First, we report a revised determination of the Boyle temperature (T_B), and Boyle-work (BW) line (p/ρ_BW = RT) with high precision. It is seen to be a perfectly linear function of density and subject only to variations in the inaccuracy of the experimental data points. This linearity for p/ρ= RT on the T(ρ) plot ( Figure 4 ) for H₂O from NIST [9] accords with the perfect quadratic parabola for the pressure of the Boyle line for T > T_c shown above in Figure 3 .

Both steam Boyle temperature and a ground-state “solid” constant of the water Boyle-work line can be obtained from the EXCEL trendline equation in Figure 4 . We obtain T_B = 1567 K and ρ_BW(0) = 82.8 mol/l from the EXCEL trendline.

If the BW line were to be perfectly linear, an alternative determination requires only one high resolution state point ρ_BW(T) along any isotherm. We obtain ρ_BW(0) values of 80.32 and 83.90 mol/l for T = T_c and 1000 K isotherms respectively. Figure 5 shows a plot of the Boyle-work lines for the two isotherms 647.1 K (T_c) and 1000 K reduced by RT. The most striking aspect of these plots is the symmetry between steam and water states on either side of the mesophase when a trendline quadratic is fitted. If this symmetry were to be perfect, it explains the linear dependence of ρ_BW(T) at w/(RT) = 1.

The conclusion to be drawn from Figure 5 is that the excellent quadratic fit for both the gas side and the liquid side of the p/ρ isotherms, w(ρ), for all T from T_c to T_B is that w/(RT) is symmetric and a quadratic function of density, and that only two coefficients are required to determine the equation-of-state p(ρ)_T plus the BW line density ρ_BW(T_B) at w/(RT) = 1. If we define a density difference Δρ = (ρ_BW – ρ)_T, then we can see that the linearity of r_BW in the limit T ◊ T_B extends to T_c given (i) the quadratic relationship between gas side and supercritical-liquid side of w/(RT) in Figure 5 , and (ii) knowing that the linearity of ρ_BW(T) has also been found to be accurate over the range T_B > T > T_c for argon [12] .

Rigidity (ω)_p,T is a defining state function that describes the distinction between gas states below the Boyle temperature and liquid or solid states [14] [15] . It is a reciprocal compressibility that also relates to Gibbs energy, and hence also to density fluctuations, by the following equalities of classical thermodunamics and Boltzmann statistics in the theory of simple liquids [16]

$\omega ={\left(\frac{\text{d}p}{\text{d}\rho }\right)}_T=-\frac{V}{K_T}=V{\left(\frac{\text{d}p}{\text{d}\ln V}\right)}_T=\rho {\left(\frac{\text{d}G}{\text{d}p}\right)}_T={\left(\frac{\text{d}\mu }{\text{d}\ln N}\right)}_{V,T}=RT/{\left(\langle \text{Δ}{N}^2\rangle \right)}_{V,T}$(1)

where K_T is the isothermal compressibility, G is Gibbs energy, μ is Gibbs chemical potential and (N/V) is a number density. It decreases with density for a gas and increases with density for a liquid or solid. The last equality reveals the statistical origin is density fluctuations that are determined by fluctuations of liquid-like clusters of molecules in gas, that are mirrored by fluctuations of gas like voids or monomers in corresponding liquid states, of same rigidity.

It is the symmetry of the rigidity, that reveals the connection to percolation transitions that can define the distinction between gaseous states, and liquid or solid states below the Boyle temperature. Accurate rigidity data have been used, inter alia, to invalidate hypotheses that assume all real gas-liquid critical points exist as a singularity with universal scaling laws [5] .

A similar line to ρ_BW(T) can be defined as the rigidity symmetry (RS) line: ρ_RS(T) connects all ω = RT state density points on the liquid side of supercritical isotherms at a somewhat lower density than ρ_BW(T), i.e. when the ω(ρ,T) state point has the same rigidity as the ideal gas i.e. ω = RT. In the following sections we will relate the RS line to the BW line and to the two percolation lines for water or steam as found previously for argon [15] . It is this symmetry between the steam densities below ρ_PB(T), and water-like densities between ρ_PA(T), and the rigidity symmetry (RS) line ρ_RS(T), that can explain the linearity of both ρ_BW(T) and ρ_RS(T).

For pure fluids, rigidity, ω(T), can be obtained from the heat capacity data in NIST thermophysical webbook [7] , given the speed of sound (c) values and the heat capacity ratio, then $\omega =c^2{C}_p/C_v$. The rigidities of isotherms below T_B to T_c are shown in Figure 6 .

