In 1890 van der Waals published an article that initiated the study of fluid mixtures. It was subsequently included as Part III of a later published version of his thesis. His essential idea was that in a binary mixture of vdw fluids described by the equations p 1 = R T v − − b 11 − − a 11 v 2 and p 2 = R T v − − b 22 − − a 22 v 2 the mixture is also a vdW fluid given by p = R T v − − b x − − a x v 2 where a x = a 11 x 1 2 + 2 a 12 x 1 x 2 + a 22 x 2 2 and b x = b 11 x 1 2 + 2 b 12 x 1 x 2 + b 22 x 2 2 Here x 1 = N 1 / N , and x 2 = N 2 / N , with N = N 1 + N 2 (so that x 1 + x 2 = 1 ) are the mole fractions of the two fluid substances. Adding the equations for the two fluids shows that p ≠ ≠ p 1 + p 2 , although for v sufficiently large p ≈ ≈ p 1 + p 2 with equality holding in the ideal gas limit. The quadratic forms for a x and b x are a consequence of the forces between molecules. This was first shown by Lorentz, and was credited to him by van der Waals. The quantities a 11 , a 22 and b 11 , b 22 in these expressions characterize collisions between two molecules of the same fluid component while a 12 = a 21 and b 12 = b 21 represent collisions between one molecule of each of the two different component fluids. This idea of van der Waals was later called a one fluid model of mixture behavior.