Full Pressure Range Research Articles

Derivation of full range vapor pressure equation from an arbitrary point

An equation was proposed to obtain the full range of vapor pressures (VPs) for a substance at an arbitrary point of VP data. The basic model is a vapor pressure equation in the form of a reduced Antoine equation derived from the van der Waals equation. Here, Antoine constant C is defined as c by dividing it into gas constant and critical temperature, and is expressed as a polynomial function of reduced temperature. Previously, the exponents were integers, but have been improved to real numbers. Edmister’s formula is a special case in which the Antoine constant C divided by the critical temperature in Chen’s equation is 0. Since this is equivalent to polynomial of c being zero, the intercept can be set to zero. The results, estimated using the derived equation, were compared with VP data for 76 substances. The error rate of the equation using the acentric factor as a variable was 0.49%. This was lower than the 0.50% error rate of the Lee-Kesler method. Meanwhile, by using the correlation between the coefficients, the VP equation can be derived at any one point of VP data other than the acentric factor. For 76 substances, the equation derived from the corresponding VPs at reduced temperatures of 0.25 ∼ 0.95 in increments of 0.05 showed a mean error rate of 0.68%. This can be considered an accuracy equivalent to the Lee-Kesler method, which has the limitation of deriving a VP equation only at the reduced temperature of 0.7.

Use of near-infrared spectroscopy for the online monitoring of natural gas composition (hydrocarbons, water and CO2 content) at high pressure

The natural gas production from Brazilian pre-salt fields imposed new challenges for petrochemical industry. Actual treatment facilities are not adequate for this new scenario and studies have been conducted to apply new adsorbents materials and membranes for water and CO2 removal from natural gas at high pressures. To better develop such investigations, sensors for on-line monitoring of natural gas properties like CO2 and water content are important, since their presence affects the quality of the final product. Near infrared (NIR) spectroscopy associated to chemometric models (partial least squares) were employed for on-line monitoring of representative natural gas systems (methane/CO2, methane/water and methane/CO2/water) and a real natural gas at temperature and pressure ranges from 20 to 60 °C and 10 to 200 bar, respectively, water content up to gas saturation and CO2 content up to 50 wt%. Water solubility values used as reference for NIR Spectrometer calibration in model systems were taken from the literature and for real natural gas calculated with Cubic Plus Association (CPA) Equation of State (EoS), while CO2 content was experimentally controlled aiming to calibrate the chemometric models in the full range of pressure and temperature. Several strategies were adopted for the chemometric model’s development to obtain the best correlation between NIR spectra and experimental data. Results indicate good correlation in both calibration and validation steps attaining linear correlation coefficients (R2) higher than 0.96 for all systems investigated. The proposed methodology is a potential tool for on-line monitoring of natural gas composition, including CO2 and water content, at high-pressures and can be applied at petrochemical industries or in laboratories, dispensing sampling or any sample preparation.

Quantifying uncertainty in analysis of shockless dynamic compression experiments on platinum. II. Bayesian model calibration

Dynamic shockless compression experiments provide the ability to explore material behavior at extreme pressures but relatively low temperatures. Typically, the data from these types of experiments are interpreted through an analytic method called Lagrangian analysis. In this work, alternative analysis methods are explored using modern statistical methods. Specifically, Bayesian model calibration is applied to a new set of platinum data shocklessly compressed to 570 GPa. Several platinum equation-of-state models are evaluated, including traditional parametric forms as well as a novel non-parametric model concept. The results are compared to those in Paper I obtained by inverse Lagrangian analysis. The comparisons suggest that Bayesian calibration is not only a viable framework for precise quantification of the compression path, but also reveals insights pertaining to trade-offs surrounding model form selection, sensitivities of the relevant experimental uncertainties, and assumptions and limitations within Lagrangian analysis. The non-parametric model method, in particular, is found to give precise unbiased results and is expected to be useful over a wide range of applications. The calibration results in estimates of the platinum principal isentrope over the full range of experimental pressures to a standard error of 1.6%, which extends the results from Paper I while maintaining the high precision required for the platinum pressure standard.

Melting and defect transitions in FeO up to pressures of Earth's core-mantle boundary.

