where formula_65 is a numerical constant of order formula_66, formula_67 is the Boltzmann constant, and formula_2 is the mean free path, which measures the average distance a molecule travels between collisions. Since formula_2 is inversely proportional to density, this equation predicts that thermal conductivity is independent of density for fixed temperature. The explanation is that increasing density increases the number of molecules which carry energy but decreases the average distance formula_2 a molecule can travel before transferring its energy to a different molecule: these two effects cancel out. For most gases, this prediction agrees well with experiments at pressures up to about 10 atmospheres. On the other hand, experiments show a more rapid increase with temperature than formula_71 (here formula_2 is independent of formula_60). This failure of the elementary theory can be traced to the oversimplified "elastic sphere" model, and in particular to the fact that the interparticle attractions, present in all real-world gases, are ignored.