Idealise nitrogen as obeying the van der Waals equation of state exactly
PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Thermal, Statistical & Complex Systems · Calculus of change and accumulation
Conditions of Validity, Ch Differentiability, the implicit function theorem and the licence to change variables
Problem
Idealise nitrogen as obeying the van der Waals equation of state exactly. One mole of nitrogen obeys the van der Waals equation of state p=V−bRT−V2a, where p is the pressure, V the molar volume, T the temperature, R=8.314Jmol−1K−1, a=0.1408Pam6mol−2 and b=3.913×10−5m3mol−1. Take V>b throughout.
((a)) State the hypothesis of the implicit function theorem that must hold at a state for this equation to define V as a differentiable function of p and T nearby. Show that the states where it fails, taken over all temperatures together, lie on one curve p=ps(V) in the (V,p) plane, and find ps(V).
((b)) Show that ps(V) is highest at V=3b. Find the temperature of the isotherm through that point, and show that ∂2p/∂V2=0 there as well, so that the point is the critical point. Give its molar volume, pressure and temperature for nitrogen, to 3 significant figures.
((c)) On the isotherm T=110K, find the two molar volumes at which V(p,T) loses its licence, to 3 significant figures. The condition is a cubic in V; solve it with a calculator or computer polynomial solver, and keep only roots with V>b.
((d)) At the state T=110K, V=1.20×10−4m3mol−1, compute (∂V/∂T)p. Decide, with a reason taken from the implicit function theorem rather than from arithmetic, whether the cyclic relation (∂V∂p)T(∂T∂V)p(∂p∂T)V=−1 holds at this state, and say what is physically unusual about the signs of the partial derivatives there.