Alexander Kud
email: alexander.e.kud@yandex.com

Christophe Coquelet
email: christophe.coquelet@mines-albi.fr

Improvements for a Fully Consistent Description of the New Semi-Empirical Vapor Density Model for Pure Compounds

Keywords: vapor density, thermodynamic consistency, fluoromethane (R41), difluoromethane (R32)

In chemical engineering, surrogate functions are commonly used in modeling thermodynamic properties of pure compounds because they are often more accurate than equations of state. In 2020, we presented a semi-empirical model describing the vapor density from the reduced triple point temperature to about a reduced temperature of 0.97 with only one temperature-dependent term. However, in many cases the temperature range up to critical temperature is necessary. Therefore, the existing semi-empirical model was extended by another temperature-dependent term. This allows the vapor density to be described from the triple point temperature up to the critical temperature. This model has the additional advantage that it can predict the vapor pressure data from vapor density data. Therefore, this model can also be used for thermodynamic consistency testing. This is not possible with conventionally known surrogate models for describing vapor density. The model was successfully tested using fluoromethane (R41) and difluo-romethane (R32) as examples.

Introduction

In chemical engineering, reactors and apparatuses are often designed by considering the presence of liquid and/or vapor phases. In order to correctly realize the mass balance for both phases, their densities must be correctly calculated. This aspect is valuable for the design of reactor where mass and volume fractions depend on density calculation. In effect, an incorrect density prediction does not only influence the calculation results of the Vapor Liquid Equilibrium (VLE) but also the result of a kinetic modeling, if chemical reactions have to be considered.

Because equations of state are not so accurate to calculate state variables at saturation like vapor and liquid densities, surrogate functions are used for modeling. Lists of such functions for pure substances are documented, for example, at the National Institute of Standards and Technology (NIST) [1] and at the Dortmund Data Bank [2]. In general, the equations for vapor density given there are empirical and have some weaknesses due to their lack of physical meanings. In our previous work (Kud et al. [3]), we presented a Semi-empirical Model (SeM) that considers compressibility factor as thermodynamic quantity and also Kud et al. [3] give an overview of the currently used surrogate models in Tab. 1, with which the new model is compared. The compressibility factor for vapor describes a graph characteristic of a compound as a function of temperature, and according to Nezbeda [4], it is called a "strong test of precision and correctness". In 2020, an empirical equation (Z model) has been developed for the compressibility factor of vapor, which represents the physical properties from the triple point to the critical temperature. The concept and ideas that have been considered for the development of the new Z model are the core of the vapor density model (SeM, ref [3]). Also, the vapor density model can be used not only for vapor density description. It can be used, according to Eggimann et al. [5], to check and evaluate the quality of molecular simulations with the transferable potentials for phase equilibria (TraPPE) model and also to check the quality of experimental data. For example, 20 compounds of different structure and polarity can be represented in a "master plot" with one graph, see abstract and Fig. 5 in their publication [5].

The semi-empirical vapor density model was successfully tested for its principal suitability on 17 compounds from different families of mole-cules. These molecules have different structure and polarity.

In many cases, however, it is necessary to calculate the vapor density and thus the compressibility factor from the triple point to the critical temperatures. This work is the continuity of our previous paper [3] and it is an extension of first model from triple point (and below) to the critical point combining pure component vapor pressure and compressibility factor equation at saturation. For this purpose, an additional, tempera-turedependent term is used in the Z-model. The thermodynamic inconsisten-cies occurring in the parameterization at the level of the triple point temperature must be eliminated by taking into account the corresponding physical property values. The results are compared with the state of the art. For this purpose, two polar compounds, namely fluoromethane (R41) and difluoromethane (R32) are used as examples. This is the subject of this work.

The basic idea of the new SeM will be briefly explained on the basis of a schematic representation in figure 1. The compressibility factor 𝑍 for the vapor state combines the two state variables vapor pressure 𝑝 and vapor density 𝜚 at the absolute temperature 𝑇 according to the equation

𝑍 = 𝑝𝑉 𝒏ℛ𝑇 = 𝑝𝜐 ℛ𝑇 = 𝑝ℳ 𝜚ℛ𝑇 , ( 1
)
where 𝑉 is the volume, 𝒏 the amount of matter, 𝜐 the molar volume, ℳ the molar mass of the compound and ℛ the universal gas constant. Introducing the reduced state variables for a pure compound according to 𝜏 = 𝑇 𝑇 𝑐 ⁄ , 𝜋 = 𝑝 𝑝 𝑐 ⁄ (where 𝑇 𝑐 is the critical temperature, 𝑝 𝑐 the critical pressure) and the constants 𝜚 𝑐 for the critical density and 𝑍 𝑐 for the critical compressibility factor, we obtain the following equation for the vapor density. 𝜚 = 𝜚 𝑐 𝑍 𝑐 𝜋 𝜏 𝑍 .

(

) 2
The Semi-empirical vapor density model is obtained by substituting a vapor pressure function 𝑓 𝑝 (𝜏, 𝛩 𝑝 ) for the reduced vapor pressure 𝜋 and by substituting the empirical 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) model

𝑍 𝑀 (𝜏, 𝛩 𝑧 ) = 𝑍 𝑐 + (1 -𝑍 𝑐 ) • { [1 -( 𝜏-𝛩 𝑧3 •𝜏 ̅ 𝑡𝑝 1-𝛩 𝑧3 •𝜏 ̅ 𝑡𝑝 ) 𝛩 𝑧1 ] 𝛩 𝑧2
𝑖𝑓 𝜏 𝑇𝑝 ≤ 𝜏 ≤ 0.96

𝑛𝑜𝑡 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑓 𝜏 > 1 (3)
for the compressibility factor 𝑍 of the vapor in equation (2) where 𝛩 𝑝 and 𝛩 𝑧 are the parameter vectors. Thus, the vapor density 𝜚 𝑀 can be calculated according to the model equation 𝐹 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 )

𝜚 𝑀 = 𝐹 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 ) = 𝜚 𝑐 𝑍 𝑐 𝑓 𝑝 (𝜏, 𝛩 𝑝 )

𝜏 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) .

The parameterization of equation ( 4) is based on the measured vapor pressures 𝑌 ̅ 𝑝 and vapor densities 𝑌 ̅ 𝜚 . Figure 1 illustrates this situation. At this point, a definition of the vapor state as it is considered in this work should be given in order to avoid misunderstandings with the gas state. The vapor state is always linked to the presence of a liquid and/or solid phase with which it is in phase equilibrium. This means that there is an exchange of substances between the two phases and the density of vapor is constant over time. Thus, there is a so-called Vapor Liquid Equilibrium (VLE) or a Vapor Solid Equilibrium (VSE). The gas state, on the other hand, exists independently of the presence of a liquid or solid state. For the sake of simplicity, all variables and quantities in this paper refer to the vapor state.

2 Theoretical background and computational modeling

The most important equations should be presented before the presentation of our strategy of modeling. The description in the arguments of the equations should be as exact as possible, so that the dependencies of the calculation results become clearly recognizable and conscious. Afterwards simplified representations in the arguments are defined. We use the vapor pressure function 𝐹 𝑝 (𝜏, 𝑇 𝑐 , 𝑝 𝑐 , 𝛩 𝑝 ) respectively 𝑓 𝑝 (𝜏, 𝑇 𝑐 , 𝛩 𝑝 ) for calculating reduced vapor pressure according on the rearranged Riedel equation [6]. 𝑓 𝑝 (𝜏, 𝑇 𝑐 , 𝛩 𝑝 ) is a semi-empirical model and is based on the corresponding state principle. This equation with the parameter vector 𝛩 𝑝 is also called DIPPR (Design Institute for Physical Properties) 101 equation and is given by Eqs. (5)(6) 𝐹 𝑝 (𝜏, 𝑇 𝑐 , 𝑝 𝑐 , 𝛩 𝑝 ) = 𝑝 𝑐 • 𝑓 𝑝 (𝜏, 𝑇 𝑐 , 𝛩 𝑝 )

𝑓 𝑝 (

The parameter Θ 𝑝4 is an integer and arbitrarily kept constant to one of the values from 1 to 6, while the remaining parameters are adjusted. The minimum of the objective function from the six parameter estimates sets the value of the optimal parameter Θ 𝑝4,𝑜𝑝𝑡 . Hogge et al. [7] have tested this equation for the temperature range considered in our work here on many compounds of different structure and they found it is a very good equation.

𝑍 𝑀 (𝜏, 𝛩 𝑧 ) ≡ 𝑍 𝑀 (𝜏, 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑧 ) . ( 14)

The first reference temperature for the vapor density Eq. ( 8) is the critical temperature. If the critical quantities 𝑇 𝑐 , 𝑝 𝑐 , 𝜚 𝑐 are kept constant on the same value for all calculations or simulations, there will be no numerical problems.

The critical temperature concerns the model equation (7)

) 17
If these equation constraints are satisfied, there is no need to consider additional constraints on the critical point to the optimization algorithm, see Section 2.1 in reference [3]. The second reference temperature is the triple point temperature, which will be analyzed in more detail in the following section.

Investigation of the temperature region around the triple point temperature

Simulations have shown that the result of parameter estimation leads to solutions that give an excellent fit, but are physically meaningless. Unfortunately, optimization algorithm cannot distinguish between a "mathematical solution" and a "physical solution". As it doesn't exist a lot of experimental vapor density data near the triple point temperature, it increases the difficulty of having a good fitting and so a good model. In order to obtain the good solution with physically meaningful fitted parameters, additional information about the thermodynamic state must be added to the optimization algorithm.

To get a better understanding of the triple point description of the model, the variation of compressibility factor of the vapor phase 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) at the vicinity of the triple point temperature as a function of the reduced temperature has to be analyzed (Fig. 2).

For temperature higher than the triple point temperature there is Vapor Liquid Equilibrium (VLE) and for temperature lower than the triple point temperature there is Vapor Solid Equilibrium (VSE). The lower limit of the model 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) lies in the VSE region and describes the vapor state as ideal.

We considered the vapor phase as ideal ( 𝑍 𝑖𝑑 = 1) at an ideal reduced temperature (𝜏 𝑖𝑑 ). This is the point (𝜏 𝑖𝑑 , 𝑍 𝑖𝑑 = 1). 𝜏 𝑖𝑑 can be calculated considering an adjustable parameter Θ ̂𝑧3 and the triple point temperature according to Eq. ( 18)

𝜏 𝑖𝑑 = 𝛩 𝑧3 • 𝜏 𝑡𝑝 . ( 18
)
Mathematically, at 𝜏 𝑖𝑑 we have the vertex of the model 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) where 𝑍 𝑀 (𝜏 𝑖𝑑 , 𝛩 𝑧 ) = 1 and the derivative is equal to zero. Therefore, the product 𝛩 𝑧3 • 𝜏 𝑡𝑝 shall be called reduced ideal gas temperature 𝜏 𝑖𝑑 in our model. On the Fig. 2, the point (𝜏 𝑖𝑑 , 1) with the reduced ideal gas temperature is considered as the weakness of the model. It is difficult to reach ideal gas behavior as pressure must be equal to zero. However, the difference of the real gas behavior from the ideal one will most probably be very small. It is possible in the temperature range 𝜏 ∈ ] 𝜏 𝑖𝑑 , 𝜏 𝑡𝑝 ] to extrapolate compressibility factor from the VLE to the VSE regions and to consider that it exists a supercooled vapor below the triple point temperature. From these considerations, additional reasonable constraints on thermodynamic state variables have to be considered. . 𝑝 𝑡𝑝 denotes the pressure at the triple point based on measurements or reliable calculations. The result of the optimization considering the formulation problem (PE 1) is the estimated parameter vector 𝛩 ̂𝑝. The objective function for parameter estimation is based on the underlying stochasticity. More information can be found in the textbooks by Larsen, Marx [8] and Seber, Wild [9]. If there are enough experimental data near the triple point temperature, the optional constraint of the state variable for the triple point pressure 𝑝 𝑡𝑝 in (PE 1) can be omitted. In such a case, the triple point pressure can be calculated with the estimated parameters according to the Eq. ( 24)

𝑝̂𝑡 𝑝 = 𝐹 𝑝 (𝜏 𝑡𝑝 , 𝛩 ̂𝑝) . ( 24
)
If the random standard deviations 𝑆 𝑖 is unknown, then Monte Carlo Simulations (MCS) can be used to simulate a random standard deviations 𝕊 𝑝𝑖 for the 𝛺 𝑝 matrix

𝛺 𝑝 = 𝑛 ̅ -1 • 𝑑𝑖𝑎𝑔 (𝕊 𝑝1 2 , 𝕊 𝑝2 2 … 𝕊 𝑝𝒩 𝑝 2 ), (25)
𝑛 ̅ is a calculated mean sample size for all measurement points. This procedure is described in detail in the supporting material of reference [3]. The triple point pressure belongs to the second reference state for the vapor density model and is needed for the calculation of the compressibility factor in the next section. If the standard deviations of the vapor density experimental data are not available, the same procedure as already described for the standard deviations of the vapor pressure measurements can be used (section 2.2.1).

