Virial Coefficients and the Classical Theory of Fluids for the Square-Well Potential

The term g2(r) proportional to the square of the density in the expansion of the radial distribution function in powers of the density has been evaluated for a square-well potential for the Born—Green—Yvon integral equation, the Kirkwood integral equation, the Percus—Yevick integral equation, the hyperchain approximation, and an integral equation derived by Meeron. The square-well potential used has a hard-core diameter of σ and attractive diameter of 2σ. Each g2(r) is then used to calculate Dp, the fourth virial co-efficient calculated from the pressure equation, and Dc, that calculated from the compressibility equation, for the above five liquid theories. This allows a comparison over the entire temperature range, rather than at infinite temperature, the only value given by the previously used hard sphere potential. It is shown that the Percus—Yevick equation yields virial coefficients in best agreement with the exact value over a large temperature range. All the results presented are low-order polynomials in f=exp(ε/kT—1), where ε is the depth of the well, and reduce to the hard-sphere results when f goes to zero.