A comparative study on the thermodynamic stability of the lying down (LD) and standing up (SU) phases of alpha,omega-butanedithiol (BDT) on unreconstructed (U) and on reconstructed (R) Au(111) surfaces is presented. The R surface is made of dithiol-Au adatom units. Density functional calculations (DFT) allow the estimation of the adsorption energy of the LD and SU BDT phases on both substrates. Surface free energies based on the DFT calculations show the coverage of the clean Au(111) surface by the ID phase, and the ID to SU phase transition as the chemical potential of the BDT molecule is increased. The LD and SU phases are more stable on R than on U substrates, suggesting that the Au(111) surface should reconstruct upon BDT adsorption. The stability analysis is extended to longer alpha,omega-dithiols. Results reveal that the to SU phase transition is favored as the hydrocarbon chain length of the dithiol molecule is increased. Changes in the hydrogen pressure affect the formation of the ID phase, while they have only minor effects on the LD to SU phase transitions. Our calculations explain the influence of the number of carbon atoms in the hydrocarbon chains, hydrogen pressure and dithiol pressure (or concentration) on dithiol adsorption, and phase transitions. This information is relevant to control the coverage, reactivity, and surface chemistry of the alpha,omega-dithiol self-assembled monolayers on Au surfaces.