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Real gas

A real gas is a gas composed of molecules or atoms that possess finite volume and experience intermolecular forces of attraction and repulsion, causing its behavior to deviate from the predictions of the ideal gas law under conditions of high pressure or low temperature.^[1] Unlike an ideal gas, which is modeled as consisting of non-interacting point particles undergoing elastic collisions, a real gas exhibits measurable non-ideal properties due to these molecular interactions and the excluded volume occupied by the particles themselves.^[2] The extent of deviation is quantified by the compressibility factor Z = \frac{PV}{nRT}, where Z = 1 for ideal behavior; for real gases, Z < 1 when attractive forces dominate (reducing effective pressure) and Z > 1 when repulsive forces or volume exclusion prevail at high densities.^[1] One of the most widely used equations of state for real gases is the van der Waals equation, derived in 1873, which modifies the ideal gas law to incorporate these effects: \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT, where a represents the strength of intermolecular attractions (in units of L²·atm·mol⁻²) and b accounts for the effective volume of the molecules (in L·mol⁻¹).^[3] The parameter a correlates with the gas's boiling point, as stronger attractions lead to higher values (e.g., 4.17 for ammonia versus 0.034 for helium), while b reflects molecular size (e.g., 0.043 for CO₂).^[3] This equation provides a more accurate description of real gas properties, such as reduced compressibility at elevated pressures, and is foundational for understanding phase transitions, critical points, and applications in thermodynamics and engineering.^[2] Real gases approach ideal behavior at low pressures and high temperatures, where molecular interactions become negligible relative to kinetic energy, but significant deviations occur near the liquefaction point or in supercritical states, influencing processes like natural gas compression and refrigerant cycles.^[1] The study of real gases extends to advanced equations of state, such as the Redlich-Kwong or Peng-Robinson models, which refine predictions for industrial applications, but the van der Waals framework remains a cornerstone for illustrating the transition from ideal to non-ideal regimes.^[3]

Fundamental Concepts

Definition and Scope

A real gas is a gaseous substance in which the finite volume of the molecules and the intermolecular attractive and repulsive forces cannot be neglected, leading to behavior that deviates from the assumptions of the ideal gas model, where molecules are treated as point particles with no interactions.^[4] This contrasts with the ideal gas approximation, which holds under conditions where molecular interactions are minimal.^[5] In the 19th century, early experimental work by Henri Victor Regnault in 1846 demonstrated that real gases deviate from Boyle's law—stating that pressure and volume are inversely proportional at constant temperature—particularly at high densities, where observed pressures exceeded predictions.^[6] Rudolf Clausius, in his 1857 revival of the kinetic theory of gases, recognized that real gases do not strictly obey ideal gas laws due to these deviations, though his model initially limited itself to dilute conditions.^[6] James Clerk Maxwell further advanced this understanding in 1860 by developing the statistical kinetic theory, highlighting how molecular velocities and collisions contribute to non-ideal properties, including the ability of real gases to liquefy under high pressure and low temperature, as observed in experiments like those of Michael Faraday starting in 1823.^[6]^[7] The scope of real gases encompasses virtually all gaseous substances under conditions beyond the dilute, high-temperature regime, where ideal behavior approximates reality. Real gases approach ideal behavior at low densities, such as pressures below 1 atm and temperatures above room temperature for most substances, where intermolecular forces become negligible relative to thermal energy.^[4]

Comparison to Ideal Gases

The ideal gas law, expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature, serves as the baseline for understanding gas behavior. This equation assumes that gas molecules are point particles with negligible volume and no intermolecular attractive or repulsive forces between them, leading to behavior where the product PV remains constant at fixed T and n. These assumptions hold well under certain conditions but fail for real gases, particularly when molecular effects become significant.^[4] Real gases exhibit quantitative deviations from this law, most commonly quantified using the compressibility factor Z = \frac{PV}{nRT}, where Z = 1 for an ideal gas and Z \neq 1 indicates non-ideality (with details on Z explored further in subsequent sections). At constant temperature, plots of PV versus P for an ideal gas yield a straight horizontal line, as PV is independent of pressure. In contrast, real gas plots show characteristic curves: at moderate pressures, PV often dips below the ideal line (indicating Z < 1), while at higher pressures, it rises above ( Z > 1 ), reflecting how actual gas volumes and pressures diverge from predictions. These deviations become pronounced as pressure increases or temperature decreases, but approach ideal behavior in the limits of low pressure and high temperature, where Z \to 1.^[3]^[4] Experimental evidence from Boyle's law investigations, which test the inverse pressure-volume relationship at constant temperature, confirms these deviations at elevated pressures. For instance, measurements on air at 300 K show Z \approx 0.993 at 100 bar (about 98.7 atm), a minor deviation of less than 1%, but Z rises to 1.033 at 200 bar and 1.067 at 250 bar, exceeding 5% deviation above approximately 250 bar (246 atm). Such data, derived from precise pressure-volume measurements, highlight the practical limits of the ideal gas approximation in high-pressure scenarios like compressed air systems.^[8]