In the (PE2) problem formulation, each equation / inequality with a constraint on the estimation parameters lead to a reduction of the degree of freedom (𝑑𝑜𝑓) calculated from all measurement points, if the constraint is reached numerically. The reduction is one 𝑑𝑜𝑓 for each parameter to be estimated. Thus, in this case, the maximum 𝑑𝑜𝑓 reduction can be 3 or 6 units for the inequalities. The 𝑑𝑜𝑓 reduction for the four equations to the state variable constraints is additional four units. In the case of an error propagation calculation for 𝑍 𝑀 , it must be taken into account that the critical quantities also have a measurement error; they must not be assumed to be constant.

If constraints are reached by the parameter estimation algorithm, then the objective function is often not in minimum. This means that an error propa-gation calculation must not be calculated according to the equations often quoted which are valid for the minimum.

2.2.3 Formulation of the parameter estimation problem (PE 3) for the evaluation of the vapor density and vapor pressure data simultaneously As already presented in the previous section, the result of the vapor density parameterization depends on the vapor pressure parameters. As a consequence, the vapor pressure 𝑝̂𝑡 𝑝 estimated from Eq. ( 24) and the vapor density 𝜚 𝑀 calculated from Eq.( 8) may result in a compressibility factor that does not match with the compressibility factor 𝑍 ̃𝑡𝑝 (Eq.( 26) or ( 27)) determined using experimental values. Moreover, two different compressibility factors exist for the same thermodynamic state at the triple point and it can signify that there is some possible thermodynamic inconsistency. The main reason is attributed to the vapor pressure model, which doesn't take into account the compressibility factor value at the triple point. To avoid this inconsistency, it is recommended to estimate the vapor pressure parameters 𝛩 𝑝 and the parameters 𝛩 𝑍 simultaneously with the model equation 𝑓 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 ) (Eqs.( 13) and ( 9)). This means to adjust simultaneously the model's parameters with both vapor pressure and vapor density data. For this purpose, it is necessary to define new vectors of state variables and matrix (Eqs. ( 28) to ( 33)).

𝑌 ̅ ≔ ( 𝑌 ̅ 𝜚 𝑌 ̅ 𝑝 ) (28)
𝛺 𝜚 = 𝑑𝑖𝑎𝑔 ( 𝑆 𝜚1 2 𝑛 𝜚1 , 𝑆 𝜚2 2 𝑛 𝜚2 … 𝑆 𝜚𝒩 𝜚 2 𝑛 𝜚𝒩 𝜚 ) ( 29
)
𝛺 𝑝 = 𝑑𝑖𝑎𝑔 ( 𝑆 𝑝1 2 𝑛 𝑝1 , 𝑆 𝑝2 2 𝑛 𝑝2 … 𝑆 𝑝𝒩 𝑝 2 𝑛 𝑝𝒩 𝑝 ) (30) 𝛺 ≔ ( 𝛺 𝜚 0 0 𝛺 𝑝 ) (31) 𝛩 ≔ ( 𝛩 𝑧 𝛩 𝑝 ) (32) 𝐹(𝜏, 𝛩) ≔ ( 𝐹 𝜚 (𝜏, 𝛩) 𝐹 𝑝 (𝜏, 𝛩) ) (33)
Since these above definitions are very important for the success of the parameterization, the most important equations for the case of two terms in the 𝑍 𝑀 model (Eq. ( 7)) shall be formulated explicitly. The parameter vector is defined by Eq. (34

The vapor pressure equation is defined by Eq. ( 35)

𝑓 𝑝 (𝜏, 𝛩) = 𝑒𝑥𝑝 [ 𝛩 7 𝑇 𝑐 (1 - 1 𝜏 ) + 𝛩 8 ⋅ 𝑙𝑛( 𝜏) + 𝛩 9 ⋅ 𝑇 𝑐 𝛩 10 (𝜏 𝛩 10 -1)] . ( 35
)
The reduced vapor density for calculating 𝐹 𝜚 (𝜏, 𝛩) in Eq. ( 33) is given by Eq. ( 36)

𝑓 𝜚 (𝜏, 𝛩) = 𝑍 𝑐 𝑓 𝑝 (𝜏,𝛩) 𝜏 • 𝑍 𝑀 (𝜏,𝛩) . ( 36
)
In Eq. ( 34), only parameters numbered 1 to 9 are estimated. The number of terms in the 𝑍 𝑀 model and the integer exponent of the DIPPR 101 equation can be taken from the separate parameterizations from formulations problems (PE 1) and (PE 2). The simultaneous parameter estimation realized in formulation problem (PE 3) will not change these values.

(PE 3) Formulation of the constrained parameter estimation problem for evaluation of vapor density and vapor pressure data simultaneously

𝑚 𝑖 𝑛 𝛩,𝐹 (𝑌 ̅ -𝐹(𝜏, 𝛩)) 𝒯 𝛺 -𝟏 (𝑌 ̅ -𝐹(𝜏, 𝛩)) s.t. 𝑍 𝑀 (𝜏 𝑖𝑑 , 𝛩) -1 = 0 (𝛩 8 = 𝒩 𝑧 = 1) ∨ (𝛩 11 = 𝒩 𝑧 = 2) 𝜏 𝑖𝑑 -𝛩 3 • 𝜏 𝑡𝑝 = 0 ( 𝜕𝑍 𝑀 (𝜏,𝛩) 𝜕𝜏 ) 𝜏 𝑖𝑑 = 0 𝑍 𝑀 (𝜏 𝑡𝑝 , 𝛩) -𝑍 ̃𝑡𝑝 = 0 𝐹(𝜏 𝑡𝑝 , Θ) 1 -𝜚 𝑡𝑝 = 0 𝑂𝑅 𝐹(𝜏 𝑡𝑝 , Θ) 2 -𝑝 𝑡𝑝 = 0 } optional ( 𝜕𝑍 𝑀 (𝜏, 𝛩) 𝜕𝜏 ) 𝜏 *
< 0 (𝜏 * = 𝜏 𝑡𝑝 , 0.6, 0.7, 0.8, 0.9)

𝐹(𝜏 𝑖 , 𝛩) 1 > 0 (𝑖 = 1, 2 … 𝒩 𝜚 ) 1.005 ≤ 𝛩 1 < 9 0.01 < 𝛩 2 < 1 0 < 𝛩 3 < 1 } 𝑖𝑓 (𝛩 8 = 𝒩 𝑧 = 1) 1.005 ≤ 𝛩 1 < 9 0.01 < 𝛩 2 < 1 0 < 𝛩 3 < 1 1.005 ≤ 𝛩 4 < 9 0.01 < 𝛩 5 < 1 0 ≤ 𝛩 6 ≤ 1} 𝑖𝑓 (𝛩 11 = 𝒩 𝑧 = 2) Θ 7 = 1 ∨ 2 … ∨ 6 𝑖𝑓 (𝛩 8 = 𝒩 𝑧 = 1) Θ 10 = 1 ∨ 2 … ∨ 6 𝑖𝑓 (𝛩 11 = 𝒩 𝑧 = 2)
where

𝑌 ̅ = (𝑌 ̅ 1 , 𝑌 ̅ 2 … 𝑌 ̅ 𝒩 𝜚 +𝒩 𝑝 ) 𝒯 is the measurement vector of means with the dimension 𝒩 𝜚 + 𝒩 𝑝 , 𝛺 ≔ ( 𝛺 𝜚 0 0 𝛺 𝑝 )
is the diagonal matrix of the corresponding variances and sample sizes.

If the experimental data for the vapor density 𝜚 𝑡𝑝 or the vapor pressure 𝑝 𝑡𝑝 at the triple point are the same regarding their confidence, then neither of the two additional equations for 𝐹 𝜚 and 𝐹 𝑝 should be used as a state variable constraint in formulation problem (PE 3). Both quantities are then consistently estimated by the optimization program.

The simultaneously estimated parameter (formulation problem (PE3)) values can differ significantly from the separately estimated parameters (formula-tion problems (PE1) and (PE2)). This is especially true for the Θ 𝑍 parameters. The standard deviations for these parameters can be considerably reduced by considering simultaneous parameterization. The reduction of the standard deviation values of the parameters depends on quality of the database. Taking into account the constraints in the adjustment of the model parameters depends on the trustworthiness of the experimental data including density at the triple point and the vicinity of the triple point.

Capability to predict vapor pressure data from vapor density data based on (PE 4)

If one looks at the vapor density model according to equations ( 8) and ( 9), then it should be possible in principle to estimate vapor pressure parameters, even if no vapor pressure data exist. Why? The semi-empirical vapor density model includes the vapor pressure model. In a virtual or mind experiment, we take one experimental vapor pressure data at a time from the database. This increases the standard deviation of the vapor pressure model parameters and also the Z model parameters. The largest standard deviation is achieved in which all experimental vapor pressures data are not considered in the data treatment. The standard deviation of the estimated vapor pressure model parameters depends only on the uncertainty of the experimental vapor density data. Such a parameterization actually works for this model and can be demonstrated via an analysis of the Fisher information matrix [11] or the variance-covariance matrix of the parameters. The interested reader can refer to SuppMat F for more details. Simulations for fluoromethane (R41) and difluoromethane (R32) have shown that the experimental vapor density data have too much uncertainties to obtain satisfactory predictive capability of our model without vapor pressure information. At least one experimental vapor pressure data is recommended with existing experimental vapor density data. Often, the boiling temperature at normal pressure is specified. This vapor pressure information should be provided. The parameter estimation problem (PE4) for this case is formulated as follows.

(PE 4) Formulation of the constraint parameter estimation problem for the prediction of vapor pressure data from vapor density data with one vapor pressure as a constraint

𝑚 𝑖 𝑛 𝛩,𝐹 (𝑌 ̅ 𝜚 -𝐹(𝜏, 𝛩)) 𝒯 Ω 𝜚 -1 (𝑌 ̅ 𝜚 -𝐹(𝜏, 𝛩)) s.t. 𝑍 𝑀 (𝜏 𝑖𝑑 , 𝛩) -1 = 0 (𝛩 8 = 𝒩 𝑧 = 1) ∨ (𝛩 11 = 𝒩 𝑧 = 2) 𝜏 𝑖𝑑 -𝛩 3 • 𝜏 𝑡𝑝 = 0 ( 𝜕𝑍 𝑀 (𝜏,𝛩) 𝜕𝜏 ) 𝜏 𝑖𝑑 = 0 𝑍 𝑀 (𝜏 𝑡𝑝 , 𝛩) -𝑍 ̃𝑡𝑝 = 0 𝐹(𝜏 𝑡𝑝 , Θ) 1 -𝜚 𝑡𝑝 = 0 Optional 𝐹(𝜏 𝑏𝑝 , Θ) 2 -𝑝 𝑏𝑝 = 0 ( 𝜕𝑍 𝑀 (𝜏, 𝛩) 𝜕𝜏 ) 𝜏 *
< 0 (𝜏 * = 𝜏 𝑡𝑝 , 0.6, 0.7, 0.8, 0.9)

𝐹(𝜏 𝑖 , 𝛩) 1 > 0 (𝑖 = 1, 2 … 𝒩 𝜚 ) 1.005 ≤ 𝛩 1 < 9 0.01 < 𝛩 2 < 1 0 < 𝛩 3 < 1 } 𝑖𝑓 (𝛩 8 = 𝒩 𝑧 = 1) 1.005 ≤ 𝛩 1 < 9 0.01 < 𝛩 2 < 1 0 < 𝛩 3 < 1 1.005 ≤ 𝛩 4 < 9 0.01 < 𝛩 5 < 1 0 ≤ 𝛩 6 ≤ 1} 𝑖𝑓 (𝛩 11 = 𝒩 𝑧 = 2) Θ 7 = 1 ∨ 2 … ∨ 6 𝑖𝑓 (𝛩 8 = 𝒩 𝑧 = 1) Θ 10 = 1 ∨ 2 … ∨ 6 𝑖𝑓 (𝛩 11 = 𝒩 𝑧 = 2) where 𝑌 ̅ 𝜚 = (𝑌 ̅ 𝜚1 , 𝑌 ̅ 𝜚2 … 𝑌 ̅ 𝜚𝒩 𝜚 )
𝒯 is the measurement vector of means with the dimension 𝒩 𝜚 ,

Ω 𝜚 = 𝑑𝑖𝑎𝑔 ( 𝑆 𝜚1 2 𝑛 𝜚1 , 𝑆 𝜚2 2 𝑛 𝜚2 … 𝑆 𝜚𝒩 𝜚 2 𝑛 𝜚𝒩 𝜚 )
is the diagonal matrix of the corresponding variances and sample sizes.

