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The complete analytical solutions to the roots of the cubic function (of one independent variable) are tabulated. For an equation of state in which the intermolecular attractive forces are described by quadratic trinomials, the derived analytical expressions for the roots of the cubic equation can be adopted to obtain the molar volumes (liquid and vapour) at a given state of temperature and pressure. The molar volumes calculated in this work are compared with the experimental data as well as other equations of state. The Guevara-Rodríguez equation of state describes the experimental data better than the other models. The equation of state has three unequal real roots of the molar volume under the extremely low pressure and temperature. However, the smallest positive root is taken as the molar volume of the liquid phase. The equation of state for the ideal gas (the ideal-gas law) is applicable only for the pressure p ≤ 20,000,000 Pa and temperature T ≤ 253 K. Compared with the experimental values, the Guevara-Rodríguez equation of state for any substance, which includes the gas, liquid and solid, is appropriate for the extra low pressure 0.000000356 ≤ p ≤ 114,000 Pa and temperature 87.8 ≤ T ≤ 266.3 K.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-025-17621-x.
Keywords: Equation of state, Molar volume, Compressibility factor, Acentric factor, One variable cubic equation
Subject terms: Analytical chemistry, Chemical engineering
Introduction
Water is ubiquitous and it has been virtually the most investigated fluid so far. An equation of state of water for industrial purpose providing the most precise representation of the thermodynamic property of the fluid phase of water substance over a wide range of definite conditions was studied. The equations of state for fluids (including gas and liquid) are very pivotal and important in correlating and analyzing the thermodynamic equilibrium state using some agreeable methods. The role of equation of state in modeling the compressive flows is intrinsic. Usually, for a given substance, the equation of state is such a relationship as the pressure is the explicit function of the temperature, the molar volume and other corresponding known parameters. Cubic equations of state have been widely used in the simulation of chemical industrial processes and in the oil extraction industry because they require only a very general knowledge of the fluid molecule and allow for a short computational time.
Grove1 discussed the general case of how to impose an equilibrium closure for a mixture based on an abstract equilibrium condition and what constraints are imposed on such a closure by thermodynamic stability and consistency. This is a mathematical discussion of methods to construct constitutive equations of state for mixtures. Nezbeda et al.2 formulated and outlined a strategy to use currently available semi theoretical tools to develop an equation of state. With respect to the absence of any experimental data, molecular modeling and simulations become this practical tool to study supersaturated steam. Both Pb and Li mixtures form the most dramatically non-ideal solutions, which have resulted in fluid properties that sometimes differ significantly from measurement values. Humrickhouse et al.3 developed an equation of state for liquid Pb83Li17 by fitting a generalized form of the Helmholtz free energy. Ghoderao et al.4 presented a five parameter cubic equation of state to predict the thermodynamic properties of pure fluids and mixtures. Cubic equations of state have been adopted very widely because of their simplicity which allows fast and reliable prediction of thermodynamic properties for pure substance and mixtures. Bell et al.5 evaluated the performance of the group contribution volume translated Peng-Robinson model when predicting the vapor-liquid equilibrium and single phase densities of 28 refrigerant mixtures with low global warming potential and zero ozone depletion potential. Cubic equations of state, particularly the Peng-Robinson equation of state, are widely applied in the refrigeration industry due to their easy applicability for new substances and their short computational time. Yang et al.6 investigated the combination of the crossover method and the multiparameter equation of state using carbon dioxide as a demonstration. The combination highlighted the procedure of applying the crossover method to a purely empirical equation of state, while leaving a major part of the equation of state unaltered but removing the fallacious non-analytical terms.
