1.627507e-27
kg
2.625006e-47
kg·m^2
—
cm^-1
—
cm^-1
—
cm^-1
1.627507e-27
kg
2.625006e-47
kg·m^2
—
cm^-1
—
cm^-1
—
cm^-1
The Rotational Spectroscopy Calculator computes the key parameters of microwave rotational spectroscopy for diatomic molecules: reduced mass, moment of inertia, rotational constant, energy levels, and transition frequencies. Rotational spectroscopy probes the rotation of molecules with exquisite precision, providing the most accurate bond lengths available for small molecules.
For a rigid diatomic molecule, the rotational energy levels are quantized: E_J = B*J*(J+1) (in cm^-1 units), where B = h/(8*pi^2*I*c) is the rotational constant and I = mu*r^2 is the moment of inertia. The quantum number J = 0, 1, 2... determines the rotational state. The selection rule for pure rotational transitions is delta_J = +1 (absorption), giving transition frequencies nu = 2B*(J+1), equally spaced by 2B.
The rotational constant B is inversely proportional to the moment of inertia: larger, heavier molecules have smaller B and more closely spaced rotational lines (in the microwave, GHz range). Hydrogen chloride (HCl) has B ~ 10.6 cm^-1 (in the far-infrared), while larger molecules have B values of 0.1-1 cm^-1 (microwave).
Measuring the spacing between rotational lines directly gives 2B, from which r = sqrt(h/(8*pi^2*mu*c*B)) gives the equilibrium bond length with picometer precision. This is how the most accurate bond lengths are determined for gas-phase molecules.
Rotational spectroscopy is used in radio astronomy to identify interstellar molecules (over 200 molecular species detected, including amino acid precursors), atmospheric sensing (monitoring greenhouse gases), and precision measurement of fundamental constants.
Reduced mass: mu = m1*m2/(m1+m2) in kg. Moment of inertia: I = mu*r^2. Rotational constant: B = h/(8*pi^2*I*c) in cm^-1. Energy of level J: E_J = B*J*(J+1) cm^-1. Transition J to J+1: nu = 2*B*(J+1) cm^-1. Convert to GHz: multiply cm^-1 by 29.98 GHz/cm^-1.
The spectrum shows equally spaced lines at 2B, 4B, 6B... cm^-1. The spacing 2B directly gives B, from which bond length r is calculated. Typical B for HCl: 10.6 cm^-1 (far-IR). For CO: 1.92 cm^-1 (microwave). For large organic molecules B can be 0.01-0.1 cm^-1 (pure microwave). Centrifugal distortion at high J causes small deviations from equal spacing.
Inputs
Results
HCl has B = 10.59 cm^-1 (literature: 10.59 cm^-1). The J=1 to J=2 transition is at 2B*(1+1) = 4B = 42.4 cm^-1. The J=0 to J=1 (shown) transition at 21.2 cm^-1 falls in the far-infrared.
Inputs
Results
CO's J=0 to J=1 transition is at 2B = 3.84 cm^-1 (115 GHz in microwave). This transition is used by radio astronomers to map CO in molecular clouds throughout the galaxy.
B = h/(8*pi^2*I*c) in cm^-1 (or B = hbar^2/(2*I) in Joules). It is inversely proportional to the moment of inertia. Larger B means faster rotation, smaller molecule, or smaller moment of inertia.
For pure rotational (microwave) absorption, delta_J = +1 and the molecule must have a permanent dipole moment (which is why N2 and O2 have no rotational spectrum in the microwave — they are homonuclear diatomics with zero dipole moment).
At temperature T, the most populated level is J_max ~ sqrt(kT/(2*hc*B)) - 1/2. At room temperature for HCl (B~10.6 cm^-1), J_max ~ 3. For larger molecules with small B, J_max can be 10-100.
Extremely accurate — bond lengths can be determined to 0.001 pm (femtometer) precision, making rotational spectroscopy the gold standard for gas-phase molecular geometry.
At high J values, the centrifugal force stretches the bond, increasing r and decreasing B. This causes the rotational lines to be slightly closer together at high J. The effect is parameterized by the centrifugal distortion constant D: nu = 2B(J+1) - 4D(J+1)^3.
Yes. Over 200 interstellar and circumstellar molecules have been identified by matching observed radio and microwave frequencies to laboratory rotational spectra. This includes simple molecules like CO, HCN, and complex organics like glycolaldehyde.
A molecule with two equal principal moments of inertia (like CH3Cl). Its rotational spectrum requires two quantum numbers J and K, giving more complex but still analyzable patterns compared to the linear molecule (diatomic) case.
Isotopically substituted molecules (e.g., H^35Cl vs H^37Cl, or 12CO vs 13CO) have the same geometry but different moments of inertia. Measuring rotational constants for multiple isotopologues allows solving for all bond lengths in polyatomic molecules.
A modern technique using short microwave pulses to create a coherent rotational polarization that then freely precays, giving a time-domain signal that Fourier transforms to a frequency-domain spectrum with very high resolution and sensitivity.
From B = h/(8*pi^2*mu*r^2*c), we get r = sqrt(h/(8*pi^2*mu*c*B)). Measuring B precisely from the line spacing gives r directly. This is the most accurate way to determine bond lengths.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Electron Configuration Calculator
Atomic & Molecular Physics Calculators
Ionization Energy Calculator
Atomic & Molecular Physics Calculators
Electron Affinity Calculator
Atomic & Molecular Physics Calculators
Electronegativity Calculator
Atomic & Molecular Physics Calculators
De Broglie Wavelength Calculator
Atomic & Molecular Physics Calculators
Heisenberg Uncertainty Calculator
Atomic & Molecular Physics Calculators