Assessment quantities

The experimental data are necessary for the parameterization of our model. (37)

𝑚𝑎𝑥𝑀𝑅𝐷%(𝑈, 𝛩 ̂ ) = 100% • 𝑚𝑎𝑥 [| 𝑌 ̅ 𝑈,𝑖 -𝐹 𝑈 (𝜏 𝑖 ,𝛩 ̂) 𝑌 ̅ 𝑈,𝑖 | 𝑖 = 1,2 … 𝒩 𝑈 ] (38) 𝐵𝑖𝑎𝑠%(𝑈, 𝛩 ̂ ) = 100% 𝒩 𝑈 ∑ ( 𝑌 ̅ 𝑈,𝑖 -𝐹 𝑈 (𝜏 𝑖 ,𝛩 ̂) 𝑌 ̅ 𝑈,𝑖 ) 𝒩 𝑈 𝑖=1 . ( 39
)
The rank (Rk) provides information whether the parameters can be estimated independently or not. Also it permits to check if there is enough experimental data to estimate all the parameters or only some of them.

In the context of evaluating the vapor pressure prediction capability, two general assessment quantities have to be defined. The first one concerns the general assessment of a regression in which the estimated state variable (e.g., vapor pressure) is obtained from a corresponding experimental data (vapor pressure experimental data). It is not only used for testing the predictive capability, but can also be used in general. The second one concerns a special assessment where the estimated state variable (e.g. vapor pressure) is obtained from another experimental data (e.g. vapor density measurement). For both quantities, it is a quotient of the number of experimental data 𝒩 𝑌,𝑐𝑜𝑛𝑓 described by the model in a confidence interval of the data value and the total number of measurements 𝒩 𝑌 of the random variable Y (e.g. vapor pressure). If there are corresponding variables, 𝒩 𝑌,𝑐𝑜𝑛𝑓 depends on the database 𝑌 and is referred to 𝒩 𝑌,𝑐𝑜𝑛𝑓 (𝑌). If the number depends on the non-corresponding database 𝑋, it is referred to 𝒩 𝑌,𝑐𝑜𝑛𝑓 (𝑋). In this case, the regression runs on 𝑋. Thus, the so-called "fit capability" and the "predictive capability" can be defined by Equations ( 40) and (41).

𝐹𝑖𝑡𝐶𝑎𝑝% ≡ 𝐹𝑖𝑡𝐶𝑎𝑝% 𝑌𝑌 ≔ 𝒩 𝑌,𝑐𝑜𝑛𝑓 (𝑌) 𝒩 𝑌 • 100% (40) 𝑃𝑟𝑒𝐶𝑎𝑝% ≡ 𝑃𝑟𝑒𝐶𝑎𝑝% 𝑋𝑌 ≔ 𝒩 𝑌,𝑐𝑜𝑛𝑓 (𝑋) 𝒩 𝑌 • 100% . ( 41
)
The number 𝒩 𝑌,𝑐𝑜𝑛𝑓 is also identical to the number of residuals whose absolute value is smaller than a given accepted deviation. Thus, this assessment quantity can be easily calculated from the residuals. 𝑃𝑟𝑒𝐶𝑎𝑝% can most likely be applied to the assessment of EoS as well. These assessment quantities are rigorous if the measurement data base is free of outliers and the accepted deviation between model and measurement is realistically chosen. The meaningfulness of these two quantities is lost by "screwing" with the ac-cepted deviation, because 100% can be reached easily. Therefore, the accepted deviation, which is incidentally related to the confidence interval 𝑐𝑜𝑛𝑓, should also be specified for this variable.

In the case of the outlier test for the experimental vapor pressure data, a CoV% value of 0.5% was assumed as the maximum relative deviation acceptable. This value was also used for the calculation of 𝐹𝑖𝑡𝐶𝑎𝑝% and 𝑃𝑟𝑒𝐶𝑎𝑝% indicators.

Using MCS, mean values and standard deviations of the parameters and the above-mentioned assessment variablescan be calculated. The estimated model parameters and the assessment quantities become more robust by using MCS, because the fluctuations of the standard deviations are taken into account.

In this context, average values of thermodynamic properties, such as boiling temperature, triple point pressure or vapor density at the triple point, can also be determined using MCS. All the details concerning the utilization Monte Carlo simulations are described in Appendix C. All calculations were made with the software Mathcad 15 [12].

Results and Discussions

Statistical analysis of the raw data estimation of standard deviations and analysis of the outliers are some results of the parametrization. The results are presented in Supplementary Material (SuppMat) B. Also, the analysis of the properties of the pure compounds are presented in SuppMat A.

First, comparison between the results obtained by separate and simultaneous evaluation are discussed in terms of model's parameter values, then the model evaluation, the calculated state variables, and finally the state of the art. We would like to focus the discussion of individual results on only one compound (fluoromethane (R41)) in order to describe the principal method and thus streamline the content. The detailed results concerning difluoromethane (R32) are presented in a Supplementary Material.

The separate evaluations for vapor pressure and vapor density were calculated according to the formulation problems (PE 1) and (PE 2). This means that the estimated vapor density parameters depend on the previously, separately estimated vapor pressure parameters. The optional constraint in the formulation problem (PE1) for vapor pressure at the triple point temperature was not enabled. The simultaneous parameter estimation was calculated according to formulation problem (PE 3) without the optionally specified restrictions at the triple point, so that degrees of freedom exist to achieve thermodynamic consistency.

The estimated model's parameters for fluoromethane (R41) are presented in Table 1. Considering the three formulation problems, we can observe that the parameters for the vapor pressure model change only slightly. The two parameter vectors are also identical in a statistical sense. Only, the parameters Θ ̅ 𝑍,1 have significant different values. The results of simultaneous evaluation also show a significantly larger number value for the parameter Θ ̅ 𝑍,6 , which is important for the description of compressibility factor close to critical point. The parameter Θ ̅ 𝑍,3 estimates the modelspecific, so-called ideal gas temperature 𝑇 𝑖𝑑 . The smaller Θ ̅ 𝑍,3 is, the smaller 𝑇 𝑖𝑑 is in comparison to the triple point temperature and the greater is the assurance to be able to extrapolate to temperature lower than the triple point temperature where we have a supersaturated liquid. It should be mentioned that the result of the parameter estimation does not assign the first term to the critical temperature range in all cases. As is well known, the result of a nonlinear parameter estimation depends very much on the starting values, which are often element of a fractal. Guess parameter values in the convergence region are suggested (see Appendix D). The same applies to the parameters of the vapor pressure equation. In the context of the assignment of the two terms in the Z function (eq (7)) to a temperature range, an analysis of the graphs can provide information.

Fig. 3 shows the first term

𝛩 𝑧6 [1 -( 𝜏-𝛩 𝑧3 •𝜏 𝑡𝑝 1-𝛩 𝑧3 •𝜏 𝑡𝑝 ) 𝛩 𝑧1 ] 𝛩 𝑧2
and second term

(1 -𝛩 𝑧6 ) [1 -( 𝜏-𝛩 𝑧3 •𝜏 𝑡𝑝 1-𝛩 𝑧3 •𝜏 𝑡𝑝 ) 𝛩 𝑧4 ] 𝛩 𝑧5
with the components 𝛩 ̅ 𝑧6 and (1 -𝛩 ̅ 𝑧6 ), respectively. It can be clearly seen that the first term is predominantly responsible for describing compressibility factor Z in the temperature range below the critical temperature. The low and medium temperature range is described by both For simulations much more digits are necessary, see appendix 4. 4) model specific temperature.

PE 1 1) PE 2 2) PE 3 3) terms. Table E.1 in Appendix E presents the model's parameters obtained for difluoromethane (R32).

Fig. 3. Compressibility factor for fluoromethane (R41) vapor vs. reduced temperature. The total compressibility factor is the sum of the first and second term according to eq (7). Term 1 is mainly responsible for the description of the critical temperature range. The fraction for the 1 st term is 𝛩 ̅ 𝑧6 = 0.555 and that for the 2 nd term is (1 -𝛩 ̅ 𝑧6 ). Compressibility factors (•) calculated from vapor densities and the vapor pressure equation ( 5) with the parameters from (PE 3) with the vapor densities from the database. 𝑍 𝑐 = 0.2462.

Table 2 shows the results for the different evaluations concerning fluoromethane (R41). The MRD% values for the vapor pressure are equally small for both evaluations of the formulation problems (PE 1) and (PE 2). Also, it shows a very good agreement between model assumption and experimental data. The maximum RD% value is acceptable, as it still belongs to the range of variation of 0.5%.

The optimal value for the model bias (𝐵𝑖𝑎𝑠% ̅̅̅̅̅̅̅̅̅ ) is a statistical zero for experimental data. Both evaluations show a zero value because the standard deviation is very large in comparison to the average value. In this case, however, it is not a statistical zero because the residuals are not randomly distributed around zero. The residuals 𝑅𝐷% ̅̅̅̅̅̅̅ (𝑝) 𝑖 values in E) show clear structures. This is due to the correlated database. A very similar residual structure can also be seen in the publication by Lemmon et al. [14]. First, the relevant, physical quantities such as triple point pressure, vapor density at the triple point and the associated compressibility factor are discussed and then the quantities for the boiling point.

For the separate evaluation according to formulation problems (PE 1) and (PE 2), the predicted data by REFPROP 10.0 software and the results from formulation problems (PE 1) and (PE 2) are compared. The results from formulation problem (PE 3) may not be included concerning the discussion of the separate evaluation. Table 3 presents the results. In this work and for the simulations physical properties (constants), parameters and other quantities are considered. It is well known that in nonlinear modeling smallest deviations of these quantities can cause large deviations in the results.

In order to have identical results between the different working groups and the same exactness, it is important to consider numerical values with many digits.

Table 3 Comparison of the results of the physical properties of fluoromethane (R41) for the different parameter estimation formulations (PE 1) to (PE 3). Results are based on Monte Carlo Simu-lations.

PE 1 (1) PE 2 (2) PE 3 (3) Quantity Database (4) Average 4) calc with the FEoS in REFPROP 10.0, except Tb which is a mean from measurements.

(5) calculated Ztp from ptp(PE 1) and vapor density (database). ( 6) vapor pressure calculated from compressibility factor and vapor density both from (PE 2). n.r. not recommended.

From the estimated vapor pressure value of 347.0777 Pa at the triple point according to formulation problem (PE 1) and the vapor density value of 0.010884 kg/m 3 from the data base, we calculate a compressibility factor value of 1.005447, which is greater than 1 and thus physically nonsensical.

With the formulation problem (PE 2), we calculate a vapor density value of 0.010948 kg/m 3 and a compressibility factor value of 0.999550, which is less than 1 and thus physically meaningful. From this, a vapor pressure value of 347.0798 Pa is calculated, which is not identical with the vapor pressure of 347.0777 Pa obtained with formulation problem (PE 1). Even if one calculates a vapor density from the vapor pressure obtained with formulation problem (PE 1) and the compressibility factor obtained with formulation problem (PE 2), the result is different from the vapor density value obtained with formulation problem (PE 2)! The reason of these differences is probably due to the restriction applied to the compressibility factor in formulation problem (PE 2). During the minimization of the sum of the deviation squares, inevitably also the compressibility factor is slightly optimized to obtain an optimal vapor density description. For both evaluations according to formulation problems (PE 2) and (PE 3), different compressibility factors are optimized as well.

There are for the separate evaluation Should one choose the result of formulation problem (PE 2) from this "rich offer", because the vapor density model is clever and involves the vapor pressure equation, Moreover, still the inconsistency concerning different vapor pressure description between formulation problems (PE 1) and (PE 2) remains.

In the case of a simultaneous evaluation according to formulation problem (PE 3), the parameters are estimated in such a way that vapor pressure and vapor density state variables lead to one compressibility factor value thermodynamically consistent. For this reason, the simultaneous evaluation is preferable.