Causes of Non-Ideal Behavior

Finite Molecular Volume

In real gases, the molecules possess a finite, non-zero volume, which reduces the effective space available for molecular motion compared to the total container volume. This effect becomes prominent at high densities or pressures, where the molecular volume is no longer negligible relative to the overall volume. As a result, the observed pressure exceeds that predicted by the ideal gas law, leading to PV > nRT.^[9] This deviation arises from the repulsive interactions due to the physical size of the molecules, which prevent them from occupying the same space. In the hard-sphere model, gas molecules are idealized as rigid, impenetrable spheres that collide elastically but cannot overlap. The excluded volume—the additional space around each molecule that other molecules cannot access—accounts for this repulsion. In the context of corrections to the ideal gas law, this excluded volume per mole is denoted by the parameter b, representing approximately four times the actual molecular volume for a system of hard spheres. Quantitatively, at high pressures approaching close packing conditions, where molecules are nearly in contact, the finite volume significantly alters the pressure-volume relationship. For instance, in helium, the excluded volume parameter b \approx 0.0238 L/mol, indicating that the correction becomes relevant when the molar volume approaches this value, enhancing the pressure by limiting compressibility. This term establishes a minimum volume limit, beyond which further compression is resisted primarily by molecular repulsion.^[10] The recognition of finite molecular volume as a key factor in non-ideal behavior was first systematically proposed by Johannes Diderik van der Waals in his 1873 doctoral thesis, where he introduced it as one essential modification to the ideal gas law to better describe real gas properties.

Intermolecular Forces

In real gases, intermolecular forces manifest primarily as van der Waals attractions, encompassing London dispersion forces present in all molecules due to temporary induced dipoles, dipole-dipole interactions between polar molecules, and hydrogen bonding in gases like water vapor where hydrogen is bonded to highly electronegative atoms. These long-range forces draw molecules inward toward one another, influencing the overall behavior away from ideality.^[11] These attractive forces reduce the pressure exerted on container walls because molecules approaching the surface are pulled back by neighboring molecules, resulting in collisions with lower momentum than predicted by the ideal gas law, where PV < nRT at moderate pressures. This pressure-lowering effect arises from the cumulative impact of attractions slowing molecular motion near boundaries.^[12] The influence of these forces exhibits strong temperature dependence: at high temperatures, molecular kinetic energy overwhelms the attractions, rendering them negligible and allowing gas behavior to approach ideality, whereas at lower temperatures, the forces dominate, facilitating phenomena like liquefaction. For instance, carbon dioxide demonstrates this through its ability to liquefy under moderate pressure at around 273 K, where intermolecular attractions enable phase transition as kinetic energy diminishes.^[13] To quantify interaction strength, models like the Lennard-Jones potential are employed, which characterize the potential energy between non-bonding neutral atoms or molecules as a function of separation distance, balancing short-range repulsion with long-range attraction dominated by dispersion forces. This model, introduced by John E. Lennard-Jones in 1924, provides a foundational framework for simulating real gas deviations.^[14]

Key Thermodynamic Properties

Compressibility Factor

The compressibility factor, denoted as Z, is a dimensionless quantity that characterizes the extent to which a real gas deviates from ideal gas behavior. It is defined by the relation

Z = \frac{PV}{nRT},

where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature. For an ideal gas, Z = 1 at all conditions, but real gases exhibit Z \neq 1 owing to finite molecular volume and intermolecular forces; specifically, Z > 1 when repulsive interactions predominate, and Z < 1 when attractive interactions are more significant.^[15] The compressibility factor can be reformulated as a function of temperature and molar density \rho = n/V, expressed as Z = Z(T, \rho). This representation arises from substituting \rho = P / (Z R T) into the definition, emphasizing that deviations depend on the gas's density and thermal state rather than pressure alone. At low densities (\rho \to 0), Z \to 1, reflecting ideal behavior, while higher densities amplify non-ideal effects through increased molecular interactions. This functional form facilitates comparisons across conditions and underpins thermodynamic modeling without relying on pressure explicitly.^[15] Generalized compressibility charts provide a graphical means to visualize and estimate Z, plotting it against reduced pressure P_r = P / P_c and reduced temperature T_r = T / T_c, with P_c and T_c as the critical pressure and temperature. These empirical charts, based on extensive PVT measurements for diverse gases, reveal isotherms where Z typically falls below 1 at low to moderate P_r for T_r > 2, then rises above 1 at high P_r due to volume exclusion effects. They enable practical predictions via the principle of corresponding states, applicable to non-polar gases like nitrogen with reasonable accuracy.^[16] Experimentally, Z is determined from precise PVT measurements, using Z = P / (\rho R T) where density \rho is obtained from volume assessments under controlled pressure and temperature. For nitrogen at 300 K, PVT data indicate Z initially near 1 at low pressures, remaining close to 1 up to moderate pressures (e.g., ≈0.997 at 10 atm), dipping to a minimum of approximately 0.8 around 100 atm as attractive forces dominate, before climbing above 1 at higher pressures where repulsions prevail.^[17]