The high-pressure melting curve of FeO controls key aspects of Earth's deep interior and the evolution of rocky planets more broadly. However, existing melting studies on wüstite were conducted across a limited pressure range and exhibit substantial disagreement. Here we use an in-situ dual-technique approach that combines a suite of >1000 x-ray diffraction and synchrotron Mössbauer measurements to report the melting curve for Fe1-xO wüstite to pressures of Earth's lowermost mantle. We further observe features in the data suggesting an order-disorder transition in the iron defect structure several hundred kelvin below melting. This solid-solid transition, suggested by decades of ambient pressure research, is detected across the full pressure range of the study (30 to 140 GPa). At 136 GPa, our results constrain a relatively high melting temperature of 4140 ± 110 K, which falls above recent temperature estimates for Earth's present-day core-mantle boundary and supports the viability of solid FeO-rich structures at the roots of mantle plumes. The coincidence of the defect order-disorder transition with pressure-temperature conditions of Earth's mantle base raises broad questions about its possible influence on key physical properties of the region, including rheology and conductivity.

Effects of Charge Density on Spread Hyperbranched Polyelectrolyte/Surfactant Films at the Air/Water Interface.

The interfacial structure and morphology of films spread from hyperbranched polyethylene imine/sodium dodecyl sulfate (PEI/SDS) aggregates at the air/water interface have been resolved for the first time with respect to polyelectrolyte charged density. A recently developed method to form efficient films from the dissociation of aggregates using a minimal quantity of materials is exploited as a step forward in enhancing understanding of the film properties with a view to their future use in technological applications. Interfacial techniques that resolve different time and length scales, namely, ellipsometry, Brewster angle microscopy, and neutron reflectometry, are used. Extended structures of both components are formed under a monolayer of the surfactant with bound polyelectrolytes upon film compression on subphases adjusted to pH 4 or 10, corresponding to high and low charge density of the polyelectrolyte, respectively. A rigid film is related to compact conformation of the PEI in the interfacial structure at pH 4, while it is observed that aggregates remain embedded in mobile films at pH 10. The ability to compact surfactants in the monolayer to the same extent as its maximum coverage in the absence of polyelectrolyte is distinct from the behavior observed for spread films involving linear polyelectrolytes, and intriguingly evidence points to the formation of extended structures over the full range of surface pressures. We conclude that the molecular architecture and charge density can be important parameters in controlling the structures and properties of spread polyelectrolyte/surfactant films, which holds relevance to a range of applications, such as those where PEI is used, including CO2 capture, electronic devices, and gene transfection.

Contact pressure dependent mechanisms of ultralow wear PTFE composites

One of the most dramatic reductions in the wear of PTFE has been achieved by compositing PTFE with as little as 1–5 wt% of alumina particles; this has been reported to produce wear rates K ∼10−7 mm3/Nm. The mechanisms associated with this reduction in wear are multifaceted, including 1) preventing crack propagation and delamination of the PTFE wear surface, 2) promoting tribochemistry and more recently 3) tribologically-induced breaking of the filler into nanoscale fragments to stabilize and reinforce tribofilms. However, in an effort to keep experiments comparable, many of the studies throughout the literature have focused on a narrow contact pressure range. In these experiments, we explored the effects of contact pressure on the tribological behavior of different PTFE and alumina composites, one of which is reported to achieve ultra-low wear (∼10−7 mm3/Nm) and another that is reported to only have mild reductions in wear (∼1 × 10−5 mm3/Nm) compared to unfilled PTFE (∼4 × 10−4 mm3/Nm). We found that with decreased contact pressures, the PTFE-alumina composite that was previously reported as high wear could achieve ultralow wear rates. The PTFE-alumina composite previously reported to achieve ultralow wear achieved ultralow wear at a range of low to high contact pressures, with a higher pressure limit corresponding to increases in wear. The friction behavior of PTFE-alumina composites was found to be highly dependent on contact pressure, with increasing pressures resulting in decreasing friction coefficients (∼0.5–0.17 over a 0.62–8.5 MPa range). This effect became more pronounced when the contact pressure was incrementally varied during testing resulting in up to a 70% decrease or increase in friction coefficient due to increasing or decreasing the pressure, respectively. IR spectra of the polymer wear surface showed that tribofilms rich in carboxylates and metal oxides form at the full range of contact pressures tested, even at the extremes. This formation of tribofilms at the sliding interface not only contributes to the ultralow wear of these materials, but plays a role in the friction behavior observed. From this, we gained new insight into the role, functionality and limitations of the alumina fillers.

