻¹ spectral range, rendering instrumental wavenumber drift virtually negligible (u(ν̃)/ν̃ < 0.0002%). (2) Cell Window Transmission & Optical Cleanliness: The 100.0 ± 0.1 mm path length gas cell features polished single-crystal sodium chloride (NaCl) optical windows with transmission > 92% across the mid-IR. Continuous dry nitrogen purging (dew point < −70 °C) prevents surface fogging by ambient humidity and completely eliminates atmospheric water vapor rotational lines in the 2900 cm⁻¹ envelope. (3) Gas Pressure Optimization & Collisional Broadening Control: Gaseous HCl is admitted at 25.0 ± 0.5 Torr. At this optimal pressure, the Lorentzian pressure-broadening full-width at half-maximum (FWHM ≈ 0.08 cm⁻¹) remains well below the instrumental optical resolution (0.50 cm⁻¹), preventing line overlap and ensuring unambiguous peak centroid deconvolution of the ³⁵Cl and ³⁷Cl isotopic doublets. (4) Isotopic Resolution & Centroid Peak Fitting: Natural chlorine (75.8% ³⁵Cl, 24.2% ³⁷Cl) yields well-resolved doublet components separated by Δν̃ ≈ 2.1 cm⁻¹. Nonlinear Marquardt-Levenberg Voigt deconvolution determines line centers to within u(ν̃_line) = ±0.008 cm⁻¹. (5) Multi-Point Least-Squares Regression & Uncertainty Propagation: Linear regression of ν̃_mean(m) against m² across six rotational pairs yields an empirical intercept ν̃₀ = 2885.900 ± 0.002 cm⁻¹ and slope −α_e = −0.30719 ± 0.00008 cm⁻¹ with R² = 1.0000000. Propagating standard uncertainties in ν̃₀, c, and reduced mass μ through the harmonic relation k = 4π²c²ν̃₀²μ yields an expanded uncertainty for the harmonic force constant of U(k) = ±0.3 N/m at the 95% confidence level (coverage factor k = 2), establishing k = 480.7 ± 0.3 N/m, in flawless concordance with the NIST and IUPAC reference benchmark of 481.0 N/m.

10. Result and discussion

The high-resolution FTIR data provide deep insights into the quantum mechanics of molecular vibrations and rotations. Hydrogen chloride possesses a permanent electric dipole moment (μ_D = 1.11 D) that changes significantly with bond elongation, conferring immense infrared absorption intensity. The fundamental absorption band originates in transitions from the ground vibrational state (v = 0) to the first excited vibrational state (v = 1). Because HCl has a ¹Σ⁺ electronic ground state with zero electronic angular momentum about the internuclear axis, conservation of total angular momentum dictates that the photon's spin (S = 1) must be absorbed into molecular rotation, strictly enforcing the rotational selection rule ΔJ = ±1. Consequently, the pure vibrational transition (ΔJ = 0) is forbidden, creating the conspicuous central gap between R(0) at 2906.17 cm⁻¹ and P(1) at 2865.02 cm⁻¹. The observed progression of line intensities in both branches reflects the thermal Boltzmann distribution of rotational states in the v = 0 level. The population of rotational level J is governed by N_J ∝ (2J + 1) · exp[−B₀ J(J + 1) h c / (k_B T)], where the (2J + 1) spatial degeneracy factor initially dominates at low J, driving intensity upward to a peak maximum at J_max ≈ 3, before the exponential Boltzmann decay causes intensities to diminish at higher J. Each rotational line is cleanly split into an asymmetric doublet due to the presence of two chlorine isotopes: ³⁵Cl (75.8%) and ³⁷Cl (24.2%). Because ³⁷Cl is heavier, its reduced mass is larger, shifting its band origin down by 2.10 cm⁻¹ and yielding a constant doublet intensity ratio of approximately 3.1:1 across all lines. The vibration-rotation coupling constant α_e = 0.3072 cm⁻¹ reflects the asymmetric anharmonic curvature of the HCl potential energy well (Morse potential). Because the potential rises steeply at short distances (nuclear repulsion) and flattens at long distances (dissociation), vibrational excitation from v = 0 to v = 1 increases the average internuclear bond length from r₀ = 1.284 Å to r₁ = 1.293 Å, reducing the effective rotational constant from B₀ = 10.44 cm⁻¹ to B₁ = 10.13 cm⁻¹. The extracted harmonic force constant k = 480.7 N/m reflects the substantial stiffness of the covalent H−Cl single bond, in outstanding agreement with theoretical ab initio potential energy surfaces.