Joule-Thomson Effect

The Joule-Thomson effect describes the temperature change experienced by a real gas during an isenthalpic expansion process, such as throttling through a porous plug or valve, where no heat is exchanged with the surroundings and no work is performed. This phenomenon arises because real gases deviate from ideal behavior due to intermolecular interactions, leading to either cooling or heating depending on the conditions. The effect was first systematically investigated through experiments conducted by James Prescott Joule and William Thomson (later Lord Kelvin) starting in 1852, using an apparatus that allowed high-pressure gas to flow through a constricted tube or porous barrier into a lower-pressure region, with temperature measured before and after the expansion. Their collaborative work, spanning from 1852 to 1862, produced extensive data on temperature changes for various gases, establishing foundational tables of experimental results that highlighted non-zero temperature shifts unlike those predicted for ideal gases. The Joule-Thomson coefficient, denoted as \mu_{JT}, quantifies this temperature change and is defined as \mu_{JT} = \left( \frac{\partial T}{\partial P} \right)_H, representing the rate of temperature variation with pressure at constant enthalpy. For most real gases at room temperature and moderate pressures, \mu_{JT} is positive, resulting in cooling upon expansion; however, its sign can change based on temperature and pressure. The inversion temperature is the specific condition where \mu_{JT} = 0, marking the boundary between cooling (positive \mu_{JT}) and heating (negative \mu_{JT}); above the upper inversion temperature, expansion causes heating, while below it but above the lower inversion curve, cooling occurs. For nitrogen, the maximum (zero-pressure) inversion temperature is 621 K, allowing effective cooling at ambient conditions for applications like air separation. Physically, the Joule-Thomson effect stems from the interplay of attractive and repulsive intermolecular forces during expansion. Attractive forces (modeled by terms like the van der Waals a parameter) pull molecules together, performing work that reduces the gas's kinetic energy and thus lowers temperature, promoting cooling. Repulsive forces (related to the excluded volume b parameter), dominant at closer molecular distances, push molecules apart, decreasing potential energy and increasing kinetic energy, which can lead to heating. The net effect depends on which force predominates, with the inversion point occurring when these contributions balance exactly. This effect is pivotal in practical applications, including refrigeration cycles and the liquefaction of gases, where repeated throttling stages exploit cooling to reach cryogenic temperatures for industrial processes like producing liquid nitrogen or oxygen. Historical and modern data tables, derived from Joule and Thomson's experiments and subsequent measurements, provide \mu_{JT} values for gases such as nitrogen (approximately 0.27 K/atm at 300 K and 1 atm), oxygen, and carbon dioxide, enabling precise thermodynamic predictions in engineering designs.

Equations of State

Van der Waals Equation

The van der Waals equation of state, proposed by Johannes Diderik van der Waals in his 1873 doctoral thesis, modifies the ideal gas law to account for the finite size of gas molecules and intermolecular attractive forces. It is expressed for one mole of gas as