Challenging Pronatalism Is Key to Advancing Reproductive Rights and a Sustainable Population

Social and environmental justice organisations have silenced discourse on human overpopulation due to fear of any association with reproductive coercion, but in doing so they have failed to acknowledge the oppressive role of pronatalism in undermining reproductive autonomy. Pronatalism, which comprises cultural and institutional forces that compel reproduction, is far more widespread, and as damaging to individual liberties as attempts to limit reproduction. The failure to recognise the enormity of pronatalism has led to the wholesale abandonment of voluntary, rights-based efforts toward a sustainable population despite widespread scientific agreement that population growth is a major driver of multiple cascading environmental crises. We examine the full range of patriarchal, cultural, familial, religious, economic and political pronatalist pressures, and argue that the reluctance to address population as a driver of the ecological crisis serves the very pronatalist forces that undermine reproductive autonomy. We posit that addressing overpopulation, and the pronatalism that drives it, must be central to international conservation and development efforts to elevate reproductive rights while also promoting planetary health.

An improved apparent permeability model considering full pore pressure range, variable intrinsic permeability and slippage coefficient

An improved apparent permeability model considering full pore pressure range, variable intrinsic permeability and slippage coefficient

Quantitative Kinetics of HO2 Reactions with Aldehydes in the Atmosphere: High-Order Dynamic Correlation, Anharmonicity, and Falloff Effects Are All Important.

Kinetics provides the fundamental parameters for elucidating sources and sinks of key atmospheric species and for atmospheric modeling more generally. Obtaining quantitative kinetics in the laboratory for the full range of atmospheric temperatures and pressures is quite difficult. Here, we use computational chemistry to obtain quantitative rate constants for the reactions of HO2 with HCHO, CH3CHO, and CF3CHO. First, we calculate the high-pressure-limit rate constants by using a dual-level strategy that combines conventional transition state theory using a high level of electronic structure wave function theory with canonical variational transition state theory including small-curvature tunneling using density functional theory. The wave-function level is beyond-CCSD(T) for HCHO and CCSD(T)-F12a (Level-A) for XCHO (X = CH3, CF3), and the density functional (Level-B) is specifically validated for these reactions. Then, we calculate the pressure-dependent rate constants by using system-specific quantum RRK theory (SS-QRRK) and also by an energy-grained master equation. The two treatments of the pressure dependence agree well. We find that the Level-A//Level-B method gives good agreement with CCSDTQ(P)/CBS. We also find that anharmonicity is an important factor that increases the rate constants of all three reactions. We find that the HO2 + HCHO reaction has a significant dependence on pressure, but the HO2 + CF3CHO reaction is almost independent of pressure. Our findings show that the HO2 + HCHO reaction makes important contribution to the sink for HCHO, and the HO2 + CF3CHO reaction is the dominant sink for CF3CHO in the atmosphere.

Static and Dynamic Performance of the RBRargo3 CTD

Abstract The static and dynamic performances of the RBRargo3 are investigated using a combination of laboratory-based and in situ datasets from floats deployed as part of an Argo pilot program. Temperature and pressure measurements compare well to co-located reference data acquired from shipboard CTDs. Static accuracy of salinity measurements is significantly improved using 1) a time lag for temperature, 2) a quadratic pressure dependence, and 3) a unit-based calibration for each RBRargo3 over its full pressure range. Long-term deployments show no significant drift in the RBRargo3 accuracy. The dynamic response of the RBRargo3 demonstrates the presence of two different adjustment time scales: a long-term adjustment O(120) s, driven by the temperature difference between the interior of the conductivity cell and the water, and a short-term adjustment O(5–10) s, associated to the initial exchange of heat between the water and the inner ceramic. Corrections for these effects, including dependence on profiling speed, are developed.

Adsorption equilibria and kinetics of ethane and ethylene on zeolite 13X pellets

To establish a design basis of the cyclic adsorption process for ethylene purification, the adsorption equilibria and kinetics of ethane and ethylene on zeolite 13X pellets were measured by a volumetric method at 303–343 K under pressure up to 600 kPa. The Sips model showed better prediction of ethane in the full pressure range, but the dual-site Langmuir (DSL) model was more accurate for ethylene in the pressure range of <250 kPa. The strong cation-π interaction between ethylene and Na+ in zeolite 13X led to higher adsorption capacity and affinity than those of ethane. It resulted in a greater isosteric heat of adsorption (Qst) of ethylene at a low loading amount, while Qst variance in ethane was almost linearly increased with a dominant lateral interaction. At 303 K, the adsorption amount and affinity of ethylene at <5 kPa were slightly greater than those of propane but lower than those of propylene. However, the adsorption isotherms of ethane/ethylene became higher than those of propane/propylene above a certain pressure. The experimental uptake curves of ethane and ethylene were well predicted by a non-isothermal sorption model, considering the adsorption thermal effects. The difference in the apparent reciprocal diffusional time constant (Dapp/R2) between ethane and ethylene was mainly attributed to the thermal effects by the heat of adsorption. The comparison of Dapp/R2 values between zeolite 13X pellet and powder indicated that macropore diffusional resistance also contributed to the adsorption kinetics.

