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A class of soft-core equations of state is constructed by introducing a temperature dependent molecular volume parameter into hard-core equations of state. Characteristic curves are calculated numerically for a soft-core version of the Frisch model, parameterized by a “softening temperature” Ts , and a repulsive potential exponent n. The entire range of behaviour from the extreme hard-core (Ts = ∞) to soft-core (Ts = 0) limit is investigated.

A general proof is given that the classical second virial coefficient satisfies the requirement for the non-existence of a termination point of any locus of Cv extrema. This validity criterion is applied to some proposed forms for the second virial coefficient. The order of the termination temperatures is verified for a fairly general intermolecular potential. In particular a proof is given that TF lies between TC and TA. Also the hard-core limit of the ratio TD/TA(∼2) is examined briefly.

Explicit formulae are presented for those characteristic curves which can be constructed for a hard-core equation of state. Excellent agreement is obtained with experimental curves for argon up to fairly high pressures (20 P c) and temperatures (10 T c). It is shown that only three termination temperatures are obtainable from a hard-core type second virial coefficient. The limitations of the hard-core equation of state are discussed briefly.

Characteristic curves and loci of Cp extrema are constructed for the Lennard-Jones 6–12 equation of state, using fitted and approximate equations of state, and the exact virial expansion at low densities. The shape of the Cp locus near its termination temperature is analyzed via the third and fourth ‘pressure’ virial coefficients for hard-core and Lennard-Jones systems.

A study is made of the internal energy of a simple fluid, and its derivative with respect to particle number at constant temperature and volume. This derivative occurs in a formula linking energy and particle number fluctuations to the specific heat, and may be calculated exactly for the Ising and certain other models along the critical isochore. Experimental results for fluid Argon are compared with the description provided by van der Waals' equation, which is able to predict within 5% the slope of the loci in the density-temperture plane along which the internal energy and its particle number derivative vanish

A detailed discussion is given of the geometrical properties and the physical significance of the characteristic curves of a simple fluid, with specific reference to experimental results for argon.

The six termination temperatures associated with the ten characteristic curves of a simple fluid are calculated for the Lennard-Jones m, n potential second virial coefficient with n ≥ m > 3. The extreme values in both the hard-core Sutherland potential limit n → x, and in the opposite limit n → m are obtained. The termination temperature ratios Tc/Tb, TF/TC and TD/TA lie within a narrow finite range, with TD/TA → 2 in the hard-core limit, independent of m. The high temperature form of the second virial coefficient is derived, and used to estimate thesoftening temperature Ts. Also, some results for the square-well potential are presented.

An elementary theory of termination temperatures of characteristic curves is developed with the aid of two simple mathematical models. The first model contains two important constant ratios involving all six termination temperatures. The second model interpolates between the first model and the hard-core limit with the aid of a parametric softening temperature, Ts The properties of the temperature ratios and the underlying models are discussed in detail, especially in the hard-core limit.

Loci of Cp extrema along isotherms are constructed for model soft-core equations of state, parameterized by an exponent N(= 3/n, where n is the repulsive potential exponent) and a softening temperature Ts . Generally the loci exhibit two branches whose geometry depends on Ts and N. In soft-core type behaviour, a locus of Cp maxima commences at the critical point and terminates on the temperature axis at a temperature Tn where the second virial coefficient has a point of inflexion, and the second branch is located at higher pressures and temperatures. In hard-core type behaviour, a locus of Cp maxima commences at the critical point and turns into a locus of minima before crossing the fusion curve, whereas the second branch, which terminates at TD , is generally a locus of minima lying at high pressures and temperatures. The values of Ts and N at which the geometry of the loci changes is studied in detail.

The six termination temperatures associated with the ten characteristic curves of a simple fluid are calculated for the Kihara potential second virial coefficient, constructed from a Lennard-Jones m, n potential augmented by a spherical hard-core. Extreme values of the termination temperatures in both the hard-core Sutherland-type potential limit n → ∞, and in the opposite limit n → m are obtained. Over the useful range of values, 0 to 1, of the ratio a∗ of the hard-core diameter to the molecular diameter in the absence of a hard-core, the termination temperature ratios TC/TF/TC and TD/TA vary only slightly, for a given value of n, with TD/TA → 2 in the hard-core limit n → ∞, independent of m and a∗.

A model soft-core equation of state is used to account qualitatively for the behaviour of the Joule-Kelvin coefficient μ and its (initial) partial derivatives at high temperatures. The two sign changes which occur in (∂μ/∂P) T 0 are present in a hard-core model, but the minimum in (∂μ/∂T) P 0 , at TD , is a soft-core effect. A hard-core model is adequate in the low temperature and density region around the inversion curve, but contours of μ at high temperatures and pressures are modified qualitatively when a soft-core is introduced.

The role of the exponent inequality β + γ > I in the continuity of (∂P/∂T), with the slope of the vaporization curve at the critical point is discussed. A connection between the divergence in the curvature of the vaporization curve and asymmetries in certain thermodynamic functions is established. It is shown, in the context of the scaling law equation of state, that the homogeneous phase specific heat C(1) V on the liquid branch of the coexistence curve is slightly lower than on the gas branch, at the same temperature.

Calculation of CP from a model soft-core equation of state reveals a line in the phase diagram on which CP is equal to its zero pressure value C P 0 . This line commences on the temperature axis where the second virial coefficient has a point of inflexion. At higher temperatures (and pressures) CP falls below C P 0 . The detailed behaviour of CP is presented via contour maps, illustrating the effects of changing the exponent N(= 3/n, where n is the repulsive potential exponent) which parameterizes the model. For soft-core fluids at high temperatures CP deviates only slightly from the ideal gas value over a wide range of temperature and density, in marked contrast to the behaviour of hard-core models.