The cubic equation of state was recommended by van der Waals7 in 1873. It has been proven that the original van der Waals cubic equation of state can not provide simultaneous accurate predictions for all the properties of pure fluids and their mixtures. Redlich et al.8 proposed an equation of state containing two individual coefficients which furnished satisfactory results above the critical temperature for any pressure. The dependence of the coefficients a and b on the composition of the gas was discussed from the critical conditions. In References 7 and 8, the coefficient a is constant and independent of the temperature T. Soave9 proposed a modified Redlich-Kwong equation of state. Vapor pressures of pure compounds can be closely reproduced by assuming the parameter a in the original equation to be temperature-dependent. Peng et al.10 outlined the development of a two-constant equation of state in which the attractive pressure term of the semiempirical van der Waals equation had been modified. The constant b was related to the size of the hard spheres. The parameter a could be regarded as a measure of the intermolecular attraction force. The Redlich-Kwong equation of state possesses a rare combination of qualities: realism and simplicity. Harmens11 chose the Redlich-Kwong equation of state as the basis for a general purpose multicomponent thermodynamic data programme for cryogenic substances. Harmens12 proposed a two-parameter cubic equation of state showing to be particularly useful for calculation of vapour-liquid equilibrium of the nitrogen-argon-oxygen system. However, the two-parameter cubic equation of state suffers from the so-called critical anomaly. Inherent with each equation of this type is an invariant value of the critical compressibility factor Zc. Schmidt et al.13 presented a cubic equation of the van der Waals type with the critical compressibility factor taken as substance dependent. Input requirements are acentric factor and the critical temperature and pressure. Parameter b is treated as independent of temperature, and the temperature dependence of parameter a is given by an expression. Harmens et al.14 proposed a cubic equation of state possessing three adjustable parameters. The three parameters were correlated in terms of the critical temperature Tc, critical pressure pc and acentric factor ω. Abbott warned against setting the calculated critical compressibility factor ζc equal to the experimental critical compressibility factor Zc. Better overall results are obtained with . Patel et al.15 presented a cubic equation of state for pure fluids. This equation requires the critical temperature and critical pressure, as well as two additional substance dependent parameters of the calculated critical compressibility factor and the fugacity slope as input parameters. Initially, ζc was set equal to 1.1Zc. Nasrifar et al.16 developed a two-parameter cubic equation of state. Both parameters are taken temperature dependent. Nasrifar et al.17 used fugacity coefficient of a cubic equation of state. Henry’s law constant of a solute in a solvent was incorporated into binary interaction parameter of the classical attractive parameter mixing rule. Twu et al.18,19 proposed an approach to locate an optimum two-parameter cubic equation of state. A methodology was also proposed to modify Twu’s CEOS/AE zero-pressure mixing rules to extend the range of application of these mixing rules. Kubic20 presented a modification of the Martin equation of state suitable for vapour-liquid equilibrium calculations. The temperature dependence of this modification is determined from the second virial coefficient correlation of Tsonopoulos and the pure-component vapour pressures. Wenzel et al.21 extended a van der Waals-type equation of state by a fluid term containing one adjustable parameter to include a solid phase.
By transforming the equation of state, one variable cubic equation in the molar volume can be obtained at given temperature and pressure. There are three roots for any one variable cubic equation in the molar volume. On one hand, which of three roots is selected to become the sole correct answer of the molar volume? On the other hand, how does one actually deduce the complete and accurate analytical solution from one variable cubic equation in the molar volume? Three goals of this work are respectively: (1) to deduce the complete analytical solution of one variable cubic equation with four arbitrary coefficients; (2) to attain the molar volume of equation of state with the attractive pressure term of universal quadratic tri-equation, adopting the complete analytical solution of one variable cubic equation; and (3) to acquire the solutions for some famous equations of state from the current references.
With three purposes in mind, the remainder of the article is arranged as follows. In section “Equation of state with the intermolecular attraction force of universal quadratic trinomial”, the molar volume of equation of state with the intermolecular attraction force of universal quadratic trinomial can be obtained according to the complete analytical solution of one variable cubic equation in Supplementary Material B. Again, in section “Some special cases of the general solution of cubic EOS”, the solutions for many well known equations of state from the current references were easily achieved in terms of these useful expressions from section “Equation of state with the intermolecular attraction force of universal quadratic trinomial”. Later, in section “Experimental verification”, the present values were compared with the experimental ones and ones in some references from section “Some special cases of the general solution of cubic EOS”. In the end, section “Conclusion” provides the conclusion from the current research study. In Supplementary Material A, the complete and precise analytical solution is presented about one variable cubic equation with two arbitrary real constants p and q: , p, q∈R, y∈C. The full and exact analytical expression is presented about one variable cubic equation with four arbitrary real constants a, b, c and d: , a, b, c, d∈R, a ≠ 0, x∈C, using variable transformation and the analytical solution from Supplementary Material A.