As with the Boyle line for various fluids [12] , the water rigidity line (black) shown is linear, the same as fluid argon [12] [15] . An inspection of the ω(T) isotherms in Figure 6 shows a symmetry that is consistent with the existence of a supercritical mesophase within which it remains constant along any isotherm and decreases from RT_B at the Boyle temperature: defined as in Figure 4 , as the isotherm above which there is no negative minimum in p/(ρRT) − 1. The rigidity (ω) is everywhere positive for equilibrium fluid states and decreases to zero within the critical divide at T_c ( Figure 6 ).

Notwithstanding the known complexities of the steam-water molecule, and the quantum effects of the H-bonds in cluster and solid structures, the thermodynamic equations-of-state for water reveals a near perfect steam-water rigidity symmetry, and hence also a linear ρ_RS(T) line as previously reported. The data, illustrated in Figure 5 and Figure 6 , can be obtained from NIST tabulations [7] which has been calculated from the Wagner-Pruss (W-P) equation-of-state [8] to represent the experimental thermophysical fluid data to within an accuracy that it has been reportedly measured. We only need an accurate description of the

equilibrium low-density steam phase to also describe the high-density water phase with the same precision. In this region, a two-term virial equation with only b₂(T) and b₃(T) terms suffices.

Given the scientific description of the phase transitions, the percolation lines PB and PA, and the BW and RS lines, this number of adjustable parameters can be reduced to zero, without loss of precision, using just the necessary physical constants. This requires the virial coefficients along any isotherm. From Figure 6 we infer that p(ρ,T), of steam below T_B, for all densities below the percolation transition density ρ_PB(T), will be continuous in all derivatives, and represented by the Mayer virial expansion for the pressure in powers of density along an isotherm [16]

$p=\rho RT\left(1+b_2\rho +b_3{\rho }^2+\cdots +b_n{\rho }^n+\cdots \right)$ (2)

The experimental pressure is evidently continuous in all its derivatives up to the first discontinuity, i.e. percolation line PB, whereupon the functional form changes as ω(T) = (dp/dρ)_T remains constant for the narrow density range within the mesophase from PB to PA.

Given the rigidity symmetry seen in Figure 6 , p(ρ,T) for water state points for densities exceeding PA can be obtained from the same virial expansion coefficients used to describe steam isotherms at the higher densities above ρ_PA(T) and for densities below the RS line on any isotherm in an expansion in powers of the density difference Δρ_RS = (ρ − ρ_RS)_T. The functions p(ρ)_T and p(Δρ_RS)_T are empirically quadratic for all T < T_B. The same virial coefficients that describe the gas phase for densities below the PB line can be used to represent the p(ρ,T) data for the supercritical liquid region between T_c and T_B for densities between the PA and RS lines at, and above the PA line, i.e. ρ_PA(T). All known experimental supercritical isotherms extend to higher temperatures, up to T_B (=1567 K) for steam, and pressures to 1000 MPa [7] .

The liquid region between T_c and T_t is amenable to expansion around the BW line ρ_BW(T) along any isotherm. The high-density liquid-side pressure equation-of-state p(ρ,T) can be obtained from the rigidity isotherms, via the equalities in Equation (1) for all T < T_B. Other thermodynamic state functions, e.g. Gibbs energies that determine phase coexistence boundaries can subsequently be prescribed in terms of the various physical constants for water at temperatures up to T_B (1567 K) and 1000 MPa [15] Real gases are composed of clusters of atoms or molecules, even at very low finite densities, to some extent, for all temperatures below T_B.

In the following section we show how the equilibrium constants between molecular clusters and monomer molecules relate to the virial coefficients that determine the p(ρ)_T that enable the equations-of-state to be prescribed without any adjustable parameters other than physical constants determined from the available experimental p(ρ,T) data, and the second virial coefficient from the triple point (T_t = 273.16 K) to T_B (1567 K) for H₂O.

