Solutions of 1,4-polybutadiene (1,4-PB, 98% cis) and of 1,2-polybutadiene (1,2-PB) in n-butane (n-C-4) were studied with respect to their vapor pressure and to their demixing into two liquid phases under isochoric conditions within the temperature range from 25 to 75 degreesC. 1,2-PB mixes homogeneously with n-C-4 at any ratio, in contrast to 1,4-PB, which exhibits a miscibility gap extending from practically pure solvent to approximately 40 wt % polymer. Corresponding to these solubility differences, the vapor pressures for the system n-C-4/1,4-PB are considerably higher than for n-C-4/1,2-PB at the same concentration and temperature. The experimental results are modeled accurately and consistently by means of a modified Flory-Huggins approach accounting explicitly for chain connectivity and conformational variability of the polymers. The vapor pressures calculated by means of the Sanchez-Lacombe theory agree very well with the experimental data for both systems; this approach fails, however, in the case of the liquid/liquid phase equilibria because it predicts similar miscibility gaps for both polymers. The modified Flory-Huggins approach explains the fundamentally different solubility of 1,2-PB and 1,4-PB in terms of pronounced dissimilarities in their conformational response to dilution, which is in the case of 1,4-PB strongly impeded by the double bonds of the main chain.