Equation of state with the intermolecular attraction force of universal quadratic trinomial
Perhaps equation of state generally expresses pressure as the sum of two terms, a repulsion pressure and an attraction pressure. Cubic equation of state is the most useful tools to predict the thermodynamic equilibrium state of simple fluids and their mixtures. Their fundamental base, which has been accepted since the pioneer work of van der Waals, is the physical notion of an ensemble of spherical particles, which interact through an attractive potential and a hard core. Cubic equations of state are not based on some particular molecules, however, they allow us to improve the treatment. By a simple considering of the intermolecular attraction forces, the cubic equations of state can be expressed as22
1
where J/(mol K) symbolises the universal gas constant, J/(molecule K) the Boltzmann’s constant, molecules/mol the Avogadro’s constant, T the temperature, v the molar volume, b the occupied volume per mole molecule, , a(T) ≥ 0 a function dependent on the temperature T, and c and d the constant parameters, c ≥ d.
Introducing and into Eqs. (93), (89), (95), (90) and (91) detects, respectively
101
102
103
104
105
In order to determine a temperature dependency for the parameter a(T), first the temperature is scaled according to
106
107
108
where Tpt is called pseudo triple point temperature, for it is the characteristic of a component and could be smaller or larger than the real triple point temperature of the component.
The experimental critical compressibility factor obeys a linear empirical relationship for non-associating substances13
136
The quantity of interest and an approximate measure are the reduced co-volume b/vc because the value of b is generally close to that of the calculated liquid molar volume vL at low reduced temperatures. Using data for 65 compounds as given in a compilation of thermophysical properties, the exact solution of Eq. (133) is compared with the approximative solution of Eq. (135) in Fig. 1. Two almost straight lines depicted by Reference 13 and present paper are completely same for the ω range considered.
Comparison of the exact and approximative solutions of b/vc, Eq. (133) and Eq. (135), respectively, calculated for 65 compounds.
For the critical compressibility factor, the calculated values of Eq. (134) are compared with the experimental ones of Eq. (136) in Fig. 2. Two almost straight lines depicted by Reference 13 and present paper are completely same for the ω range considered. In Fig. 2, since the two trends are parallel (equality of slopes), I can suggest a new corrected term 0.331 different than 0.291 in Eq. (136) related to the experimental critical compressibility factor obeying a linear empirical relationship for non-associating substances. Or it’s related to the degree of association!
Experimental and calculated critical compressibility factors for 65 compounds.
Use the acentric factor ω as input data. Table 1 lists two parameters the reduced co-volume, and the critical compressibility factor. Table 1 shows that ζc is always greater than Zc and the difference is approximately constant 0.042.
Table 1.
Reduced co-volume and the critical compressibility factor.
Introducing Eqs. (180) and (182) into Eq. (24) obtains
183
184
185
where
186
187
The parameter Zc in Eqs. (186) and (187) is the experimental critical compressibility factor. Equation (185), while it does not predict the experimental critical compressibility factor, does predict a value which is linearly dependent on the experimental value by introducing Eqs. (186) and (187) into Eq. (185)
188
189
The calculated values of Eq. (188) are compared with the experimental critical compressibility factors in Fig. 3, which shows that ζc is always bigger than Zc. Further, the difference is approximately constant 0.043. In Fig. 3, likewise, I can suggest an interesting linear relationship between experimental results and the Kubic model. It’s a kind of correction and adaptation, or prediction for other non treated compounds. I can suggest a new corrected term 0.331 different than 0.291 in Eq. (136) related to the experimental critical compressibility factor obeying a linear empirical relationship for non-associating substances.
Experimental and calculated critical compressibility factors for 65 compounds.
Experimental verification
Some thermophysical properties and data will be experimentally determined for 65 compounds as given in Table 1. In cases where data points are given in temperature steps of 4℃, as is the case for some refrigerants. These experimental data must include the four critical data (critical pressure pc, critical molar volume vc, critical temperature Tc, and critical compressibility factor Zc) and acentric factor ω. Of course, other related parameters, such as the reduced molar volume of the saturated vapor phase and the hard sphere diameter σ, are to be given, too. Experimental devices used in the test are used by changing the pressure and temperature. Closed-loop control of vapor slip can be achieved by sensor. The cross-sensitivity of sensor to vapor finally becomes an advantage, and the average of vapor leakage is further reduced.