Steam in any equilibrium thermodynamic state below T_B is a multicomponent mixture of monomers, dimers, trimers etc. This general revision of cluster physics of all real gases was first suggested by Sedunov [17] . Each species obeys the ideal gas law of partial pressures in the low-density limit. The equilibrium constants k_n for multiple association can be expressed as sequence of gas phase chemical reactions according to the equilibria: n(H₂O) ⌠ (H₂O)_n. The total pressure can be expanded as a series when (H₂O)₁ is the monomer mole fraction and (H₂O)_n fraction of cluster size n. Defining the equilibrium constants,

$k_n={\left({\text{H}}_{\text{2}}\text{O}\right)}_n/{\left({\text{H}}_{\text{2}}\text{O}\right)}_1^n$ (3)

Using the square brackets for densities, the Dalton-law partial pressures can be summed as a series for the cluster particle density species n(ρ_p) at total pressure (p),

$p={\rho }_{p}RT=1+k_2{\left[{\left({\text{H}}_{\text{2}}\text{O}\right)}_1\right]}^2+\cdots +k_n{\left[{\left({\text{H}}_{\text{2}}\text{O}\right)}_1\right]}^n+\cdots$ (4)

and $\rho =\left[{\left({\text{H}}_{\text{2}}\text{O}\right)}_1\right]+2k_2{\left[{\left({\text{H}}_{\text{2}}\text{O}\right)}_1\right]}^2+\cdots +n{k}_n{\left[{\left({\text{H}}_{\text{2}}\text{O}\right)}_1\right]}^n+\cdots$ (5)

From Equations (4) and (5) the equilibrium constant between monomers and dimers relate to the Mayer virial coefficients: k₂ = −b₂ and for trimers k₃ = (2k₂)² − b₃ are obtained. These simple relations between b_n and k_n [5] [17] show there is a significant cancellation, to some unknown extent, of the higher virial terms in the Mayer cluster expansion Equation (2). All higher virial coefficients contain a dependence upon k₂ and k₃ (T). We find that for all steam equilibrium state points for T < T_B, only b₂(T) and b₃(T) are required for accurate pressures in the range from T_B > T > T_c at densities below the percolation line ρ_PB(T), and for all isotherms below T_c, and for all equilibrium gas states at densities below both the gas-liquid and gas-solid coexistence line. This delineated (by PB) definition, and empirically accurate simplification of the gaseous state below T_B, requiring only b₂(T) and b₃(T), has also been found to apply to gaseous argon [15] .

There is an extensive literature on the second- and third-virial coefficients of steam although there is a dearth of original data for b₃(T), notwithstanding many computer-model pairwise Hamiltonian predictions. The most comprehensive data for b₂(T) from experimental data is by Harvey and Lemon [18] . The experimental correlation they report up to T = 1273 is parameterised in an equation that extrapolates to b₂ = 0 at the Boyle temperature, using a guidance from model pair potential computer calculations. They obtain a value of T_B = 1538 K, that compares favourably with our result from the 2024 NIST Boyle-line interpolation in Figure 4 (T_B = 1567 K). A combined uncertainty of less than ±2% could accommodate both results. We have recalculated the values of b₂(T), and from the NIST isotherms using a parameterisation of p(ρ)_T in the range from T_t to T_B the results are plotted in Figure 7 .

The Mayer virial series Equation (2), is a Taylor expansion for pressure derivatives in powers about zero density, whence “bond” lengths within clusters that determine k_n become negligible compared to divergent correlation lengths between clusters that determine p(ρ)_T in the virial expansion Equation (2). The rigidity symmetry along an isotherm means that the liquid-side supercritical properties can be determined from the corresponding-state gas phase given only these lower virial coefficients. We can now proceed to specify the thermodynamic state functions. notably, the Boyle and critical and triple point temperatures, BW and/or RS lines, and the coexisting PB and PA densities at T_c by physical constants required to specify the delineation lines between equations for p(ρ)_T from T_t to T_B.

The best available data, including our own data for b₂(T) is shown in a reduced form in Figure 8 and is amenable to a linear parametrisation with a single parameter over the entire range from Boyle T_B to triple-point T_t. Preliminary results for b₃(T) [19] , show that for steam-water above T_c, as with argon [15] , shows a maximum b₂(T) and b₃(T) confirm a linear relationship to the BW line for T > T_c by -b₂ = b₃ρ_BW(T) exactly in the limit T ◊ T_B as noted by Powles 40 years ago [12] . b₃ and also b₄, that goes to zero at T_c, appear to contain information on T_c. b₃(T) shows a maximum at T_c, whereas b₄ goes to zero for all T > T_c. The higher virial coefficients n > 4 must either cancel, similarly, or will become negligible at the low densities below the ρ_PB(T) line. Hence, we can proceed to specify the requisite physical constants for thermodynamic equations-of-state from only the lower virial coefficients b₂(T) and b₃(T) for all gaseous states below T_B.

