The description of components that associate or hydrogen-bond is difficult. An alternative model to the one considered in the previous problem is the continuous association model, in which 

Assume that this associating fluid is described by the van der Waals equation, and in the equation-of-state representation the parameters of the j-mer are gotten from

 Using the notation 

N₀ = number of moles if no association occurs 

N_T = number of moles when association occurs and assuming the binary interaction parameters are all zero since the species are so similar, 

a. Obtain expressions for a and b for the mixture in terms of a₁, b₁, N₀, and N_T only. 

b. Show that the ratio

is independent of index j and obtain an explicit expression for this ratio. (Note that ∅̅_j is the fugacity coefficient of species j.) 

c. Obtain an expression for the equation of state of this associating fluid that contains only P, V , T, N₀, a₁, b₁, and the equilibrium constant K. (Does your equation reduce to the van der Waals equation for a nonassociating one-component system in the limit of K → 0?)

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A₁ + A₁ = A2 A₁ + A₂ = A3 A₁ + An = An+1 etc. here all integers to ∞ are included