The critical pressure, critical molar volume vc, critical temperature, acentric factor and critical compressibility factor of 16 substances are presented in Table 2. The acentric factors of the present paper are close to the values in Reference 22. The critical compressibility factors of the present paper approach the values in Ref.22.
Table 2.
Acentric factor and critical compressibility factor.
A lot of theoretical values and the experimental ones are shown in Fig. 6 about reduced pressure pr and reduced molar volume vr. The theoretical values of this study are in better agreement with the experimental ones than other ones from some typical Refs.7–10,12,13,16,18,20. Especially, in Fig. 6a–e, the van der Waals values from Reference 7 are in bigger disagreement with the experimental ones. In Fig. 6c and d, the pr–vr behavior at a given temperature appears as a bell-shaped curve, which is not correct since it will indicate unstable. A bell-shaped pr–vr curve typically indicates the equilibrium. Further clarification is needed to justify this behavior.
The theoretical values of the occupied volume per mole molecule labeled b are compared with the experimental ones listed in Table 4. The theoretical values of b are very close to the experimental ones. From Eq. (68), the important constant b is directly determined by the diameter of the hard core σ.
Table 4.
Experimental and theoretical values of b (mL/mol).
In Tables 5, 6 and 7, the pressure p and test value v of molar volume are monotonously increasing. The pressure is much lower than the standard atmospheric pressure 101,325 Pa. The changing range of the pressure is very large from 0.000000356 Pa to 114,000 Pa. The temperature T is monotonously increasing only in Table 7 except for one datum 123.43 labeled ↓ decreasing (this experimental data point does not seem valid, or it seems that the data point 123.43 K is a typographical error and probably it should be 132.43 K). The changing range of the temperature is large from 87.8 K to 266.3 K, too. One variable cubic Eq. (2) has three unequal real roots labeled v1, v2, v3. Hold and . According to the test values and theoretical ones of molar volume, the smallest positive root v1 is selected as the molar volume. In Reference 15, one variable cubic equation in Ωb has one real positive root and two conjugate imaginary roots. Therefore, Ωb is the smallest positive real root in the subsequent calculations. According to Tables 5, 6 and 7, a residual plot, such as percentage molar volume error versus pressure or temperature, can be obtained, as it provides more informative insight into the model’s accuracy.
Conclusion
The parameters in any equation of state with two different real roots for any substance, which includes the gas, liquid and solid, are given in terms of intermolecular attractive force. Using the analytical formulas of one variable cubic equation, a complete analytical solution is deduced to obtain the molar volume of equation of state for any given pressure and temperature.
The values in the present article are compared with the experimental ones and ones in some references. The theoretical values of this study are in better agreement with the experimental ones than ones in other references.
It is clear that the equation of state has three unequal real roots of the molar volume under the extremely low pressure and temperature. When there are 3 real roots in the molar volume, then the smallest positive root is the liquid phase, and the biggest positive root is the vapour phase. The equation of state for the ideal gas is applicable only for the pressure p ≤ 20,000,000 Pa and temperature T ≤ 253 K. Compared with the experimental data, the equation of state for any substance, which includes the gas, liquid and solid, is applicable for the extremely low pressure 0.000000356 ≤ p ≤ 114,000 Pa and temperature 87.8 ≤ T ≤ 266.3 K.
Letting the product of two cube roots hold a constant, the three real root formulas of one variable cubic equation are deduced. Two kinds of analytical formulas for the solutions of one variable cubic equation are completely listed in two tables.
