P-V-T Equations of State

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There are two simple paths in P-T space to calculate the volume at simultaneous and  T:




The path in red, is the "isothermal path". The properties such as V0T and K0T at P = 0 are calculated at the temperature of interest, and then used in an isothermal EoS at the needed temperature.  In EosFit7 any thermal expansion model can be combined with any isothermal equation of state and two types of models for the variation of the bulk modulus with temperature, which can be selected with the cross command in the input utility.


The path in blue, is the "thermal pressure path". The pressure at the final volume is calculated from the isothermal EoS at the reference temperature. Then the increase in pressure due to isobaric (fixed-volume) heating to the temperature of interest is added in. The idea of thermal pressure is therefore that the total pressure at a given V,T can be expressed as the sum of two terms:



P(V,Tref) is determined only from the isothermal equation of state for the material at the reference temperature, but with the ‘observed’ volume from P and T.


Pth(T) is the pressure that would be created by increasing the T from Tref to T at constant volume, along an isochor. The thermal pressure at Tref is thus zero, so at Tref the EoS reduces to the isothermal EoS, as required. The thermal pressure at other temperatures clearly depends on the bulk modulus.


Because an isochor in P-T space has the slope

  

the thermal pressure induced by heating along the isochor is

 


There is nothing theoretically superior about thermal-pressure EoS over the isothermal approach, because they also require assumptions to be made about the EoS parameters and behaviour; first along the isothermal compression path in exactly the same way as used in the isothermal approach, and then also along the isochor.  For example, one can assume that αVKTR is constant along the isochor, so that Pth = αVKTR(T-T0), , but this is clearly thermodynamically wrong because it will still give (dKTR/dT)P non-zero at low T. See Angel et al. (2018) for further details.



EosFit7 provides three thermal-pressure models.