In view of the small differences, one can think that it does not matter whether the separate or simultaneous evaluation is calculated. But it is a question of recognizing the principal differences of the evaluation in the light of the thermodynamic consistency. The differences between the results obtained from formulation problems (PE1) and (PE2) turn out to be so small numerically here because the data basis is very good in the range of the triple point temperature. If the data basis is more uncertain near the triple point temperature, then most likely the differences between the evaluations will also be larger.

Table 3 still lists the physical properties at the fluoromethane (R41)'s boiling point. Considering the evaluations according to formulation problems (PE 1) and (PE 3), the same boiling temperature of 194.84 K are calculated. The value is also identical to that calculated by REFPROP 10.0 software. The vapor density at the boiling point is about 2% lower than that calculated by REFPROP 10.0. Table E.3 in SuppMat E shows the results obtained for difluoromethane (R32).

Often thermodynamic properties in the temperature range below the critical temperature are of great interest. But many equations of state, including fundamental equations, show weaknesses in the prediction of the thermodynamic properties in this temperature range. For comparison, compressibility factor was also calculated using the FEoS implemented in REFPROP 10.0 and the very commonly used PR EoS involving the Mathias Copeman (MC) α function [START_REF] Mathias | Extension of the Peng-Robinson Equation-of-state To Complex Mixtures: Evaluation of the Various Forms of the Local Composition Concept[END_REF]. The MC parameters for the PR EoS were fitted to the vapor pressure equation and are listed in Appendix E for Fig. E.4. The model equations were compared with experimental data that exist in this temperature range. Fig. 4 shows the results for fluoromethane (R41). In addition to the graphical representation, the statistical assessment quantities, presented in Table 4, were calculated for this temperature range. From Table 4, we can observe that only the semi-empirical vapor density model prediction are in good agreement with experimental data. The FEoS available in REFPROP 10.0 software shows the largest deviations, followed by the PR EoS. Now to the results of the exciting question whether vapor pressure data can be predicted from vapor density data. The following strategy is recommended for this:

1. Take all reliable vapor density data and only one vapor pressure data, e.g. at the boiling point or better activate a contraint for this point. Calculate the parameterization for all parameters based on the problem formulation (PE 4) for all parameters. Result is 𝛩 ̂.

2. Extract the parameter part from 𝛩 ̂ used for the vapor pressure prediction.

3. Calculate with this part of parameter vector the predicted vapor pressure data only and compare the values with the experimental vapor pressure data.

It is recommended to select a vapor pressure value in the middle temperature range of the vapor pressure curve to provide information about the convexity of the vapor pressure model to the optimization algorithm.

The results for fluromethane and difluoromethane are listed in Table 5. For better comparison, the results from formulation problem (PE 3) are also listed. First of all, it can be stated that in principle it is possible to predict the vapor pressure from vapor density using the empirical vapor density model. The condition is that only a trustworthy vapor pressure information in the middle of temperature range should be provided for the present database. This is not possible with the empirical vapor density models known so far like Wagner type equations.

From

1) maximal accepted deviation: 0.5 %. 2) Expected full rank of the variance-covariance matrix: 9

For the assessment according to formulation problem (PE 3), the Fit Capability 𝐹𝑖𝑡𝐶𝑎𝑝% shows that more than 99% of the experimental vapor pressure data can be described by the model. In contrast, the predictive power 𝑃𝑟𝑒𝐶𝑎𝑝% predicts 53% of the vapor pressure data for fluoromethane and only 16% for difluoromethane. The reason for the much lower predictive capability compared to the fit capability is the quality of the experimental vapor density data. The measurement errors are larger for difluoromethane than for fluoromethane.

The predictive capability 𝑃𝑟𝑒𝐶𝑎𝑝% can be significantly increased, but this has its price. Either experimental vapor density data with better accuracy and precision are measured in the laboratory, or either two trustworthy experimental vapor pressure data are provided. Qualitatively better experimental vapor density data are not available for the simulations within the scope of this work. Therefore, the predictions are calculated using two experimental vapor pressures data as constraints at two reduced temperatures 𝜏 1 = 0.5 and 𝜏 2 = 0.8, which were set arbitrarily. These two vapor pressures should come from very trustworthy measurements or calculations. In this case, experimental vapor pressure data were selected from the database with very small RD%(p) values. The results are listed in Table 6.

With only one more experimental vapor pressure data, more than 96% of the 293 or 716 experimental vapor pressure data can be predicted with a maximum error of ±0.5%. Now, three vapor pressure values could be formulated as constraints to achieve even greater accuracy. However, this must not be done, because there is no degree of freedom remaining to estimate the third vapor pressure parameter. Rank of cov matrix 2) of the parameter 9 9

1) maximal accepted deviation: 0.5 %. 2) Expected full rank: 9

Three unknown vapor pressure parameters are determined by three equations (constraints) and are not estimated. In this case, the number of degrees of freedom is zero. This should not be allowed in the statistical sense. In the number of degrees of freedom should be as large as possible. If more than two vapor pressure data are used, then the evaluation must be done according to the problem formulation (PE 3). Using Model-based Design of Experiments, the optimum temperatures of the vapor pressure data can be calculated.

The predictive capability according to the formulation problem (PE 4) can be used to check experimental vapor density data and also SeM for thermodynamic consistency. The following questions can be answered:

-Do the vapor density data match with the experimental vapor pressure data? -Are the vapor density and vapor pressure data thermodynamically consistent? -With known thermodynamic consistency of the experimental vapor density and experimental vapor pressure data, is the semi-empirical vapor density model appropriate for the compound under investigation?

Conclusions

The semi-empirical model can now be used to describe vapor densities for pure compounds for the full range from the triple point to the critical temperature. It was successfully verified using fluoromethane (R41) and difluoromethane (R32). This model can also be used to verify the reliability of vapor density data. The model can predict vapor pressure from vapor density data with only one reliable vapor pressure information. The predictive capability can be highly improved by providing two vapor pressure data at two different temperatures. In this case, for fluoromethane and difluoromethane, more than 96% of the vapor pressure data can then be predicted with a relative error of ≤0.5%. The predictive capability of the vapor pressure is strongly dependent on the quality of the vapor density data. For a supercooled liquid, the compressibility factor for the vapor phase could be extrapolated to a near temperature range below the triple point temperature. The uncertainty can be given by error propagation. Its value depends on the quality of the vapor density and vapor pressure data at the vicinity of the triple point temperature. In any case, the predictions need to be verified experimentally which is not in the scope of this work.

The compressibility factor for the vapor of fluoromethane and difluoromethane according to the Z-model agrees well with the average compressibility factor based on molecular simulations according to Eggimann et al. [5] within the limits of the mean deviations. In principle, molecular simulations according to Eggiman et al. can also be used to obtain additional information on whether the semi-empirical vapor density model is suitable for modeling a compound. We are optimistic that this semi-empirical model with two terms can be applied to other compounds as well. The precondition is that there are no large fractions of oligomers in the vapor phase. • There are homogeneous and heterogeneous variances.

𝜏

• There is a homogeneous and heterogeneous structure in CoV.

• There are unrealistically small CoV values (3.3•10 -5 ) that most likely relate to apparatus uncertainties and not to uncertainties of a whole experiment replicate. • There are most likely also assumed and no calculated uncertainties.

• There is most likely also calculated vapor pressure data.

• There are measurement data whose uncertainties or standard deviations are much larger than the measured vapor pressure.

Experimental data with unrealistically small CoV values were replaced by a realistic CoV value of about 0.008. This value is based on a CoV analysis of experimental vapor pressure data and experimental pressure data at the critical point and agrees very well with experience values for pressure data over a wide temperature range. A mean sample size 𝑛 ̃ of about 17 can be estimated from the Student t distribution (see eq (2D-8) in SupMat 2D in ref [3]). In this way, standard deviation and sample size can be estimated. We are aware that, strictly speaking, this is not possible because of an underdetermined equation system. But with plausible, practical assumptions, a good estimate of an unknown mean standard deviation and mean sample size can be made. In this way, parameter estimation with standardized residuals becomes possible.

The next step is to detect the outliers. For this purpose, it is assumed that the Riedel vapor pressure equation [7], also known as the DIPPR 101 equation, is the "true" function describing the vapor pressure data, since it describes the vapor pressure data very well compared to other known equations (see list of equations in ref [2]).

Since the raw data set has vapor pressure data near the triple point temperature, it will be checked especially for outliers. For this purpose, the vapor pressure calculated with FEoS [A.5] is assumed to be very trustworthy with REFPROP 10.0 at the triple point. Therefore, the first parameter estimation is calculated according to formulation problem (PE 1) with constraint on the triple point pressure. All vapor pressure data whose relative deviations are greater than 0.5% in absolute value or whose weighted or standardized residuals are greater than two standard deviations are eliminated. For ~75 % of the 33 eliminated outliers, both exclusion criteria apply together. The remaining 25% are excluded only by the relative error of 0.5%. Now, the database for the parameterization consists of 293 experimental data points.

If a residuals analysis of the database shows stochastic deviations, then random variables are present in the experimental data and the database can be used directly for parameterization. If, however, systematic deviations are recognizable in the residuals, as e.g. Fig. E.5 shows, then correlated data are present which have been calculated with equations. The sum of the weighted correlated residual squares (𝑆𝑊𝑆) calculated in this way is not 𝝌 𝟐 distributed. However, the test quantity 𝑆𝑊𝑆 should have the character of a 𝝌 𝟐 distributed quantity as much as possible. This can be achieved with a pocketknife trick by calculating a mean CoV (𝐶𝑜𝑉 ̃) for all experimental data points such that the test quantity 𝑆𝑊𝑆 is 𝝌 𝟐 distributed, knowing full well that this is not the case. This correction, described below, can only partially improve so that the parameterization algorithm computes the most likely solution of a 𝝌 𝟐 distributed quantity. A clever transformation of the general definition equation for computing the 𝑆𝑊𝑆, together with the assumptions 𝑆 𝑖 ≅ 𝑆 ̃𝑖 and 𝑛 ̃≔ 𝑛 𝑖 , yields the equation (B.1)

𝑆𝑊𝑆(𝑌 ̅ 𝑝 , 𝜏 , 𝐶𝑜𝑉 ̃, 𝑛 ̃, 𝛩 𝑝 ) = 𝑛 C𝑜𝑉 ̃2 ∑ ( 𝑌 ̅ 𝑝𝑖 -𝐹 𝑝 (𝜏 𝑖 ,𝛩 𝑝 ) 𝑌 ̅ 𝑝𝑖 ) 2 𝒩 𝑝 𝑖=1 = 𝑛 C𝑜𝑉 ̃2 𝑆𝑅𝐷(𝑌 ̅ 𝑝 , 𝜏 , 𝛩 𝑝 ) . (B.1)
The sum 𝑆𝑅𝐷 in (B.1) is the well-known objective function of the relative residuals and is easy to calculate. The most likely value of the probability density function 𝑝𝑑𝑓 𝝌 𝟐 of the 𝝌 𝟐 distribution is the maximum 𝑝𝑑𝑓 𝝌 𝟐 ,𝒎𝒂𝒙 . The quantile 𝜒 𝑚𝑎𝑥 2 for this maximum can be calculated according to the simple equation 𝜒 𝑚𝑎𝑥 2 = 𝑑𝑜𝑓 -2 = 𝒩 𝑝 -𝒩 𝛩 𝑝 -2 and is identical to S𝑊𝑆(𝑌 ̅ 𝑝 , 𝜏 , 𝐶𝑜𝑉 ̃, 𝑛 ̃, 𝛩 𝑝 ) (see chapter 7.5 in ref [8] or other textbooks of statistics). The mean (𝐶𝑜𝑉 ̃) is thus given by the equation (B. In the next iteration, the sample size 𝑛 ̃ is checked with the quantile of the maximum of the 𝑝𝑑𝑓 𝝌 𝟐 ,𝒎𝒂𝒙 . For this purpose, the quotient 𝜆 is calculated according to the equation (B.4)

𝜆 = 𝑆𝑊𝑆 1 𝜒 𝑚𝑎𝑥 2 . (B.4)
If 𝜆 differs significantly from 1 (about ±0.05), then a corrected sample size 𝑛 * is calculated according to the equation (B.5)

𝑛 * = 𝑛 λ . (B.5)
The new objective function is given by Eq. (B.6)

𝑆𝑊𝑆 2 (𝑌 ̅ 𝑝 , 𝑆 ̃, 𝜏, 𝛩, 𝑛 * ) = 𝑛 * ∑ ( 𝑌 ̅ 𝑝𝑖 -𝐹 𝑝 (𝜏 𝑖 ,𝛩 𝑝 ) 𝑆 ̃𝑖 ) 2 𝑁 𝑖=1 (B.6)
and will be used for MCS. For fluoromethane, this approach calculates a mean relative error 𝐶𝑜𝑉 ̃% for all experimental data of about 1.45 % from the triple point temperature to the critical temperature, and the mean sample size 𝑛 * is about 23.

