Peng-Robinson EOS: Acentric Kappa & Alpha(T) Cohesion Function

Separate the three quantities

The acentric factor omega is a dimensionless property defined from reduced saturation pressure at T/Tc = 0.7: omega = −log10(psat/Pc) − 1. It is not a direct measurement of molecular polarity, and methane should not simply be assigned zero. Obtain omega and Tc from a suitable property source for the actual fluid.

This worksheet uses supplied illustrative values omega = 0.224 and Tc = 304.2 K. It evaluates the quadratic kappa correlation used in the lower-acentric-factor branch of Cantera’s Peng–Robinson implementation. The cutoff 0.491 selects that implementation’s correlation; it is not a universal physical-validity boundary. Higher values require an explicitly selected alternative rather than silently reusing this branch.

Calculate kappa in B6

Kappa depends only on omega. The numeric guard rejects blank and text inputs. Negative omega is not automatically invalid, but this worksheet does not establish the empirical range for any substance. Use property data and validate the chosen equation of state for the intended conditions. Keep the unrounded kappa value for the next calculation.

=IF(ISNUMBER(B2),IF(B2>0.491,"Choose a high-acentric-factor correlation",0.37464+1.54226*B2-0.26992*B2^2),"Enter numeric acentric factor")

Calculate reduced temperature in B7

Both temperatures must be numeric and greater than zero in kelvin. Tr = T/Tc is dimensionless. The initial Tr is about 1.150558843. A value above one only says that temperature exceeds Tc; without pressure and a phase calculation it does not establish a supercritical state.

=IF(COUNT(B3:B4)<>2,"Enter numeric temperatures",IF(OR(B3<=0,B4<=0),"Temperatures must be positive kelvin",B4/B3))

Calculate alpha in B8

Alpha = [1 + kappa(1 − sqrt(Tr))]^2 scales the attractive term in this model. It is not the complete equation of state and does not calculate pressure, density, fugacity, phase equilibrium or mixture behavior.

With the supplied inputs kappa is about 0.706562734 and alpha at 350 K is about 0.899983372. At T=Tc, alpha=1 algebraically, without implying that pressure is also critical. At 250 K alpha is about 1.136420595. Format for display without rounding B6 before using it.

For positive kappa the expression initially decreases as temperature rises above Tc, but it is not always below one for every T>Tc: the squared bracket can turn upward at extreme extrapolation. This is a mathematical behavior, not validation of those extreme temperatures. The familiar above/below-one trend also cannot be asserted unconditionally for every kappa.

=IF(ISNUMBER(B6),IF(ISNUMBER(B7),(1+B6*(1-SQRT(B7)))^2,B7),B6)

Further reading

• Cantera 3.1 — PengRobinson implementation (kappa and alpha)
• CoolProp — EquationOfState acentric-factor definition