The complete analytical solution of one variable cubic equation with four arbitrary coefficients has been receiving wide attention during the past 480 years since there are two accounts of this breakthrough by Girolamo Cardano of Pavia now best known for his Ars Magna in 1545, that gives methods for finding real solutions of cubic and quartic equations, and by Nicolo Tartaglia of Brescia regarding the solution of cubic equations in 154627. The full potential of the complete analytical solution of one variable cubic equation with four arbitrary coefficients is still far from being realized. In the present instance, however, it would require the solution of one variable cubic equation readily available elsewhere to introduce too many fragmentary, incomplete even premature formulae. With this fact in my mind the only sole and important question is: After all, what is the complete analytical solution of one variable cubic equation with four arbitrary coefficients? I myself have derived and tabulated the complete analytical solutions, which cannot be found anywhere, for the roots of the cubic equation with arbitrary coefficients. The complete analytical solution of one variable cubic equation with four arbitrary coefficients is, for solving the cubic equation of state proposed by Guevara-Rodríguez22of utmost importance. As the accuracy and efficiency requirements for solving the cubic equation of state proposed by Guevara-Rodríguez22 are getting increasingly higher, I ought to be better prepared for the complete analytical solution of one variable cubic equation with four arbitrary coefficients. Using the complete analytical solution of one variable cubic equation with four arbitrary coefficients, I myself have solved the cubic equation of state proposed by Guevara-Rodríguez22. A vast digital twin, used by the team of Wang Xiaopeng, a hydrologist at the China Three Gorges University, Yichang, China, is at the heart of the project and a tool for using data on engineering modelling28. There are few global examples that show how to develop digital twins in a robust way with good standards. In general, the concept of digital twin is more advanced in industrial manufacturing than in river management. Improving openness could allow the data and models from several global famous equations of state to be used for the research and development of noncommercial applications. Such as, using a vast digital twin and comparing with several global famous equations of state, I myself have demonstrated the effectiveness and universality of the cubic equation of state proposed by Guevara-Rodríguez22. Open access to the present paper including Supplementary Material A and Supplementary Material B could be to all individuals.
Supplementary Information
Below is the link to the electronic supplementary material.
Ensure that all authors are individually mentioned in the author contribution statement. Kindly mention that this manuscript only includes a sole author named Hongliang Tian. I myself have written the main manuscript text, reviewed the manuscript, analyzed the data and edited the revised version of the manuscript.
Funding
This work was supported by the National Natural Science Foundation of China under grant number 51275273. Sponsor: Hongliang Tian provided design ideas and paper direction guidance.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Competing interests
The authors declare no competing interests.
Footnotes
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. Patel et al.15 presented a cubic equation of state for pure fluids. This equation requires the critical temperature and critical pressure, as well as two additional substance dependent parameters of the calculated critical compressibility factor and the fugacity slope as input parameters. Initially, ζc was set equal to 1.1Zc. Nasrifar et al.16 developed a two-parameter cubic equation of state. Both parameters are taken temperature dependent. Nasrifar et al.17 used fugacity coefficient of a cubic equation of state. Henry’s law constant of a solute in a solvent was incorporated into binary interaction parameter of the classical attractive parameter mixing rule. Twu et al.18,19 proposed an approach to locate an optimum two-parameter cubic equation of state. A methodology was also proposed to modify Twu’s CEOS/AE zero-pressure mixing rules to extend the range of application of these mixing rules. Kubic20 presented a modification of the Martin equation of state suitable for vapour-liquid equilibrium calculations. The temperature dependence of this modification is determined from the second virial coefficient correlation of Tsonopoulos and the pure-component vapour pressures. Wenzel et al.21 extended a van der Waals-type equation of state by a fluid term containing one adjustable parameter to include a solid phase.
, p, q∈R, y∈C. The full and exact analytical expression is presented about one variable cubic equation with four arbitrary real constants a, b, c and d: 
, a, b, c, d∈R, a ≠ 0, x∈C, using variable transformation and the analytical solution from Supplementary Material A.
J/(mol K) symbolises the universal gas constant, 
J/(molecule K) the Boltzmann’s constant, 
molecules/mol the Avogadro’s constant, T the temperature, v the molar volume, b the occupied volume per mole molecule, 
, a(T) ≥ 0 a function dependent on the temperature T, and c and d the constant parameters, c ≥ d.