Of course, 𝑆𝑊𝑆 2 can no longer be used as an evaluation variable, since the statistics of the 𝝌 𝟐 distribution have already been plugged in as information to use an optimal 𝝌 𝟐 estimator as an objective function. Thus, the parameterization is based on equation (B.6) and the evaluation is based on relative deviations RD% using the 𝑆𝑅𝐷 objectiv function. With each EoS, 16 data points were simulated. The relative error is assumed to be twice as large as for the experiments. This corresponds to a factor of 4 in the variance. The four EoS should be weighted together with the same variance as the experiments, hence the doubling in standard deviation.

To get an idea of how well the vapor densities at the triple point are described by known EoS, Table B.1 lists the calculated vapor densities that agree very well. The FEoS shows the lowest vapor density deviations in the measurement range and is therefore selected for the extrapolation of simulated experimental data. Table B.2 lists the calculated vapor densities at the triple point temperature. Based on the MCS, not only the mean values of the estimated parameters and the assessment quantities 𝑀𝑅𝐷%, 𝑚𝑎𝑥𝑅𝐷% and 𝐵𝑖𝑎𝑠% are obtained, but also their standard deviations. Thus, statements about their certainty or uncertainty can also be made.

For simplification, the abbreviation 𝑈 = 𝑝, 𝜚 will be used for the state variables vapor pressure and vapor density.

The coefficient of variation 𝐶𝑜𝑉 and the standard deviation 𝑆 belong to the evaluation of the results, are detailed forthe parameters and the evaluation variables in the context of the MCS. .

(C.7)

The standard deviation 𝕊 of 𝑀𝑅𝐷%(𝑈) gives information about the mean deviation of 𝜅 and is calculated according to the known equation for standard deviation which is written here in abbreviated form (Eq. (C.8))

𝕊 𝑀𝑅𝐷%(𝑈) = 𝑠𝑡𝑑𝑒𝑣(𝑀𝑅𝐷% ̅̅̅̅̅̅̅̅̅ (𝑈), 𝑀𝑅𝐷%(𝑈) 𝜅 𝜅 = 1,2 … 𝒩 𝑀𝐶𝑆 ).

(C.8)

In context with the average relative deviation, the maximum of the absolute relativeof the vector element in the vector 𝑅𝐷% ̅̅̅̅̅̅̅ (𝑈) should be included as an additional assessment variable. This quantity is more sensitive to outliers in the data base or to an extreme "model bias" than the average 𝑀𝑅𝐷% ̅̅̅̅̅̅̅̅̅ (𝑈). It is defined for the iteration 𝜅 as follows (Eq. (C.9))

𝑚𝑎𝑥𝑅𝐷%(𝑈) 𝜅 = 𝑚𝑎𝑥[|𝑅𝐷%(𝑈) 𝑖,𝜅 | 𝑖 = 1,2 … 𝒩 𝑈 ] . (C.9)
The average of all MCS can be calculated according to the Eq. (C.10)

𝑚𝑎𝑥𝑅𝐷% ̅̅̅̅̅̅̅̅̅̅̅̅̅ (𝑈) = 1 𝒩 𝑀𝐶𝑆 ∑ 𝑚𝑎𝑥𝑅𝐷%(𝑈) 𝜅 𝒩 𝑀𝐶𝑆 𝜅=1 . (C.10)
The calculation of a standard deviation for an extreme value 𝑀𝐴𝑋% ̅̅̅̅̅̅̅̅̅ (𝑈), which is always at the same experimental temperature in case of a systematic deviation for each MCS, is no longer random. Since there are unfortunately not only random variables in a experimental data set, but also mean values simulated with models, such a value can lead to a systematic deviation at always the same measurement temperature. Therefore, the specification of a standard deviation is not recommended. With the initial parameter 𝛩 𝑝,𝑖𝑛𝑖 ≔ (3.5 -0.78 1.83) 𝒯 , every compound converged so far. The results of the parameter estimation are converted and given for Eq. ( 6).

SupMat E Supplementary results The vapor pressure parameters for both evaluations according to formulation problems (PE 1) and (PE 3) are equal in a statistical sense. This can also be assumed for the Z parameters, since the parameter Θ ̅ 𝑧1 is estimated with a much larger CoV for the evaluation according to formulation problem (PE 3) than in the case of a separate evaluation formulation problem (PE 2). The reason for this is unknown. In the case of difluoromethane (R32), the parameters Θ ̅ 𝑧3 and Θ ̅ 𝑧4 are described more precisely (lower values of CoV%) than for fluoromethane (R41). Θ ̅ 𝑧3 and Θ ̅ 𝑧4 parameters are necessary to rep- For simulations much more digits are necessary, see appendix 4. 4) model specific temperature PE 1 1) PE 2 2) PE 3 3) resent compressibility factor Z in the temperature range near the triple point temperature. With R32, the dispersion of the vapor density data is not so large in comparison to R41's values. In this case, there are no or only very slight differences in the ratings for the separate and simultaneous evaluations. Both evaluations give good ratings. 4) calc with the FEoS in REFPROP 10.0, except Tb which is a mean from measurements. 5) calculated Ztp from ptp(PE 1) and vapor density (database). 6) vapor pressure calculated from compressibility factor and vapor density both from (PE 2). n.r. not recommended.

The conclusions obtained from the results in Tab. E.3 are basically the same for difluoromethane (R32) as for fluoromethane. SupMat F Why is it possible to estimate vapor pressure parameters considering vapor density data?

The derivation shall be made on the basis of the Jacobian matrix, since it is used in many Newton based parameter estimation programs. Also a direct correlation can be shown.

We start with the development of the Jacobian matrix, whose definition is given in many textbooks of mathematics on regression, such as in ref [9]. The standard deviation of the experimental data gives information about the quality of the database. Therefore, the Jacobian matrix should be developed based on the standardized residual 𝑟( 𝜏, 𝑌 ̅ 𝜚 , 𝑆 ̅ 𝜚 , 𝛩) with the standard deviation 𝑆 ̅ 𝜚 of the mean 𝑌 ̅ 𝜚 according to the equation (F. The parameter vector 𝛩 is defined according to Eq (33). The Jacobian matrix is then defined by Eq. (F.2) where 𝑌 ̅ 𝜚 and 𝑆 ̅ 𝜚 are supposed to be scalar quantities.

𝐽( 𝜏

Fig. 1 .

 1 Fig. 1. Schematic representation of the development of the semi-empirical vapor density model for a pure compound as a function of the reduced temperature 𝜏 from reduced triple point temperature 𝜏 𝑇𝑝 to the critical temperature 𝜏 𝑐 = 1. 𝑓 𝑝 (𝜏, 𝛩 𝑝 ) is the vapor pressure function, 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) is the function for the compressibility factor of the vapor, 𝐹 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 ) is the semi-empirical model equation for the vapor, 𝛩 𝑝 , 𝛩 𝑧 are the model parameters, 𝑌 ̅ 𝑝 , 𝑌 ̅ 𝜚 are the experimental data.

  𝑍 𝑀 (𝜏, 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑧 ) = 𝑍 𝑐 + (1 -𝑍 𝑐 )

Fig. 2 .

 2 Fig. 2. Variation of the compressibility factor 𝑍 for the vapor as a function of the reduced temperature 𝜏 at the vicinity of the reduced triple point temperature 𝜏 𝑡𝑝 . The objective is to create physically meaningful constraints for the parameterization 𝛩 𝑧 of the 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) model. ▬▬ Graph of the compressibility factor model 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) for the vapor, ▬▬ First temperature derivative of the model function 𝑍 𝑀 (𝜏, 𝛩 𝑧 ) at 𝜏 𝑖𝑑 and 𝜏 𝑡𝑝 . 𝜏 𝑖𝑑 "Ideal gas temperature", model specific temperature for which 𝑍 𝑀 (𝜏 𝑖𝑑 , 𝛩 𝑧 ) = 1. 𝑍 ̃𝑡𝑝 Calculated compressibility factor based on reliable data such as measured vapor pressure and vapor density at 𝑇 𝑡𝑝 . 𝛩 𝑧3 Model parameter which determines 𝜏 𝑖𝑑 , it must be valid: 𝛩 𝑧3 < 1.

  The data are analyzed, checked for outliers and not take into account if necessary. The results of the analysis of the data base used in this work is presented in Appendix B. The databases for the evaluation are provided and are listed in "Data References" from [dataset 1] to [dataset 4]. The definition of the objective function results in the parameter values, with which the following evaluation variables are calculated: a) Mean of Relative Deviation in % (𝑀𝑅𝐷%(𝑈)) b) Maximum of Mean of Relative Deviation in % (𝑚𝑎𝑥𝑅𝐷%) c) Model Bias (𝐵𝑖𝑎𝑠%(𝑈)) d) Rank (𝑅𝑘) of variance-covariance matrix (𝑐𝑜𝑣) of the estimated parameter 𝑈 denotes the vapor density or vapor pressure. These indicators are calculated in general way according to the Eqs. (37) to (39) 𝑀𝑅𝐷%(𝑈, 𝛩 ̂ ) = 100% 𝒩 𝑈 ∑ | 𝑌 ̅ 𝑈,𝑖 -𝐹 𝑈 (𝜏 𝑖 ,𝛩 ̂

Fig. E. 5

 5

-

  three different triple point pressure values (345.04, 347.0777, 347.0798) Pa, -at least two different vapor density values (0.010884, 0.010948) kg/m 3 and -three different compressibility factor values (0.999544, 1.005447, 0.999550).

Fig. 4 .

 4 Fig. 4. Comparison of experimental data from fluoromethane (R41) and predicted compressibility factor for vapor from different models vs. absolute temperature near the crititical temperature (𝑇 𝑐 = 317.454 𝐾, 𝑍 𝑐 = 0.24623). PR (MC) denotes Peng Robinson EoS with Mathias Copeman 𝛼 function. The uncertainty for the new model (𝑍 𝑀 ) is ~0.03 which is calculated by error propagation. Z(exp) was calculated from the vapor density measurements and the vapor pressure function with the parameters according to formulation problem (PE 3).

  Experimental vapor densities data of difluoromethane can be described very well by our new model. The relative deviations are presented in Fig.E.8 in SuppMat E.

  Fig. E.6 and E.7 in SubbMat E show the corresponding residuals RD%(p).

2 ) 2 . (B. 2 )

 222 𝐶𝑜𝑉 ̃= √ 𝑛 ̃ 𝑆𝑅𝐷(𝑌 ̅ 𝑝 ,𝜏 ,𝛩 𝑝 ) 𝒩 𝑝 -𝒩 𝛩 𝑝 -Equation (B.2) can be used to calculate a mean standard deviation 𝑆 ̃𝑖 = 𝐶𝑜𝑉 ̃• 𝑌 ̅ 𝑝𝑖 for each experimental data point. The new objective function is given by Eq. (B.3). 𝑆𝑊𝑆 1 (𝑌 ̅ 𝑝 , 𝑆 ̃, 𝜏, 𝛩, 𝑛 ̃) = 𝑛 ̃∑ ( 𝑌 ̅ 𝑝𝑖 -𝐹 𝑝 (𝜏 𝑖 ,𝛩 𝑝 )

B. 1 . 2

 12 Vapor density data The raw data [B.2], [B.3] consist of a total of 18 measured data. From the experimental data of Cawood and Patterson [B.3] a mean CoV of 0.02534 can be calculated from experiment repetitions at four temperatures (303.163, 308.162, 315.139, 316.139)K. The experimental data point at 314.425 K is an outlier and is removed. From the experimental vapor density data, the compressibility factors were calculated using the vapor pressure equation and compared with the compressibility factors of the FEoS in REFPROP 10.0 as a function of absolute temperature. It is noticeable that the experimental data point at 269.1 K from ref [B.2] is significantly too small, which means that the vapor density is significantly too large. This experimental data point was also removed. The experimental data data base consists of 16 experimental data points from the temperature range 𝑇 ∈[291.172, 317.119] K or 𝜏 ∈ [0.917, 0.9989]. These data are not sufficient for parameterization of the vapor density model (8) over the entire temperature range. Therefore, vapor density data are simulated with trustworthy EoS in the temperature range 𝑇 ∈ [T 𝑡𝑝 , 290] K. The following EoS were selected for this purpose: 1. FEoS implemented in Refprop10 [A.5] 2. Peng Robinson EoS [B.4] with Mathias Copeman (MC) α function [B.5] 3. Peng Robinson with α function based on Acentric Factor ώ [B.4] 4. modified PT EoS with MC α function [B.6]

a)

  FEoS in REFPROP 10.0 [A.6]. b) PR EoS with acentric factor [B.4]. c) PR EoS with MC α function [B.5]. d) Patel Teja EoS with acentric factor [B.13]. e) Modified PT with MC α function [B.5].

