Engineering & Science
Rackett Equation: Saturated Liquid Molar Volume & Density
Predict saturated liquid molar volume V_sat = V_c * Z_c^((1 - T / T_c)^(2/7)) (cm^3/mol) and mass density ρ_L = MW / V_sat (g/cm^3) for LPG storage tanks and pressurized liquid pipelines.
Why cubic gas equations miss liquid densities—and how Rackett’s (2/7) exponent nails them!
Standard cubic equations of state (Redlich-Kwong, SRK, Peng-Robinson) are notorious for missing saturated liquid densities by 8%–15%, creating costly custody-transfer and relief-valve sizing errors in LPG and LNG storage! Harold G. Rackett (Journal of Chemical & Engineering Data, 1970) discovered a remarkably accurate corresponding-states correlation requiring only critical volume V_c, critical temperature T_c, and critical compressibility Z_c: V_sat = V_c * Z_c^((1 - T / T_c)^(2 / 7))!
Check pressurized liquid n-butane (MW = 58.12 g/mol, T_c = 425.1 K, V_c = 255.0 cm^3/mol, Z_c = 0.274): at ambient storage T = 298.15 K (25 °C), Rackett predicts V_sat = 101.97 cm^3/mol and liquid density ρ_L = 0.5700 g/cm^3 (experimental = 0.573 g/cm^3, within 0.5%!); heating the vessel to T = 380.0 K swells molar volume 26.4% to 128.93 cm^3/mol (ρ_L = 0.4508 g/cm^3)—showing why LPG bullets must never be liquid-filled past ~80–85% ullage!
| n-Butane Storage Temperature T (Tc = 425.1 K) | Reduced Temp Tr = T / Tc | Rackett V_sat (cm^3/mol) | Liquid Density ρL (g/cm^3) |
|---|---|---|---|
| Ambient Sphere Storage (298.15 K) | 0.7014 | 101.97 cm^3/mol | 0.5700 g/cm^3 |
| Hot Process Feed Drum (380.00 K) | 0.8939 | 128.93 cm^3/mol | 0.4508 g/cm^3 (-20.9% Swell!) |
| Critical Point Limit (T = Tc = 425.1 K) | 1.0000 | 255.00 cm^3/mol (= Vc) | 0.2279 g/cm^3 |
Rackett Saturated Liquid Molar Volume V_sat (cm^3/mol)
Evaluates V_c * (Z_c ^ ((1 - T / T_c) ^ (2 / 7))) for subcritical liquid temperatures T <= T_c.
=IF(OR(B2 <= 0, B3 <= 0, B2 > B3, B4 <= 0, B5 <= 0, B5 >= 1), "Invalid Rackett Inputs", IF(B2 = B3, ROUND(B4, 2), ROUND(B4 * (B5 ^ ((1 - B2 / B3) ^ (2 / 7))), 2)))Saturated Liquid Mass Density ρ_L = MW / V_sat (g/cm^3)
Divides molecular weight B6 (g/mol) by unrounded Rackett molar volume to return liquid specific gravity / density in g/cm^3.
=IF(OR(B2 <= 0, B3 <= 0, B2 > B3, B4 <= 0, B5 <= 0, B5 >= 1, B6 <= 0), "Invalid Rackett Inputs", ROUND(B6 / (B4 * (B5 ^ (IF(B2 = B3, 0, (1 - B2 / B3) ^ (2 / 7))))), 4))Try the example
Use the original values or change them to test your understanding.