Problem S9.1. Assume at low density the direct correlation function can be approximated by C(q) = (e-v

(a) - 1). For a low-density hard-sphere gas (radius R), use the Ornstein-Zernicke equation to obtain the first-order correction (in density) to the ideal gas expression for the isothermal compressibility. Show that your result agrees with the result obtained when starting from the virial expansion, P/nkBT = 1+nB2, where n N/V and B2 is the second virial coefficient.