For

  Fig. E.4. Average Relative Deviation% (𝑅𝐷% ̅̅̅̅̅̅̅ (𝜚)) for vapor density of fluoromethane (R41) vs. reduced temperature from the triple point temperature (𝜏 𝑡𝑝 = 0.409) to the critical temperature. The evaluation is based on formulation problem (PE 3). 𝑅𝐷% ̅̅̅̅̅̅̅ (𝜚) are calculated from the semi-empirical vapor density model and the following density data used for parameterization: (•) real experimental data,(•) simulated experimental data with REFPROP 10.0, (•) simulated experimental data with PR EoS and MC α function, (•) simulated experimental data with PR EoS and the α function with acentric factor ώ, • simulated experimental data with the modified PT EoS and MC α function [B.6].

Fig E. 5 .

 5 Fig E.5. Average relative deviation % (𝑅𝐷% ̅̅̅̅̅̅̅ (𝑝)) vs. reduced experimental vapor pressure for fluoromethane (R41). 𝜏 𝑡𝑝 = 0.409 𝜋 𝑡𝑝 = 5.87 • 10 -5 . The evaluation is based on formulation problem (PE 3).

Fig. E. 6 .Fig. E. 7 .Fig. E. 9 .

 679 Fig. E.6. Average relative deviation % (𝑅𝐷% ̅̅̅̅̅̅̅ (𝑝)) of the predicted vapor pressures compared to the known database of vapor pressure for fluoromethane (R41). For the prediction of the vapor pressure data, vapor density data and the vapor pressure of 101325 Pa at the boiling temperature were used. The evaluation is based on formulation problem (PE 4) with one constraint at 𝜏 𝑏𝑝 = 0.6138, 𝜋 𝑏𝑝 = 0.0172. 𝜏 𝑡𝑝 = 0.409, 𝜋 𝑡𝑝 = 5.87 • 10 -5 .

1 )

 1 𝑟( 𝜏, 𝑌 ̅ 𝜚 , 𝑆 ̅ 𝜚 , 𝛩) = 𝑌 ̅ 𝜚 -𝐹 𝜚 (𝜏,𝛩)

  𝜏, 𝑇 𝑐 , 𝛩 𝑝 ) = 𝑒𝑥𝑝 [

	𝛩 𝑝1 𝑇 𝑐	(1 -	1 𝜏 ) + 𝛩 𝑝2 ⋅ 𝑙𝑛( 𝜏) + 𝛩 𝑝3 ⋅ 𝑇 𝑐 𝛩 𝑝4 (𝜏 𝛩 𝑝4 -1)] .

  𝜚 𝑀 = 𝐹 𝜚 (𝜏, 𝑇 𝑐 , 𝜚 𝑐 , 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑝 , 𝛩 𝑧 ) = 𝜚 𝑐 • 𝑓 𝜚 (𝜏, 𝑇 𝑐 , 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑝 , 𝛩 𝑧 ) (8) 𝜔 = 𝑓 𝜚 (𝜏, 𝑇 𝑐 , 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑝 , 𝛩 𝑧 ) = 𝑍 𝑐 𝑓 𝑝 (𝜏,𝑇 𝑐 ,𝛩 𝑝 ) 𝜏•𝑍 𝑀 (𝜏,𝑍 𝑐 ,𝜏 𝑡𝑝 ,𝛩 𝑧 ,) . 𝑓 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 ) ≡ 𝑓 𝜚 (𝜏, 𝑇 𝑐 , 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑝 , 𝛩 𝑧 )

			(9)
	The critical compressibility factor 𝑍 𝑐 is calculated according to the known
	relation 𝑍 𝑐 =	ℳ𝑝 𝑐 (ℛ 𝑇 𝑐 𝜚 𝑐 ) ⁄	from the critical properties 𝑇 𝑐 , 𝑝 𝑐 𝑎𝑛𝑑 𝜚 𝑐 . For fur-
	ther developments, the following identities (10 to 14) are applied due to
	a simplify notations:
	𝐹

(7)

𝛩 𝑧4 and 𝛩 𝑧7 are the numbers of terms. The model for calculating the vapor density 𝜚 𝑀 or reduced vapor density 𝜔 in reduced state variables is given by Eqs. (8-9) 𝑝 (𝜏, 𝛩 𝑝 ) ≡ 𝐹 𝑝 (𝜏, 𝑇 𝑐 , 𝑝 𝑐 , 𝛩 𝑝 )

[START_REF] Taylor | Guidelines for Evaluating an Expressing the Uncertainty of NIST Measurement Results[END_REF]

𝑓 𝑝 (𝜏, 𝛩 𝑝 ) ≡ 𝑓 𝑝 (𝜏, 𝑇 𝑐 , 𝛩 𝑝 )

(11)

𝐹 𝜚 (𝜏, 𝛩 𝑝 , 𝛩 𝑧 ) ≡ 𝐹 𝜚 (𝜏, 𝑇 𝑐 , 𝜚 𝑐 , 𝑍 𝑐 , 𝜏 𝑡𝑝 , 𝛩 𝑝 , 𝛩 𝑧 )

(12)

  for 𝑍 𝑀 (𝜏, 𝛩 𝑧 ). If during a calculation a simulated critical temperature 𝑇 𝑠𝑖𝑚,𝑐 becomes larger than the critical temperature 𝑇 𝑐 for any reason, then no solution exists or the result is a complex number. The Eqs. (15 to 17) must be valid for the critical state,

	𝑓 𝜚 ( 𝑇 𝑠𝑖𝑚,𝑐 ⁄ , 𝛩 𝑝 , 𝛩 𝑧 ) = 1 𝑇 𝑐 ,	(15)
	𝑓 𝑝 ( 𝑇 𝑠𝑖𝑚,𝑐 ⁄ , 𝛩 𝑝 ) = 1 𝑇 𝑐 ,	(16)
	𝑍 𝑀 ( 𝑇 𝑠𝑖𝑚,𝑐 ⁄ , 𝛩 𝑧 ) = 𝑍 𝑐 . 𝑇 𝑐 ,	(

  At the reduced ideal gas temperature 𝜏 𝑖𝑑 , Eqs. (19) to (21) 𝑍 𝑀 (𝜏 𝑖𝑑 , 𝛩 𝑧 ) = 1(𝛩 𝑧4,𝑜 = 𝒩 𝑧 = 1) ∨ (𝛩 𝑧7,𝑜 = 𝒩 𝑧 = 2) 𝑝 denotes the experimental data vector with the random variables 𝑌 ̅ 𝑝𝑖 , 𝑖 = 1, 2 … 𝒩 𝑝 . Each 𝑌 ̅ 𝑝𝑖 is the average calculated value considering the experimental data at the temperature 𝑇 𝑖 . The 𝑌 ̅ 𝑝𝑖 are normally distributed 𝑌 ̅ 𝑝𝑖 ~𝓝 (𝜇 𝑝𝑖 ,

	(PE 1) Formulation of the constrained parameter estimation
	problem for vapor pressure evaluation
	( 𝜕𝑍 𝑀 (𝜏,𝛩 𝑧 ) 𝜕𝜏 m i n ) 𝜏 𝑖𝑑 𝛩 𝑝 , 𝐹 𝑝 (𝑌 ̅ 𝑝 -𝐹 𝑝 (𝜏, 𝛩 𝑝 )) = 0	𝒯 𝛺 𝑝 -1 (𝑌 ̅ 𝑝 -𝐹 𝑝 (𝜏, 𝛩 𝑝 ))	(19) (20)
	𝜏 𝑖𝑑 = 𝛩 𝑧3 • 𝜏 𝑡𝑝 . s.t. 𝛩 𝑝4 = 1 ∨ 2 … ∨ 6 𝐹 𝑝 (𝜏 𝑡𝑝 , 𝛩 𝑝 ) -𝑝 𝑡𝑝 = 0 optional	(21)
	At the triple point, Eq. (3) and the derivative of compressibility factor ver-sus reduced temperature Eq. (20) are considered as constraints where 𝑌 ̅ 𝑝 is the measurement vector of
	𝑍 𝑀 (𝜏 𝑡𝑝 , 𝛩 𝑧 ) -𝑍 ̃𝑡𝑝 = 0						means with the dimension 𝒩 𝑝	(22)
	( 𝜕𝑍 𝑀 (𝜏, 𝛩 𝑧 ) 𝜕𝜏 𝛺 𝑝 = 𝑑𝑖𝑎𝑔 ( ) 𝜏≥ 𝜏 𝑡𝑝 < 0 . 𝑆 𝑝1 2 𝑛 𝑝1 ,	𝑆 𝑝2 2 𝑛 𝑝2	…	2 𝑆 𝑝𝒩 𝑝 𝑛 𝑝𝒩 𝑝	)	is the diagonal matrix of the cor-	(23)
							responding variances and sample
							sizes
	the parameter estimation pro-blem for the evaluation of the vapor pressure data should have to be for-mulated. In effect, vapor pressure model parameters are needed in the vapor density Eqs. (8) and (9). 2 𝑌 ̅ 𝜎 𝑝𝑖 𝑛 𝑝𝑖 )
	with the sample sizes 𝑛 𝑝𝑖 , the unknown true pressures 𝜇 𝑝𝑖 and the corres-
	ponding unknown true variances 2.2 Parameterization of the semi-empirical vapor density model 2 𝜎 𝑝𝑖 𝑛 𝑝𝑖 . 𝛺 𝑝 denotes the variance matrix with the 2.2.1 Formulation of the parameter estimation problem (PE 1) for the variances of the 𝑌 ̅ 𝑝𝑖 which are Chi-squared distributed (𝑛 𝑝𝑖 -1)𝑆 𝑝𝑖 2 𝑛 𝑝𝑖 𝜎 𝑝𝑖 2 ~𝝌(𝑛 𝑝𝑖 -1) 2
	evaluation of vapor pressure data
	The formulation problem is given by (PE 1).

𝑍 ̃𝑡𝑝 is the calculated compressibility factor based on trustworthy data. Accurate experimental data or predictions with very well parameterized equations of state like Fundamental Equation of State (FEoS) can be used. With these constraints, we can usefully complete the parameter estimation problem for the vapor density model. But before that,

  ).

	𝛩 =	𝛩 1 𝛩 2 𝛩 3 𝛩 4 𝛩 5 𝛩 6 𝛩 7 𝛩 8	≡	𝛩 𝑍1 𝛩 𝑍2 𝛩 𝑍3 𝛩 𝑍4 𝛩 𝑍5 𝛩 𝑍6 𝛩 𝑝1 𝛩 𝑝2
		𝛩 9		𝛩 𝑝3
	(	𝛩 10 𝛩 11 )		𝛩 𝑝4 ( 𝒩 𝑧 )

Table 1 Estimated

 1

parameters based on the formulation of the parameter estimation problems (PE 1) to (PE 3) for fluoromethane (R41). Results are based on Monte Carlo Simulations.

Table 2

 2 Regression assessment for the different kind of parameter estimation formulations (PE 1) to (PE 3) for fluoromethane (R41). The variance-covariance matrix of the parameters has full rank in any case. The maximum MRD% value is about five times larger than the MRD% value for both evaluations. However, the residuals analysis in Fig.E.4 shows that this deviation belongs to a measurement point that is not an outlier. Thus, we can consider that it is not a systematic deviation. The bias is close to zero value for both analyses, since the standard deviation is much larger than the mean itself. This zero value signifies that we have realized a very good and balanced regression. Both evaluations indicate a very good score for the regression. TableE.2, Fig. E.8 and Fig. E.9 in SuppMat E show the results for difluoromethane.

