Here are two facts about a substance:

  • The entropy is the following function of T and V:
S = R{{V_0 } \over V}\left( {{T \over {T_0 }}} \right)^{\alpha  - 1}
  • At constant temperature T_0 the work the substance does on its surroundings as it expands from V_0 to V is
\left. W \right|_{T_0 }  = RT_0 \ln {V \over {V_0 }}

(a)Find the Helmholtz free energy F, assuming that it is zero at the state values specified by the subscript 0.

(b)Find the equation of state of this stuff. Is there a point in parameter space where it is ideal?

(c)For this part we’ll simplify the algebra by assuming that \ln {V \over {V_0 }} = 1 and also \alpha = 1. At what temperature T' would the system do twice as much work in going from V_0 to V as it does at T = T_0?