A pure-component pressure p i for species i in a gas mixture may be defined as the

Question:

A pure-component pressure p_i for species i in a gas mixture may be defined as the pressure that species i would exert if it alone occupied the mixture volume. Thus,
where y_i is the mole fraction of species i in the gas mixture, Z_i is evaluated at p_i and T, and V is the molar volume of the gas mixture. Note that p_i as defined here is not a partial pressure y_iP, except for an ideal gas. Dalton’s “law” of additive pressures states that the total pressure exerted by a gas mixture is equal to the sum of the purecomponent pressures of its constituent species: P = Σ_ip_i . Show that Dalton’s “law” implies that Z = Σ_iy_iZ_i , where Z_i is the compressibility factor of pure species i evaluated at the mixture temperature but at its pure-component pressure.