		PE 1 1)	PE 2 2)		PE 3 3)
	Assessment Quantity	Average	Stdev	Average	Stdev Average	Stdev
	MRD% for vapor pressure	0.166	0.0004			0.166	0.0005
	maxRD% for vapor pressure	0.510	n.r.			0.516	n.r.
	Bias% for vapor pressure	-0.011	0.0036			-0.008	0.004
	MRD% for vapor density			1.091	0.015	1.113	0.012
	maxRD% for vapor density			4.781	n.r.	4.734	n.r.
	Bias% for vapor density			-0.0003	0.070	-0.034	0.063

1) vapor pressure data only. 2) vapor density data only. 3) vapor pressure and vapor density data simultaneously. n.r. is not recommended Concerning the vapor density, the two mean errors (MRD%) are equal considering a statistical point of view and are also in good agreement with the known uncertainty of the experimental vapor density data. The residual plots (Fig. E.4 and Fig. E.5) for vapor density and vapor pressure are detailed in SuppMat E.

Table 4

 4 Assessment comparison of the new model with the state of the art in the temperature range below the critical temperature. The basis for the comparison is the com-pressibility factor for fluoromethane (R41) calculated from measured (exp) vapor densities. The values given are average values.

	Assessment	New Model	RP 1) 10.0	PR (MC) 2)
	Average MRD%	1.79	10.6	3.81
	Average maxRD%	4.69	29.0	6.51
	Average Bias%	0.086	-10.6	0.74
	1) REFPROP. 2) PR EoS with Mathias Copeman α function

Table 5

 5

it can be seen that the MRD% values for the formulation problem (PE 4) evaluation become larger, but are still in a range that is classified as good or satisfactory. The Bias% values indicate a systematic deviation for both compounds (see Fig. E.6 and E.7), which can also be seen in the residual plots.

Table 5

 5 Assessment of predictive capability 𝑃𝑟𝑒𝐶𝑎𝑝% of vapor pressure data based on vapor density data and the evaluation according to formulation problem (PE 4) with one constraint at 𝜏 𝑏𝑝 . The number of vapor pressure data to be predicted is 𝒩 𝑝,𝑅41 = 293 and 𝒩 𝑝,𝑅32 = 716.

		Fluoromethane (R41) Difluoromethane (R32)
	Assessment Quantity	(PE 3)	(PE 4)	(PE 3)	(PE 4)
	Average MRD% for vapor pressure	0.166	0.465	0.171	0.989
	Average maxRD% for vapor pressure	0.516	1.230	0.512	1.911
	Average Bias% for vapor pressure	-0.008	0.387	-0.017	-0.755
	FitCap% 1)	99.3		99.9
	PreCap% 1)		53.2		16.1
	Rank of cov matrix 2) of the parameter	9	9	9

Table 6

 6 Assessment of predictive capability 𝑃𝑟𝑒𝐶𝑎𝑝% of vapor pressure data based on vapor density data and two vapor pressure data at 𝜏 1 = 0.5 and 𝜏 2 = 0.8 as constraints formulated in formulation problem (PE 4). The number of vapor pressure data to be predicted is 𝒩 𝑝,𝑅41 = 293 and 𝒩 𝑝,𝑅32 = 716.

		Fluorome-	Difluorome-
		thane (R41)	thane (R32)
	Assessment Quantity
	Average MRD% for vapor pressure	0.193	0.178
	Average maxRD% for vapor pressure	0.720	0.566
	Average Bias% for vapor pressure	0.013	0.009
	PreCap% 1)	96.6	99.4

Table B . 1

 B1 Calculated fluoromethane (R41) vapor density ϱ from different EoS at triple point temperatureTo the raw database[A.3] of 766 data points, the triple point pressure calculated with FEoS in REFPROP 10.0 is also added, since there is no measured data near the triple point temperature. After removing the outliers and the data treatment (described in Section B.1.1). The database consists of 716 data points obtained after data treatment and removing of the outliers. A mean relative experimental error of 𝐶𝑜𝑉 ̃ is estimated to be 0.0161 and a mean sample size of 30 from the triple point temperature to the critical temperature are estimated.Since the available experimental vapor density data do not cover the necessary temperature range from the triple point to the critical temperatures, three equations of state with different alpha functions are used for testing in the temperature range 𝑇 ∈ [𝑇 𝑡𝑝 , 253] 𝐾 or 𝜏 ∈ [𝜏 𝑡𝑝 , 0.72]. The tests consist of the selection of the more suitable EoS for this purpose. Selection criterion for extrapolation of EoS outside the temperature measurement range shall be the smallest vapor density deviation in the temperature range of the measurements.The following EoS are used for the test.

		ϱ / kg/m 3
	FEoS in REFPROP 10.0	0.010888
	PR EoS (MC)	0.010882
	PR EoS (ώ)	0.010882
	PT EoS (ώ)	0.010882
	Modified PT EoS (MC)	0.010887
	Mean	0.010884
	Ideal Gas for comparison 1)	0.010879
	1) at saturated pressure
	SupMat B.2 Difluoromethane (R32)
	B.2.1 Vapor pressure
	B.2.2 Vapor density
	The raw data base was collected by four working groups [B.8 to B.11] in the
	temperature range 𝑇 ∈ [253.07, 351.263] 𝐾 and reduced temperature 𝜏 ∈ [0.72 , 0.9999],
	respectively. A literature search yielded additional measured data from De-
	fibaugh [B.12] in the temperature range of 𝑇 ∈ [220 , 351] 𝐾, which were used
	to parameterize an MBWR EoS [B.14]. This EoS was used to calculate vapor
	densities, which can be found in Table 8 in ref [B.12]. An analysis shows
	that the MBWR EoS parametrized with Defibaugh data predicted density in a
	very good agreement with experimental data from [B.8-11]. Of these data,
	only those in the temperature range of 𝑇 ∈ [220, 253] 𝐾 or 𝜏 ∈ [0.63 , 0.72] were
	selected to complete the above temperature range 𝜏 ∈ [0.72 , 0.9999].

Table B

 B For this reason, random standard deviations are generated using Monte Carlo simulations (see Appendix 2D of the Supporting Information in reference[3]). It should be mentioned that the experimental average values of the database don't absolutely change. These values are of course directly used for parameterization.

	.2
	Calculated difluoromethane (R32) vapor density ϱ from different EoS at triple
	point temperature.
		ϱ / kg/m 3
	FEoS in REFPROP 10.0	0.0022033
	PR EoS (MC)	0.0022030
	PR EoS (ώ)	0.0022030
	PT EoS (ώ)	0.0022030
	Modified PT EoS (MC)	0.0022030
	Mean	0.0022030
	Ideal Gas for comparison 1)	0.0022029
	1) at saturated pressure
	SupMat C Definition of the assessment quantities and Monte Carlo
	Simulations
	The NIST database has an experimental mean value and an associated uncer-
	tainty for each measured temperature [C.1]. The specification of the uncer-
	tainty is based on an infinitely large sample size [C.1]. A standard devia-
	tion can be calculated from the uncertainty. However, this standard devia-
	tion no longer has a stochastic character, since this has been lost by error
	propagation calculations, e.g. with equations related to the experimental
	device. However, stochastic standard deviations are needed for the weighted
	parameterization.

  The Parameters Θ ̂𝑈,𝑗,𝜅 denotes the estimated parameter 𝑗 for the state variable 𝑈 at MCS loop number 𝜅. The average parameter vector Θ ̅ 𝑈 is calculated according to Eq. 𝑖,𝜅 ) for measurement No. 𝑖 at 𝑇 𝑖 in the Monte Carlo iteration number 𝜅 is calculated according to Eq. (C.3) This quantity is used to calculate the graphs for the diagrams. The Mean Relative Deviation% 𝑀𝑅𝐷%(𝑈) 𝜅 for the 𝜅 th iteration is defined by Eq. (C.6) 𝒩 𝑈 is the number of all experimental data for the state variable 𝑈. The average 𝑀𝑅𝐷% ̅̅̅̅̅̅̅̅̅ over all numbers 𝒩 𝑀𝐶𝑆 is defined by Eq. (C.7)

	𝑀𝑅𝐷%(𝑈) 𝜅 =	100% 𝒩 𝑈	∑ | 𝒩 𝑈 𝑖=1	𝑌 ̅ 𝑈,𝑖 -𝑈 ̂𝑖𝜅 𝑌 ̅ 𝑈,𝑖	|	=	100% 𝒩 𝑈	∑ |𝑅𝐷%(𝑈) 𝑖,𝜅 | 𝒩 𝑈 𝑖=1	,	(C.6)
	where 𝑀𝑅𝐷% ̅̅̅̅̅̅̅̅̅ (𝑈) =	1 𝒩 𝑀𝐶𝑆	∑ 𝒩 𝑀𝐶𝑆 𝜅=1	𝑀𝑅𝐷%(𝑈) 𝜅
	(C.1)
	Θ ̅ 𝑈,𝑗 =	1 𝒩 𝑀𝐶𝑆	∑ 𝒩 𝑀𝐶𝑆 𝜅=1		Θ ̂𝑈,𝑗,𝜅	.	(C.1)
	The Coefficient of Variation in per cent (CoV%) is calculated as follows
	(Eq. (C.2))
	𝐶𝑜𝑉%(Θ ̅ 𝑈,𝑗 ) =	𝑠𝑡𝑑𝑒𝑣(Θ ̅ 𝑈,𝑗 ,Θ ̂𝑈,𝑗,𝜅 𝜅=1,2…𝒩 𝑀𝐶𝑆 ) |Θ ̅ 𝑈,𝑗 |	• 100% ,	(C.2)
	where 𝑠𝑡𝑑𝑒𝑣 denotes the known equation for calculation of the standard devia-
	tion.
	The Fit Assessment Quantities
	The Relative Deviation% (𝑅𝐷%(𝑈) 𝑅𝐷%(𝑈) 𝑖,𝜅 = 𝑌 ̅ 𝑈,𝑖 -𝑈 ̂𝑖𝜅 𝑌 ̅ 𝑈,𝑖 • 100% ,	(C.3)
	where 𝑈 ̂𝑖𝜅 is the estimated vapor state variable which is identical to Eq.
	(C.4)
	𝑈 ̂𝑖𝜅 ≡ 𝐹 𝑈 (𝑇 𝑖 , Θ ̂𝑈,𝜅 )					(C.4)
	and 𝑌 ̅ 𝑈,𝑖 denotes the experimental mean at 𝑇 𝑖 . The average Relative Deviation%
	at 𝑇 𝑖 for all MCS is defined by Eq. (C.5)
	𝑅𝐷% ̅̅̅̅̅̅̅ (𝑈) 𝑖 =	1 𝒩 𝑀𝐶𝑆	∑ 𝒩 𝑀𝐶𝑆 𝜅=1	𝑅𝐷%(𝑈) 𝑖,𝜅	.	(C.5)

  The boiling point temperature 𝑇 ̂𝑏 is calculated from average estimated parameter vector Θ ̅ 𝑝 according to the implicit vapor function 𝐹 𝑝 (Eq. (C.14))From this, the mean deviation (standard deviation) 𝕊 𝑇 𝑏 around the estimated mean 𝑇 ̅ 𝑏 can be calculated by Eq. (C.17)𝕊 𝑇 𝑏 = 𝑠𝑡𝑑𝑒𝑣(𝑇 ̅ 𝑏 , 𝑇 𝑏,𝜅 𝜅 = 1,2 … 𝒩 𝑀𝐶𝑆 ).(C.17)Since the average 𝑇 ̅ 𝑏 is almost identical to the estimated 𝑇 ̂𝑏, this is also the standard deviation for the estimated 𝑇 ̂𝑏 ((Eq. (C.18))It is not recommended to calculate the 𝑠𝑡𝑑𝑒𝑣 of 𝑝̂𝑡 𝑝 and 𝜚 ̂𝑡𝑝 considering MCS because the optimization algorithm reaches the limit of the constraint (𝑝̅ 𝑡𝑝 or 𝜚̅ 𝑡𝑝 ) or is in a range very close to this limit. In effect, the standard deviation is a random variable and close to a constraint value the optimiza-tion algorithm uses penalty terms that destroy the random character of a variance.The determination of the minimum number of MCS is based on the statistics for control charts for the estimated parameters. For each parameter, averages of 1000 MCS are calculated for such a control chart, i.e. a block consists of 1000 MCS. A confidence range is calculated based on the Student t probability density function. Experiences show that a confidence interval of 0.809 is strict to detect and abort convergence. At least 15 blocks must be calculated at the beginning of the MCS. The observation space for the current mean calculation is 10 blocks retrospectively. Within 10 blocks the parameter value must not exceed or fall below the confidence limits. If this condition is met, the MCS is aborted.SupMat D Estimated Parameter Values for SimulationsTable D provides the average values of estimated parameter with more digits for simulations. Results are based on MCS.