11. Precautions and limitations

• Anhydrous hydrogen chloride is a toxic, severely corrosive gas that forms dense white fumes of hydrochloric acid on contact with moist air; handle only within an approved chemical fume hood with certified scrubber ventilation.
• Protect the polished sodium chloride (NaCl) gas cell optical windows from atmospheric humidity and moisture; never touch window surfaces with bare fingers or breathe directly onto the cell.
• Verify that the gas cell is evacuated below 0.05 Torr prior to filling and inspect PTFE O-rings for leaks to prevent corrosive gas escape into the laboratory.
• Maintain continuous dry nitrogen purge of the FTIR spectrometer optical bench to prevent hygroscopic degradation of the KBr beam splitter and eliminate atmospheric water vapor absorption lines.
• Limit HCl gas cell fill pressure to ≤ 30 Torr; excessive pressure induces pressure-broadening (collisional broadening) that blurs the ³⁵Cl/³⁷Cl isotopic doublet splitting.
• Wear certified eye protection, lab coat, and chemical-resistant neoprene gloves when connecting gas manifold cylinders or operating cryogenic traps.

['The harmonic oscillator force constant k is extracted from the fundamental band origin ν̃₀, which includes minor anharmonic correction terms (ν̃₀ = ω_e − 2ω_e x_e).', 'At sample pressures exceeding 50 Torr, self-broadening obscures the narrow rotational lines, precluding baseline deconvolution of the ³⁵Cl and ³⁷Cl isotopologue peaks.', 'Centrifugal distortion constants (D_e ≈ 5.3 × 10⁻⁴ cm⁻¹) introduce minor high-J deviations (J > 8) that are neglected in the truncated quadratic model.'] The physical chemistry simulation model evaluates gas-phase H³⁵Cl at 25.0 °C (sample pressure 25.0 Torr, 100.0 mm path length). System parameters: reduced mass μ(H³⁵Cl) = 1.626652 × 10⁻²⁷ kg; fundamental band origin ν̃₀ = 2885.90 cm⁻¹; equilibrium rotational constant B_e = 10.5934 cm⁻¹; vibration-rotation coupling constant α_e = 0.3072 cm⁻¹; ground state rotational constant B₀ = 10.4398 cm⁻¹; excited state rotational constant B₁ = 10.1326 cm⁻¹. Across the six rotational pairs m = [1, 2, 3, 4, 5, 6], measured mean transition wavenumbers are ν̃_mean = [2885.593, 2884.671, 2883.135, 2880.985, 2878.220, 2874.841] cm⁻¹. Plotting ν̃_mean against m² = [1, 4, 9, 16, 25, 36] yields a linear slope of −α_e = −0.30719 cm⁻¹ (α_e = 0.3072 cm⁻¹) and Y-intercept ν̃₀ = 2885.900 cm⁻¹ with R² = 1.0000000. Applying the harmonic relation k = 4π²c²ν̃₀²μ extracts an empirical force constant k = 480.68 N/m (difference = 0.0660% from the 481.0 N/m literature benchmark), confirming quantum mechanical harmonic and anharmonic oscillator models.

12. Conclusion

High-resolution FTIR spectroscopy of gas-phase hydrogen chloride at 25.0 °C successfully resolved the ro-vibrational structure of the fundamental vibrational band at 0.50 cm⁻¹ resolution. Linear regression of the rotational mean transition model ν̃_mean(m) = −α_e · m² + ν̃₀ across six rotational pairs for H³⁵Cl yielded an empirical slope of −α_e = -0.30719 cm⁻¹ (α_e = 0.3072 cm⁻¹) and a band origin Y-intercept of ν̃₀ = 2885.900 cm⁻¹ with R² = 1.0000000. From ν̃₀ and the reduced mass μ = 1.62665 × 10⁻²⁷ kg, the harmonic bond force constant was determined to be k = 480.7 N/m (differing by only 0.0660% from the accepted literature benchmark of 481.0 N/m), and the equilibrium bond length was determined to be r_e = 1.275 Å, rigorously validating diatomic quantum mechanics.