The COSMO-RS method allows to calculate the chemical potential of a compound in the liquid phase, as well as in the gas phase, see the the COSMO-RS theory that was discussed before and Ref.[2]. In the ADF COSMO-RS implementation the following equations were used to calculate properties using these chemical potentials.
| ∑i xi | = | 1 |
| pipure | = | exp {(μipure-μigas)/RT} |
| pivapor | = | xi exp {(μisolv-μigas)/RT} |
| pvapor | = | ∑i pivapor |
| yivapor | = | pivapor/pvapor |
| γi | = | exp {(μisolv-μipure)/RT} |
| ai | = | γi xi |
| GE | = | ∑i xi (μisolv-μipure) |
| HE | = | ∑i xi (EiHB+Eimisfit-EiHB pure-Eimisfit pure) |
| kH | = | 1/Vsolvent exp {(μigas-μisolv)/RT} |
| kHcc | = | kH RT |
| ΔGsolv | = | μisolv-μigas + RT ln(Vsolvent/Vgas) |
| xiSOL | = | 1/γi (T>Tm) |
| xiSOL | = | 1/γi exp {ΔHfus(1/Tm - 1/T)/R - ΔCp(ln(Tm/T) - Tm/T + 1)/R} (T<Tm) |
| log10Psolv1/solv2 | = | 1/ln(10) (μisolv2-μisolv1)/RT + log10(Vsolv2/Vsolv1) |
The above equations are not always exact, some assume ideal gas behavior, for example.
The molar fraction xi of each compound i of the solvent should add up to 1.
With the COSMO-RS method it is possible to predict vapor pressures. However, it is also possible to use experimental vapor pressures of pure compounds as input data for the calculation. This may increase the accuracy of the calculated vapor pressures in a mixture, for example.
In the COSMO-RS method the volume of 1 molecule in the liquid phase is approximated with the volume of the molecule shaped cavity, that is used in the COSMO calculations. In this way it is possible to calculate the volume of 1 mole of solvent molecules in the liquid phase. However, for properties that depend on such volumes, one can also use (related) experimental data as input data for the calculation.
The calculation of the boiling temperature of a solvent is performed with an iterative method. The temperature is varied until the calculated vapor pressure is within a certain threshold of the desired pressure.
Also the calculation of solubility of compound i is performed with an iterative method, since the activity coefficient γi depends on the molar fraction of this compound. The COSMO-RS method does not predict ΔHfus, ΔCp, or Tm. These can be given as input data for the calculation of solubility calculations of solid compounds.
| Quantity | Meaning |
| R | Gas constant |
| T | Temperature |
| μigas | The chemical potential of the pure compound i in the gas phase |
| μipure | The chemical potential of the pure compound i in the liquid phase |
| μisolv | The chemical potential of compound i in a liquid solution |
| xi | The molar fraction of compound i in a liquid solution |
| yi | The molar fraction of compound i in the gas phase |
| γi | Activity coefficient of compound i in a liquid solution |
| ai | Activity of compound i in a liquid solution |
| pipure | The vapor pressure of the pure compound i |
| pivapor | The partial vapor pressure of compound i |
| pvapor | The total vapor pressure |
| GE | The excess Gibbs free energy |
| HE | The excess enthalpy |
| EiHB pure | The hydrogen bond energy of the pure compound i in the liquid phase, see Ref.[2] |
| EiHB | The partial hydrogen bond energy of compound i in a liquid solution |
| Eimisfit pure | The misfit energy of the pure compound i in the liquid phase, see Ref.[2] |
| Eimisfit | The partial misfit energy of compound i in a liquid solution |
| kH | Henry's law constant: ratio between the liquid phase concentration of a compound and its partial vapor pressure in the gas phase |
| kHcc | dimensionless Henry's law constant: ratio between the liquid phase concentration of a compound and its gas phase concentration |
| ΔGsolv | The solvation Gibbs free energy (at 1 atm) |
| Vsolvent | Volume of 1 mole of solvent molecules in the liquid phase |
| Vgas | Volume of 1 mole of molecules in the gas phase (at 1 atm, ideal gas) |
| xiSOL | Solubility of compound i in a solvent (molar fraction) |
| ΔHfus | The enthalpy of fusion of compound i |
| ΔCp | The Δ heat capacity of fusion of compound i |
| Tm | The melting temperature of compound i |
| log10Psolv1/solv2 | The logarithm of the partition coefficient P, which is the ratio of the concentrations of a compound in two immiscible solvents, solvent 1 and solvent 2 |