Pressure-Induced Structural Phase Transition, Anomalous Insulator-to-Metal Transition, and n-p Conduction-Type Switching in Defective, NiAs-Type Cr1-δTe.

Cr1-δTe, as a unique series of defective compounds with a NiAs-type structure and periodic metal vacancies, has attracted intensive research interest because of its diversity in structure and property dependent on the tunable stoichiometric ratio. Another feature of these compounds is their ability to switch between NiAs- and MnP-type structures, in which there often exist composition-, temperature-, or pressure-induced phase transitions accompanied by intriguing physical property switching. Herein, we report the syntheses of a series of Cr1-δTe compounds with similar compositions but distinct crystal structures, their phase transitions, anomalous transport, and photoelectric conduction behaviors under high pressure (HP). For the three Cr1-δTe compounds with δ = 0, 0.25, 0.375, CrTe undergoes pressure-induced NiAs-to-MnP phase transition at around 15 GPa, while Cr3Te4 and Cr5Te8 undergo isostructural phase transitions at around 12 and 11 GPa, respectively. Electrical transport measurements indicate anomalous conduction behaviors: CrTe undergoes a semiconductor-to-metal transition at around 24 GPa, while Cr3Te4 and Cr5Te8 show unexpected metal-semiconductor-metal transitions sequentially upon compression. Besides, CrTe and Cr5Te8 exhibit pressure-induced n-p conduction-type switching at around 9 and 3 GPa, respectively, while Cr3Te4 shows p-type conductivity in the full pressure range. Local structure analyses based on in situ HP Raman spectra are performed to understand the structure and property evolutions of Cr1-δTe under HP. Defective transition-metal chalcogenides with pressure-induced NiAs-to-MnP phase transition and conduction-type conversion provide a potential platform for the rational design of photoelectric conversion devices.

Bioinspired engineering of gradient and hierarchical architecture into pressure sensors toward high sensitivity within ultra-broad working range

Introducing hierarchical microstructure on the soft material surface can effectively improve the sensitivity of the pressure sensor, however, the high sensitivity generates only at a low pressure range because of the limited compressibility of the microstructure and the incompressibility of the soft matrix. Herein, we first report a facile strategy of bioinspired engineering of hierarchical and gradient structures that can simultaneously improve the sensitivity and broaden the pressure working range. Such hierarchical and gradient structures feature hemispherical arrays and gradient pores that allow the structural deformation from tiny pressure to high pressure, significantly boosting the sensitivity over the full pressure range. By integrating these unique structures on one sensor, the sensor exhibits an ultrahigh sensitivity over a broad pressure regime (i.e., from 102.3 kPa−1 within 0–1.9 kPa to 0.34 kPa−1 within 169.6–400 kPa), a fast response time (35 ms), low limit detection (0.4 Pa) and excellent stability (>5000). The facile strategy and structure design achieve high sensitivity and broad pressure working range for an advanced pressure sensor, endowing it with wide applications, including pressure/weight monitoring, real-time human pulse wave measurements, joint motion detection and human-computer interaction.

Kicks and their significance in pore pressure prediction

Knowledge of subsurface formation pressures is critical for the calibration of predictions and models needed for safe drilling of deep wells, historically for oil and gas wells. The same details apply to the sequestration of CO 2 , ephemeral storage of gases such as hydrogen and for geothermal power. An estimated 10–14% of wells globally experience an unexpected influx of formation fluid, indicative of the controlling mud in the borehole at that time having a lower pressure than the surrounding formation. The drilling events, known as kicks and wellbore breathing, lead to, at best, downtime on the drilling rig which might affect the economic viability of the well, or in the extreme its safety with possible loss of life such as in the case of an uncontrolled blowout. Not all kicks are of equivalent value: dynamic and static kicks can be classified with a high degree of confidence and may become values for true formation pressure. Other types of fluid influx during drilling, including swab kicks and wellbore breathing, need to be identified and will not be accepted in a kick database. These types of influx may be eliminated as potential formation pressure values but, along with mud weights, can be valuable data to constrain the range of possible formation pressures, of significant where no other data exist. A new, rigorous evaluation procedure for determining formation pressure is presented, and compared with direct pore pressure measurements (e.g. RFT, MDT, RCI values). The comparison shows that the proposed methodology illustrates typical uncertainty of about 10 bar (145 psi) pressure over the full range of pressures for which data are available in this study. Thematic collection: This article is part of the Geopressure collection available at: https://www.lyellcollection.org/cc/geopressure

Hierarchical Network Enabled Flexible Textile Pressure Sensor with Ultrabroad Response Range and High-Temperature Resistance.