	The Physical Quantities
	𝐹 𝑝 (𝑇 ̂𝑏, Θ ̅ 𝑝 ) -𝑝 0 = 0			(C.14)
	where 𝑝 0 is the standard pressure at 1 atm (𝑝 0 = 1.01325 • 10 5 Pa). The corres-
	ponding standard deviation is calculated by solving the implicit Eq. (C.15)
	𝐹 𝑝 (𝑇 𝑏,𝜅 , Θ ̂𝑝,𝜅 ) -𝑝 0 = 0		(C.15)
	for the boiling point at each temperature 𝑇 𝑏,𝜅 where the average boiling tem-
	perature from all MCS is as follows (Eq. (C.16))
	𝑇 ̅ 𝑏 =	1 𝒩 𝑀𝐶𝑆	∑ 𝒩 𝑀𝐶𝑆 𝜅=1	𝑇 𝑏,𝜅	.		(C.16)
	𝕊 𝑇 ̂𝑏 ≈ 𝕊 𝑇 𝑏 .				(C.18)
	The boiling point density for vapor is calculated using the average of esti-
	mated parameter according to the Eq. (C.19)
	𝜚 ̂𝑏 = 𝐹 𝜚 (𝑇 ̂𝑏, Θ ̅ 𝑝 , Θ ̅ 𝑍 ).			(C.19)
	The standard deviation 𝕊 𝜚 ̂𝑏 is calculated similar to 𝕊 𝑇 𝑏 and the equations
	are as follows (Eqs. (C.20 to C.22))
	𝜚 𝑏,𝜅 = 𝐹 𝜚 (𝜏, Θ ̂𝑝,𝜅 , Θ ̂𝑧,𝜅 )		(C.20)
	𝐵𝑖𝑎𝑠%(𝑈) 𝜅 = 𝜚̅ 𝑏 = 1 𝒩 𝑀𝐶𝑆 ∑ 𝒩 𝑀𝐶𝑆 100% 𝒩 𝑈 𝜚 𝑏,𝜅 ∑ 𝒩 𝑈 𝑖=1 𝜅=1	𝑌 ̅ 𝑈,𝑖 -𝑈 ̂𝑖𝜅 𝑌 ̅ 𝑈,𝑖	(C.11) (C.21)
	and the corresponding average quantity and standard deviation are as expect-𝕊 𝜚 ̂𝑏 ≈ 𝑠𝑡𝑑𝑒𝑣(𝜚̅ 𝑏 , 𝜚 𝑏,𝜅 𝜅 = 1,2 … 𝒩 𝑀𝐶𝑆 ) . (C.22)
	ed ((Eqs. (C.12 and C.13))
	𝐵𝑖𝑎𝑠% ̅̅̅̅̅̅̅̅̅ (𝑈) = The triple point pressure and its corresponding density value are calculated 1 ∑ 𝐵𝑖𝑎𝑠%(𝑈) 𝜅 𝒩 𝑀𝐶𝑆 𝜅=1 , (C.12) 𝒩 𝑀𝐶𝑆 directly from the model equations according to Eqs. (C.23 and C.24) 𝕊 𝐵𝑖𝑎𝑠%(𝑈) = 𝑠𝑡𝑑𝑒𝑣(𝐵𝑖𝑎𝑠% ̅̅̅̅̅̅̅̅̅ (𝑈), 𝐵𝑖𝑎𝑠%(𝑈) 𝜅 𝜅 = 1,2 … 𝒩 𝑀𝐶𝑆 ). (C.13) 𝑝̂𝑡 𝑝 = 𝐹 𝑝 (𝑇 ̅ 𝑡𝑝 , Θ ̅ 𝑝 ) (C.23)
	In this context, characteristic quantities such as boiling temperature, triple point pressure or vapor density at the triple point can also be de-𝜚 ̂𝑡𝑝 = 𝐹 𝜚 (𝑇 ̅ 𝑡𝑝 , Θ ̅ 𝑝 , Θ ̅ 𝑍 ). (C.24)
	termined using MCS.

The model bias for iteration 𝜅 is defined as follows (Eq. (C.11))

Table D

 D Average values of estimated parameter with more digits for simulations.Parameter estimation with one term converges very quickly. The recommended parameter vectors for two terms (𝒩 𝑧 = 2) are very robust. If problems still occur, another proposal can be tested for success. An equation for testing the necessary precondition of convergence capability in the case of nonlinear parameter estimation problems is given from Marquardt (see page 439 in [D.1]) as an example. More recent findings in this area are described for example in the work ofClesse et al. [D.2].

	For parameter estimation and calculation of the cov matrix of parameters,
	a specially scaled form 𝑓 𝑝 * (Eq. (D.2)) is used instead of the vapor pressure
	function given by eq (6). This ensures that the values of the parameters
	are in the numerical range of [0.1, 10]. The calculation does not fail.
	The scaled form is given by Eq. (D.2).
	𝑓 𝑝 * (𝜏, 𝑇 𝑐 , 𝛩 𝑝 ) = 𝑒𝑥𝑝 [	𝛩 𝑝1 •1000 𝑇 𝑐	(1 -	1 𝜏	) + 𝛩 𝑝2 ⋅ 10 • 𝑙𝑛( 𝜏) + 𝛩 𝑝3 ⋅ (𝜏 𝛩 𝑝4 -1)] .	(D.2)
				Parameter values for
	Parameter	Fluoromethane (R41)	Difluoromethane (R32)
	𝛩 ̅ 𝑧1		1.626 221 818 488	3.938 050 095 271
	𝛩 ̅ 𝑧2		0.234 926 702 594	0.439 081 284 486
	𝛩 ̅ 𝑧3		0.944 612 459 040	0.895 572 259 879
	𝛩 ̅ 𝑧4		4.249 065 063 034	3.136 381 522 713
	𝛩 ̅ 𝑧5		1.824 148 272 542	1.796 494 026 212
	𝛩 ̅ 𝑧6		0.555 004 927 438	0.660 228 583 214
	𝛩 ̅ 𝑝1		3023.800 649 449 500	3933.332 225 338 400
	𝛩 ̅ 𝑝2		-5.976 977 380 797	-8.105 025 423 986
	𝛩 ̅ 𝑝3		1.567 545 795 817	1.638 388 300 915
	𝛩 𝑝4						2	2
	𝒩 𝑍						2	2
	We use the following parameter vectors as guess values for a successful pa-
	rameterization of the Z model (Eq. (D.1))
		(2.6 0.5 0.95) 𝒯			𝑖𝑓 𝒩 𝑧 = 1
	𝛩 𝑍,𝑖𝑛𝑖 ≔ {	(1.5 0.2 0.95 2.5 0.53 0.5) 𝒯	𝑖𝑓 𝒩 𝑧 = 2	.	(D.1)

Table E

 E

.1 Estimated parameter for difluoromethane (R32). Results are based on MCS.

Table E . 2

 E2 Regression assessment for the different kind of parameter estimation formulations (PE 1) to (PE 3) for difluoromethane (R32). Results are based on MCS. pressure data only. 2) vapor density data only. 3) vapor pressure and vapor density data simultaneously. n.r. is not recommended

		PE 1 1)	PE 2 2)		PE 3 3)
	Assessment Quantity	Average	Stdev	Average	Stdev Average	Stdev
	MRD% for vapor pressure	0.171	0.0001			0.171	0.0002
	maxRD% for vapor pressure	0.513	n.r.			0.512	n.r.
	Bias% for vapor pressure	-0.019	0.0021			-0.017	0.002
	MRD% for vapor density			0.864	0.014	0.862	0.015
	maxRD% for vapor density			6.410	n.r.	6.391	n.r.
	Bias% for vapor density			0.0136	0.040	-0.017	0.002
	1) vapor

Table E . 3

 E3 Comparison of the results of the physical properties of difluoromethane (R32) for the different parameter estimation formulations (PE 1) to (PE 3). Results are based on MCS. pressure only. 2) vapor density only. 3) vapor pressure and vapor density data simultaneously.

			PE 1 1)		PE 2 2)	PE 3 3)
	Quantity	Database 4)	Average Stdev	Average Stdev Average	Stdev
	Boiling point temperature / K	221.38	221.42 0.001		221.43	0.001
	Boiling point temperature / °C	-51.78	-51.73 0.001		-51.72	0.001
	Boiling point vapor density / kg/m 3	2.988			2.988 0.001	2.985	0.001
	Triple point vapor pressure / Pa	47.99983 48.05325	n.r. 48.05328 6)	48.03778	n.r.
	Triple point vapor density / kg/m 3	0.002203			0.002205	n.r. 0.002205	n.r.
	Compressibility factor at triple point	0.999917 1.001026 5)		0.999917	n.r. 0.999920	n.r.
	"Ideal gas temperature" / K				122.03 1.98	122.10	0.60
	1) vapor

  The partial derivatives for the semi-empirical vapor density model 𝐹 𝜚 (𝜏, 𝛩) are then obtained as follows (eq. (F.3 and F.4))

		0.6
		0.4
	RD%(p)	-0.2 0 0.2
		-0.4
		-0.6
			1.0E-06	1.0E-05	1.0E-04	1.0E-03	1.0E-02	1.0E-01	1.0E+00
											Reduced Measured Vapor Pressure
	, 𝑆 ̅ 𝜚 , 𝛩) ≔	𝜕𝑟( 𝜏,𝑌 ̅ 𝜚 ,𝑆 ̅ 𝜚 ,𝛩) 𝜕𝛩 𝒯	= -	1 𝑆 ̅ 𝜚	•	𝜕𝐹 𝜚 (𝜏,𝛩) 𝜕𝛩 𝒯	.	(F.2)
	𝐺 𝑗 (𝜏, 𝛩) ≔ (	𝜕𝐹 𝜚 (𝜏,𝛩) 𝜕𝛩 𝑗	) 𝑗=1...6	=	𝜚 𝑐 𝑍 𝑐 𝑓 𝑝 (𝜏,𝛩) 𝜏	(	𝜕𝑍 𝑀 (𝜏,𝛩) -1 𝜕𝛩 𝑗	)	𝑗=1...6	(F.3)
	𝐺 𝑖 (𝜏, 𝛩) ≔ ( 𝜕𝐹 𝜚 (𝜏,𝛩) 𝜕𝛩 𝑖 ) 𝑖=7...9	=	𝜚 𝑐 𝑍 𝑐 𝜏 𝑍 𝑀 (𝜏,𝛩) ( 𝜕𝑓 𝑝 (𝜏,𝛩) 𝜕𝛩 𝑖	) 𝑖=7...9	.	(F.4)

Acknowledgment

The authors give their sincere thanks for valuable discussions to Dr. Stefan Maixner (retired engineer from department of research, BASF SE, Germany).

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

SupMat B Data treatment

An EXCEL file is provided for all experimental data that have been considered in this work, checked and cleared of outliers. The first column represents the independent variable (e.g., mean temperature), the second column indicates the mean of the measured value of the dependent variable, the third column indicates the standard deviation of the mean of the dependent variable, the fourth indicates the covariance between the measured mean of the dependent and independent variables, and the fifth column indicates the sample size for the measurement of the dependent variable. The databases for the evaluation are provided and are listed in "Data References" in the article. -All data points with the remark "not accepted" were removed.

-The calculated standard deviations 𝑆 ̃𝑖 can be figured out from the uncertainty data according to 𝑆 ̃𝑖 = 𝑢 𝑖 2 ⁄ . Then the Coefficient of Variation (CoV)

The last two equations show impressively that each matrix element of the Jacobian matrix holds the vapor pressure parameters 𝛩 𝑖 (𝑖 = 7. . .9). This is the answer to the question of the heading for this SupMat. It is the necessary but not sufficient condition that must be met for the parameters to be estimated numerically.

For this, a second condition must be satisfied in order to estimate the vapor pressure parameters as well, namely, the full rank of the variancecovariance matrix 𝑐𝑜𝑣( 𝝉, 𝑺 ̅ 𝝔 , 𝛩 ̂) of the estimated parameters 𝛩 ̂.

For this purpose, the Jacobian matrix of the experimental data for the individual experimental temperatures 𝝉 = (𝜏 If full rank exists, then all parameters can be estimated. The full rank depends only on the quality of the experimental vapor density data. Thus the estimateability of the vapor pressure parameters also depends on it.

Full rank is available for fluoromethane and difluoromethane. The 𝑐𝑜𝑣( 𝝉, 𝕊 ̅ 𝝔 , 𝛩 ̂) matrix with the standard deviations from the Monte Carlo simulations also have full rank for both compounds.