Consider a gas of N particles in volume V. Around each atom there are two zones, as pictures. The inner zone (of volume b) is a hard-core repulsion into which no other atom can tread (infinite positive potential energy for another atom there). The region A has volume a and is associated with an attractive potential energy of -E_1 if another atom is in that region. Work in the dilute gas limit, thus compute the partition function of the gas as Z_N = 1/N! Z^N_1 where Z_1 is the partition function of a single gas atom in the environment of a chamber of volume V filled with N other atoms. Write down the partition function Z_N in terms of V, a, b, E_1 etc. Use (a) to determine the pressure as a function of volume and temperature... that is, derive the associated equation of state. Show that in the small a and b/V limit the the result of (b) is indistiguishable to the Van der Waals equation of state (!!)