The percolation pressures, of bound clusters in the gas, and of gaseous voids in the liquid, with decreasing T, intersect to trigger a critical point coexistence line at T_c and a supercritical mesophase with linear hybrid properties in the mesophase. The percolation transitions are higher-order phase transitions that delineate the gas phase below the Boyle temperature when the clusters of “liquid-in-gas” diverge to percolate the phase volume. There are no phase transitions for all gas densities below the PB-line density ρ_PB(T). The gas phase equation-of-state is given by the Mayer virial expansion, for given T. For all ρ < ρ_PB(T), we have the following, formally exact virial series, up to the first thermodynamic discontinuity, virial expansions to describe the BL and RS lines and (dω/dρ)_T “solidity” line [5] [15] ).

work $w={\left(\frac{p}{\rho }\right)}_T=RT\left(1+b_2\rho +b_3{\rho }^2+\cdots +b_n{\rho }^{n-1}\right)$ (6)

rigidity $\omega ={\left(\frac{\text{d}p}{\text{d}\rho }\right)}_T=RT\left(1+2b_2\rho +3b_3{\rho }^2+\cdots +n{b}_n{\rho }^{n-1}\right)$ (7)

solidity $\sigma ={\left(\frac{{\text{d}}^{2}p}{\text{d}{\rho }^2}\right)}_T=RT\left(2b_2+6b_3\rho +\cdots +n\left(n-1\right)b_n{\rho }^{n-2}\right)$ (8)

The Boyle work ratio for both the critical isotherm (647.1 K) and T = 1000 K, of H₂O in Figure 5 show that both work (ω) and rigidity (ω) decrease with increasing density for steam and mesophase densities below PA; both increase with density for the liquid phase above a minimum near to ρ_PA(T_c). The RS line, ω = RT is more convenient reference point in describing equations-of-state and Gibbs energies for phase transitions. Its derivative, σ, Equation (8), is negative for a gas, zero in the mesophase, and positive for liquid-side states. Equations (6)-(8) show that there must exist a maximum coexisting gas-phase density along the percolation line PB, and minimum coexisting liquid density, respectively, since both ω(T) and σ(T) go to zero at T_c.

For all terms n > 3 negligible, all points on the gas-phase percolation line PB, including the maximum coexisting gas density at T_c, occur when σ is zero in Equation (8), and PA, including minimum coexisting liquid density, likewise, can be obtained from w is a minimum in Equation (6).

$3{\rho }_{\text{PB}}\left(T\right)=2{\rho }_{\text{PA}}\left(T\right)=-{\left(b_2/b_3\right)}_T$ (9)

For the coexisting densities at T_c, Equation (9) can be shown to be very close to experimental reality. We can calculate the ratio −(b₂/3b₃)_T [19] over a range of T from T_c to T_b. Preliminary results are shown in Figure 9 . Over this temperature range, the BW line can be represented by a single ground-state density constant that determines −(b₂/b₃)_T more accurately by using the limiting region T ◊ T_B, a result first noted by Powles [12] .

Thus, if the terms b_n > 3 in Equations (9) and (10) are negligible for the gas phase below the PB line, the result can be used to confirm the experimental observation that the ω goes to a constant and that the solidity (σ) goes to zero at –(b₂/3b₃)_T. The virial ratio value at T_c is obtained from Figure 8 at the zero-density intercept −(b₂/b₃)₀ = −0.1162T + 50 mol/l. This is accurate to within the uncertainty that we can extract the values in Figure 8 from the NIST data [7] , as 3-times the maximum reported experimental coexisting gas density (ca. 16 mol/l) and minimum coexisting liquid density (ca. 22 mol/l) as seen in Figure 1 , Figure 2 , Figures 4 - 6 , from PB and PA at T_c.

Likewise, Equation (6) for the BW line can be used to obtain the minimum coexisting water density. From Figure 5 , the Boyle-work w(ρ)_T goes to a minimum at the density ρ_PA(T_c). Then, b₂ + b₃ρ_PA = 0 at T_c, resulting in the simple equality again that the minimum coexisting liquid density at T_c is given by ρ_PA(T_c) = −1/2 (b₂/b₃)_Tc. Using the ground-state constant 82.8 mol/l to define the T_B asymptotic slope of −(b₂/b₃)_T data in Figure 9 , we obtain a value ρ_PA(T_c) = 21.5 mol/l. This result is also reassuringly close to the values we have obtained from NIST thermodynamic steam water coexistence data [7] in Figure 1 , Figure 2 , Figures 4 - 6 .

