Determine pressure-molar volume and temperature of a gas

Question

Response to the following problem:

If P, v, and T are the pressure, molar volume, and temperature of a gas and P_c, v_c, and T_c are the critical pressure, critical molar volume, and critical temperature, then the reduced pressure P_R, the reduced molar volume V_R, and the reduced temperature T_R are defined as

P_R = P/Pc ,      v_R =v/vc ,   T_R = T/Tc

(a) Show that, in terms of reduced quantities, the van der Waals equation becomes

(P_R + 3/v²_R) (v_R-1/3) = (8/3) T_R

When the van der Waals equation is in this form, the material constants a and b do not appear explicitly. Thus, all gases that obey the van der Waals equation may be considered in the same state when the values of P_R, V_R, and T_R are the same (i.e., each gas is measured in units of its particular values of P_c, v_c, and T_c). This is the principle of corresponding states, which is a principle of universal similarity established first by van der Waals.

(b) Plot three curves for P_R as a function of V_R, one for T = ½T_c, one for T = Tcand one for T = 2T_c. What happens physically when the equation indicates three allowed values of VR for a single PR and T?