Thin, lightweight, and flexible textile pressure sensors with the ability to detect the full range of faint pressure (<100 Pa), low pressure (≈KPa) and high pressure (≈MPa) are in significant demand to meet the requirements for applications in daily activities and more meaningfully in some harsh environments, such as high temperature and high pressure. However, it is still a significant challenge to fulfill these requirements simultaneously in a single pressure sensor. Herein, a high‐performance pressure sensor enabled by polyimide fiber fabric with functionalized carbon‐nanotube (PI/FCNT) is obtained via a facile electrophoretic deposition (EPD) approach. High‐density FCNT is evenly wrapped and chemically bonded to the fiber surface during the EPD process, forming a conductive hierarchical fiber/FCNT matrix. Benefiting from the large compressible region of PI fiber fabric, abundant yet firm contacting points and high elastic modulus of both PI and CNT, the proposed pressure sensor can be customized and modulated to achieve both an ultra‐broad sensing range, long‐term stability and high‐temperature resistance. Thanks to these merits, the proposed pressure sensor could monitor the human physiological information, detect tiny and extremely high pressure, can be integrated into an intelligent mechanical hand to detect the contact force under high‐temperature.

Large Pressure Effects Caused by Internal Rotation in the s-cis-syn-Acrolein Stabilized Criegee Intermediate at Tropospheric Temperature and Pressure.

Criegee intermediates are important atmospheric oxidants, and quantitative kinetics for stabilized Criegee intermediates are key parameters for atmospheric modeling but are still limited. Here we report barriers and rate constants for unimolecular reactions of s-cis-syn-acrolein oxide (scsAO), in which the vinyl group makes it a prototype for Criegee intermediates produced in the ozonolysis of isoprene. We find that the MN15-L and M06-2X density functionals have CCSD(T)/CBS accuracy for the unimolecular cyclization and stereoisomerization of scsAO. We calculated high-pressure-limit rate constants by the dual-level strategy that combines (a) high-level wave function-based conventional transition-state theory (which includes coupled-cluster calculations with quasiperturbative inclusion of quadruple excitations because of the strongly multiconfigurational character of the electronic wave function) and (b) canonical variational transition-state theory with small-curvature tunneling based on a validated density functional. We calculated pressure-dependent rate constants both by system-specific quantum Rice-Ramsperger-Kassel theory and by solving the master equation. We report rate constants for unimolecular reactions of scsAO over the full range of atmospheric temperature and pressure. We found that the unimolecular reaction rates of this larger-than-previously studied Criegee intermediate depend significantly on pressure. Particularly, we found that falloff effects decrease the effective unimolecular cyclization rate constant of scsAO by about a factor of 3, but the unimolecular reaction is still the dominant atmospheric sink for scsAO at low altitudes. The large falloff caused by the inclusion of the stereoisomerization channel in the master equation calculations has broad implications for mechanistic analysis of reactions with competitive internal rotations that can produce stable rotamers.

Temperature-dependent kinetics of the atmospheric reaction between CH2OO and acetone.

Criegee intermediates are important oxidants produced in the ozonolysis of alkenes in the atmosphere. Quantitative kinetics of the reactions of Criegee intermediates are required for atmospheric modeling. However, the experimental studies do not cover the full relevant range of temperature and pressure. Here we report the quantitative kinetics of CH2OO + CH3C(O)CH3 by using our recently developed dual strategy that combines coupled cluster theory with high excitation levels for conventional transition state theory and well validated levels of density functional theory for direct dynamics calculations using canonical variational transition theory including tunneling. We find that the W3X-L//DF-CCSD(T)-F12b/jun-cc-pVDZ electronic structure method can be used to obtain quantitative kinetics of the CH2OO + CH3C(O)CH3 reaction. Whereas previous investigations considered a one-step mechanistic pathway, we find that the CH2OO + CH3C(O)CH3 reaction occurs in a stepwise manner. This has implications for the modeling of Criegee-intermediate reactions with other ketones and with aldehydes. In the kinetics calculations, we show that recrossing effects of the conventional transition state are negligible for determining the rate constant of CH2OO + CH3C(O)CH3. The present findings reveal that the rate ratio between CH2OO + CH3C(O)CH3 and OH + CH3C(O)CH3 has a significant negative dependence on temperature such that the CH2OO + CH3C(O)CH3 reaction can contribute as a significant sink for atmospheric CH3C(O)CH3 at low temperature. The present findings should have broad implications in understanding the reactions of Criegee intermediates with carbonyl compounds and ketones in the atmosphere.

