A generalized temperature and acentric factor dependent function of the attractive term, called the alpha function, of the Twu-Coon-Cunningham (TCC) cubic equation of state (CEOS) (1992) has been developed. The approach developed in this work makes the alpha function become a linear function of the acentric factor at a constant reduced temperature, not a fourth order or a sixth order function as suggested by Soave and other researchers. The advantage of a linear function in the acentric factor is obvious in the extrapolation of the alpha function to heavy hydrocarbons and petroleum fractions. The new generalized alpha function, when used with the TCC CEOS, allows the accurate reproduction of the vapor pressure data from the triple point to the critical point for hydrocarbons. The new CEOS provides much more reliable and accurate vapor pressure prediction than the SRK (Soave, 1972) or Peng and Robinson (PR) (1976) equations for both light and heavy hydrocarbons.