, 
, and 
, write the virial expansion according to the infinite decreasing geometric sequence
and 
and 
into Eqs. (93), (89), (95), (90) and (91) detects, respectively
and 
into Eq. (83) detects
into Eqs. (89), (90), (91), (93) and (95) detects, respectively
and 
completely satisfying 
into Eqs. (126) and (130) obeys, respectively
is approximately constant 0.042.
or 
satisfying for 65 compounds in Table 1.
and 
into Eqs. (128) and (129) discovers
leads to the inequality for ever existing
and 
satisfies Eq. (149).
and 
and 
complying with 
in Sect. 2.2 and Eq. (149) into Eqs. (125)–(129) detects, respectively
and 
and 
obeying 
in Sect. 2.2 and Eq. (149) into Eqs. (125)–(129) detects, respectively
and 
and 
satisfying 
in Sect. 2.2 and Eq. (149) into Eqs. (125)–(129) detects, respectively
and 
and 
satisfying 
in Sect. 2.2 and Eq. (149) into Eqs. (125)–(129) detects, respectively
and Eq. (180) into Eq. (29) obtains
is approximately constant 0.043. In Fig. 3, likewise, I can suggest an interesting linear relationship between experimental results and the Kubic model. It’s a kind of correction and adaptation, or prediction for other non treated compounds. I can suggest a new corrected term 0.331 different than 0.291 in Eq. (136) related to the experimental critical compressibility factor obeying a linear empirical relationship for non-associating substances.
of the saturated vapor phase and the hard sphere diameter σ, are to be given, too. Experimental devices used in the test are used by changing the pressure and temperature. Closed-loop control of vapor slip can be achieved by sensor. The cross-sensitivity of sensor to vapor finally becomes an advantage, and the average of vapor leakage is further reduced.
is depicted in Fig. 5.
of 16 substances.
22
22
atm, 
K, 
bar, 
K. For the van der Waals equation of state23–25.
Pa m6 mol− 2.
m3 mol− 1.
L/mol at 
atm and 
K.
and 
. According to the test values and theoretical ones of molar volume, the smallest positive root v1 is selected as the molar volume. In Reference 15, one variable cubic equation in Ωb has one real positive root and two conjugate imaginary roots. Therefore, Ωb is the smallest positive real root in the subsequent calculations. According to Tables 5, 6 and 7, a residual plot, such as percentage molar volume error versus pressure or temperature, can be obtained, as it provides more informative insight into the model’s accuracy.
with four arbitrary coefficients has been receiving wide attention during the past 480 years since there are two accounts of this breakthrough by Girolamo Cardano of Pavia now best known for his Ars Magna in 1545, that gives methods for finding real solutions of cubic and quartic equations, and by Nicolo Tartaglia of Brescia regarding the solution of cubic equations in 154627. The full potential of the complete analytical solution of one variable cubic equation 
with four arbitrary coefficients is still far from being realized. In the present instance, however, it would require the solution of one variable cubic equation readily available elsewhere to introduce too many fragmentary, incomplete even premature formulae. With this fact in my mind the only sole and important question is: After all, what is the complete analytical solution of one variable cubic equation 
with four arbitrary coefficients? I myself have derived and tabulated the complete analytical solutions, which cannot be found anywhere, for the roots of the cubic equation 
with arbitrary coefficients. The complete analytical solution of one variable cubic equation with four arbitrary coefficients is, for solving the cubic equation of state proposed by Guevara-Rodríguez22of utmost importance. As the accuracy and efficiency requirements for solving the cubic equation of state proposed by Guevara-Rodríguez22 are getting increasingly higher, I ought to be better prepared for the complete analytical solution of one variable cubic equation with four arbitrary coefficients. Using the complete analytical solution of one variable cubic equation with four arbitrary coefficients, I myself have solved the cubic equation of state proposed by Guevara-Rodríguez22. A vast digital twin, used by the team of Wang Xiaopeng, a hydrologist at the China Three Gorges University, Yichang, China, is at the heart of the project and a tool for using data on engineering modelling28. There are few global examples that show how to develop digital twins in a robust way with good standards. In general, the concept of digital twin is more advanced in industrial manufacturing than in river management. Improving openness could allow the data and models from several global famous equations of state to be used for the research and development of noncommercial applications. Such as, using a vast digital twin and comparing with several global famous equations of state, I myself have demonstrated the effectiveness and universality of the cubic equation of state proposed by Guevara-Rodríguez22. Open access to the present paper including Supplementary Material A and Supplementary Material B could be to all individuals.