Iron-wüstite revisited: A revised calibration accounting for variable stoichiometry and the effects of pressure

We present thermodynamic and empirical calculations for the iron-wüstite (IW) buffer applicable from 100 kPa to 100 GPa and from 1000 to 3000 K. The thermodynamic calculation self-consistently accounts for changing stoichiometry of iron-saturated wüstite as a function of temperature and pressure. In contrast to some previous models for calculating IW at high pressure, the model incorporates a thermodynamically valid representation of the free energy of stoichiometric FeO at 100 kPa. Earlier high pressure models that relied on the JANAF thermochemical tables (Chase, 1998) were compromised because JANAF has erroneous values for the properties of FeO. This resulted in predicted oxygen fugacities buffered by IW that are between 0.2 and 1.1 log units too reducing at 3000 and 1000 K, respectively. The revised thermodynamic calculations indicate that iron-saturated wüstite becomes more nearly stoichiometric with increasing pressure, but that this shift depends on temperature. Near-stoichiometric FeO (y < 0.01, for Fe1-yO) is reached close to 8 GPa at 1000 K and 17 GPa at 2000 K. An empirical function is presented that accurately reproduces the thermodynamic calculation and facilitates easy quantification of the fO2 of IW over the full range of temperatures and pressures covered by the model. Some caution is warranted in calculation of IW at high pressures, after FeO undergoes a transition from an insulator to a conductor and where low spin Fe2+ is stabilized, as the present model does not incorporate the effects of these transitions on the IW buffer.

Pressure-Driven Symmetry-Preserving Phase Transitions in Co(IO3)2

High-pressure synchrotron X-ray diffraction studies of cobalt iodate, Co(IO3)2, reveal a counterintuitive pressure-induced expansion along certain crystallographic directions. High-pressure Raman and infrared spectroscopy, combined with density-functional theory calculations, reveal that with increasing pressure, it becomes energetically favorable for certain I–O bonds to increase in length over the full range of pressure studied up to 28 GPa. This phenomenon is driven by the high-pressure behavior of iodate ion lone electron pairs. Two pressure-induced isosymmetric monoclinic–monoclinic phase transitions are observed at around 3.0 and 9.0 GPa, which are characterized by increasing oxygen coordination of the iodine atoms and the probable formation of pressure-induced metavalent bonds. Pressure–volume equations of state are presented, as well as a detailed discussion of the pressure dependences of the observed Raman- and infrared-active modes, which clarifies previous inconsistencies in the literature.

General Cluster Sorption Isotherm

Abstract Adsorption isotherms are an essential tool in chemical physics of surfaces. However, several approaches based on a different theoretical basis exist and for some isotherms existing approaches can fail. Here, the sorbate-sorbate interaction is not limited to an orientation vertical to the adsorbent surface. Instead, a formal orientation-independent clustering, also comprising the possibility of multilayers as a special case, is underlain. The flexible clusters are described by classical thermodynamics including a free energy relationship depending on the cluster size. In this paper a rigorous unified treatment of the adsorbate-adsorbate-adsorbent interaction is shown to result in a general isotherm equation which is applied to literature data both concerning type IV isotherms of argon and nitrogen in ordered mesoporous silica, and type II isotherms of disordered macroporous silica. The new isotherm covers the full range of partial pressure (10−6 - 0.7). The determination of surface areas is not possible by this isotherm because the cross-sectional area of a cluster is unknown. Based on the full description of type IV isotherms, most known isotherms including BET are accessible by respective simplifications. The presented model is an extension of the classical derivation of the ζ-isotherm which was shown to describe type IV isotherms restricted to adsorbates with very strong tendency of clustering such as water on hydrophobic surface (Phys. Chem. Chem. Phys. 21 (2019) 5614).