Engineering & Science
Lorentz-Lorenz Molar Refractivity (RM) & Clausius-Mossotti Polarizability Formula
Calculate phase-invariant molar refractivity R_M = ((n^2 - 1) / (n^2 + 2)) * (MW / rho) in cm^3/mol and electronic molecular polarizability alpha_0 (in Angstrom^3) from optical refractive index n and density rho.
Bridge macroscopic refractive index n directly to single-molecule electron-cloud volume!
Compress water vapor into liquid water and its density jumps ~1,250-fold while refractive index jumps from 1.00025 to 1.333—yet Hendrik Lorentz and Ludvig Lorenz (1880, extending the electrostatic Clausius-Mossotti local-field equation (eps_r - 1)/(eps_r + 2) via Maxwell's n^2 = eps_r) proved that Molar Refractivity R_M = ((n^2 - 1) / (n^2 + 2)) * (MW / rho) stays virtually CONSTANT across gas, liquid, and solid phases!
Even more remarkably, dividing R_M (in cm^3/mol) by (4 * PI() / 3) * N_A = 2.5228e24 mol^-1 and multiplying by 10^24 Angstrom^3/cm^3 gives the microscopic Gaussian polarizability volume alpha_0 = R_M / 2.5228 in cubic Angstroms—the literal deformable volume of one molecule's electron cloud! Liquid water (n = 1.333, MW = 18.015 g/mol, rho = 0.998 g/cm^3) has R_M = 3.713 cm^3/mol (alpha_0 = 1.472 A^3), whereas pi-conjugated toluene (n = 1.496, MW = 92.14, rho = 0.867) jumps to R_M = 31.045 cm^3/mol!
| Pure Liquid / Dielectric | Refractive Index n, MW, Density | Molar Refractivity R_M (cm^3/mol) | Polarizability Volume alpha_0 (A^3) |
|---|---|---|---|
| Liquid Water (H2O, 20 °C) | n = 1.333, MW = 18.015, rho = 0.998 | 3.713 cm^3/mol | 1.472 Angstrom^3 |
| Toluene (C7H8 Aromatic) | n = 1.496, MW = 92.14, rho = 0.867 | 31.045 cm^3/mol | 12.306 Angstrom^3 |
| Invalid Index n <= 1 or Density <= 0 | Vacuum / Domain Guard | Guarded Input Error | "Invalid Optical/Density Inputs" |
Lorentz-Lorenz Molar Refractivity R_M (cm^3 / mol)
Evaluates (((n^2) - 1) / ((n^2) + 2)) * (MW_g_mol / rho_g_cm3).
=IF(OR(B2 <= 1, B3 <= 0, B4 <= 0), "Invalid Optical/Density Inputs", ROUND((((B2 ^ 2) - 1) / ((B2 ^ 2) + 2)) * (B3 / B4), 3))Electronic Polarizability Volume alpha_0 in Cubic Angstroms (A^3)
Divides R_M by (4 * PI() / 3) * 0.602214076 (= 2.522548... exact SI/CGS Avogadro factor).
=IF(OR(B2 <= 1, B3 <= 0, B4 <= 0), "Invalid Optical/Density Inputs", ROUND(((((B2 ^ 2) - 1) / ((B2 ^ 2) + 2)) * (B3 / B4)) / ((4 * PI() / 3) * 0.602214076), 3))Try the example
Use the original values or change them to test your understanding.