Some useful expressions for deriving component fugacity coefficients from mixture fugacity coefficient

It is proved that the fugacity coefficient (ϕ) of a mixture and the fugacity coefficient of component i ( ϕ ˆ i ) follow the relation ln   ϕ ˆ i = ∂ ( n ln   ϕ ) / ∂ n i T , P , n j ( j ≠ i ) , = ∂ ( n ln   ϕ ) / ∂ n i T , P , V t , n j ( j ≠ i ) , where ni is the amount-of-substance of component i, and T, P, Vt and n are the temperature, pressure, total volume and total amount-of-substance of the system, respectively. This relation is very useful for the derivation of ϕ ˆ i . It allows the ϕ function to contain both P and V, so ϕ can be expressed with various equations of state (either volume-explicit or pressure-explicit). This approach can also use the similarity of the ϕ expressions of pure fluids and mixtures to simplify the derivation of ϕ ˆ i . Besides, ϕ ˆ i can also be directly derived from equation of state through a variation of the approach above. These approaches are much easier than the commonly used ones when the equation of state for fluids is complex.