The condition (p/ρ_BW)_T = RT defines the BW line that connects all these state points of both the super- and sub-critical liquid along any isotherm below T_B. The linearity can be explained by considering the form of a rigidity symmetry line. The RS line, defined by Equations (1) and (10) ω(T) = RT has its origins in the symmetry of gas and liquid density fluctuations which are equivalent to ideal gas density variance along this line in accord with Equation (1). Again, from the rigidity virial expansion Equation (10) for H₂O we see a simple prediction. If the isotherm were to be perfectly symmetric, then the virial expansion that defines the rigidity for the gas below PB-line can be applied for liquid-state densities between the PA-percolation line and the rigidity symmetry line ρ_RS(T). Then, the rigidity for ρ_PA < ρ < ρ_RS

$\omega =RT\left(1+2b_2\text{Δ}\rho +3b_3\text{Δ}{\rho }^2+\cdots +n{b}_n\text{Δ}{\rho }^{n-1}\right)$ (10)

where Δρ = (ρ_RS − ρ)_T. The RS line ρ_RS(T), when Δρ = 0, decreases linearly with T. Along any isotherm the symmetry ( Figure 5 ) extends to the BW line density ρ_BW(T). Since, for all densities beyond ρ_PA(T), there are no phase transitions of any order, both the Boyle-work ratio w(T) and rigidity ω(T) are continuous in all their density derivatives along the respective lines to the first-order phase transitions on condensation. If all the higher terms n ≥ 4 in Equations (6)-(8) are negligible by cancellation for densities below ρ_PB, then, for the Boyle work ratio

$w=RT\left(1+b_2\text{Δ}\rho +b_3\text{Δ}{\rho }^2\right)$ (11)

where Δρ = (ρ_BW − ρ)_T. Then from (10) and (11), neglecting all b_n terms n > 3,

${\rho }_{\text{RS}}\left(T\right)=2{\rho }_{\text{BW}}\left(T\right)/3$ (12)

Equation (12) suggests that the BW line (p/ρ_B) = RT) will relate to the RS line for all supercritical liquid-side states. For H₂O, as with argon [15] , this simple relationship between BW and RS lines is found to be in accord with the experimental p(ρ,T) data.

Using the experimental co-existence envelope data from NIST [9] , for the present, we can construct an accurate phase diagram that enables the bounds of the p(ρ,T) equations-of-state to be quantified. There are no arbitrary parameters other than the virial coefficients b₂(T) and b₃(T) and the physical constants that they determine. The Boyle temperature T_B is determined by b₂(T): when it changes sign from negative to positive. The BW line ground state density is then equivalent to the (-b₂/b₃)_T line.

This ground-state constant, determined also from the virial coefficients, can be used to sketch an accurate fluid-state diagram using the various simple equalities derived above. The ratio –(b₂/b₃)_T, below T_B, determines the BW line ground-state constant for water. Given the simple relationships between PA, PB, RS, and BW for all isotherms T < T_B, we can determine the critical temperature T_c (whence w(T) ◊ zero at ρ_PA(T) or ρ_PB(T), and then fill in the isotherms of p(T), w(T), or ω(T) over the temperature range from the triple point to the BW line and beyond, to the state points above the BW line below T_B ( Figure 10 ). At lower temperatures below T_c, water has a curved Boyle work line [20] . The Boyle line (T > T >T_c) bifurcates below the linear –(b₂/b₃)_T value at a density around 55 mol/l at a temperature below T_c around 600 K.

The second observation from the water-steam delineated phase diagram is that the equilibrium regions of the BW and RS lines completely exist only within a single Gibbs supercritical “liquid” phase with P = 1. Whereas the percolation

transitions delineate the supercritical gas and liquid regions by higher-order phase transitions, the BW and RS lines have no phase transitions and are therefore continuous in all their derivatives and the expansions of Equations (10)-(12) for the liquid-side of the mesophase are exact in the vicinity of BW and RS lines and accurate over the range of the respective regions in Figure 9 . Since both lines are defined by their ground-state physical constant densities obtained in linear interpolation limit that T ◊ 0, but given T_B and T_c, only ρ_BW(0) is required to determine PB,PA,RS, BW(T) and given b₂(T) then ρ_BW(0) also determines b₃(T) for use in virial equations-of-state.