\left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT,

where P is the pressure, V_m is the molar volume, R is the gas constant, T is the temperature, a is the attraction parameter, and b is the volume exclusion parameter.^[18]^[19] The derivation begins with the ideal gas law PV_m = RT, but introduces corrections for real gas behavior. For the pressure term, the measured pressure P is lower than the ideal pressure because molecules near the container walls are pulled back by attractive forces from other molecules in the bulk gas. This attractive force is proportional to the square of the molecular density, leading to a correction where the effective pressure is P + \frac{a}{V_m^2}, with the term \frac{a}{V_m^2} representing the mean-field contribution of these attractions.^[19] The volume correction addresses the finite size of molecules, which occupy space and reduce the effective free volume available for molecular motion. In the ideal gas law, V_m assumes point particles, but for real gases, the actual volume for movement is V_m - b, where b accounts for the excluded volume per mole due to molecular repulsion at close range. Combining both corrections yields the full van der Waals equation \left(P + \frac{a}{V_m^2}\right)(V_m - b) = RT.^[19] The parameters a and b are gas-specific constants determined experimentally, often from critical point data or direct measurements. The parameter a (in units of L² atm mol⁻²) quantifies the strength of intermolecular attractions, increasing with molecular polarity and size, while b (in L mol⁻¹) represents the effective volume occupied by one mole of molecules, roughly four times the actual molecular volume assuming spherical particles. For carbon dioxide, a = 3.658 L² atm mol⁻² and b = 0.0429 L mol⁻¹.^[20] At the critical point, where the distinction between liquid and gas phases vanishes, the van der Waals equation predicts an inflection point in the isotherm (\frac{\partial P}{\partial V_m} = 0 and \frac{\partial^2 P}{\partial V_m^2} = 0). Solving these conditions gives the critical molar volume V_c = 3b, critical pressure P_c = \frac{a}{27b^2}, and critical temperature T_c = \frac{8a}{27Rb}. These relations allow estimation of a and b from experimentally measured critical constants.^[21] A key application of the van der Waals equation is predicting gas liquefaction. Below the critical temperature, the isotherms exhibit a non-physical loop, indicating instability; the horizontal portion of the corrected isotherm corresponds to the coexistence of liquid and vapor phases, enabling calculation of liquefaction conditions under varying pressures. This qualitative feature explained the continuity between gaseous and liquid states, a central insight in van der Waals' work.^[19] For example, consider one mole of CO₂ at 273 K and a molar volume of 22.4 L (approximating standard conditions). Using the ideal gas law, P = \frac{RT}{V_m} = 1 atm. With the van der Waals equation and CO₂ constants (a = 3.658 L² atm mol⁻², b = 0.0429 L mol⁻¹, R = 0.0821 L atm mol⁻¹ K⁻¹), the corrected pressure is P = \frac{RT}{V_m - b} - \frac{a}{V_m^2} \approx 0.995 atm, showing a small deviation due to attractions dominating at low density. At higher density, such as V_m = 0.05 L, the ideal law gives P \approx 448 atm, but the van der Waals equation yields P \approx 1620 atm, highlighting the volume exclusion effect.^[22]^[20] Despite its foundational role, the van der Waals equation has limitations, particularly at high densities where it fails to accurately capture repulsive interactions beyond simple exclusion, sometimes predicting unphysical negative pressures or volumes below b. It performs well near the critical point for qualitative predictions but deviates significantly from experimental data at extreme pressures or low temperatures, necessitating more advanced models for precise quantitative work.^[23]

Redlich-Kwong Equation

The Redlich–Kwong equation of state, proposed in 1949 by Otto Redlich and J. N. S. Kwong, improves upon the van der Waals equation by introducing temperature dependence in the attractive parameter to enhance accuracy for real gas behavior, particularly in vapor-liquid equilibria at elevated temperatures. This empirical model retains the cubic form while addressing shortcomings in high-temperature predictions through a modified attractive term.^[24] The equation is expressed as

P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T} \, V_m (V_m + b)},

where P is pressure, T is temperature, V_m is molar volume, R is the universal gas constant, and a and b are substance-specific parameters accounting for intermolecular attractions and the finite volume of molecules, respectively. The key modification lies in the temperature-dependent attractive term a / \sqrt{T}, which replaces the constant a in the van der Waals equation, yielding better representation of decreasing attractive forces with increasing temperature. This adjustment stems from empirical fitting to experimental data, ensuring improved fidelity for gases above their critical temperatures across wide pressure ranges.^[24] The parameters are derived from the critical temperature T_c and critical pressure P_c to satisfy critical point conditions:

a = 0.42748 \frac{R^2 T_c^{2.5}}{P_c}, \quad b = 0.08664 \frac{R T_c}{P_c}.

These expressions ensure the equation reproduces the critical compressibility factor and other critical properties with reasonable accuracy.^[24] The Redlich–Kwong equation demonstrates superior performance for non-polar hydrocarbons, providing reliable predictions of phase behavior and thermodynamic properties compared to earlier models. For instance, it accurately captures the vapor-liquid equilibrium in the phase diagram of propane, aligning closely with experimental saturation curves over a broad range of conditions.^[24]^[25]

Peng-Robinson Equation

The Peng-Robinson equation of state, introduced in 1976 by Ding-Yu Peng and Donald B. Robinson, refines the Redlich-Kwong equation by modifying the repulsion term and incorporating temperature-dependent functionality to yield more accurate predictions of liquid densities and vapor pressures near the critical point.^[26] The equation takes the form

P = \frac{RT}{V_m - b} - \frac{a \alpha}{V_m(V_m + b) + b(V_m - b)},

where P is pressure, T is temperature, V_m is molar volume, R is the gas constant, and a, b, and \alpha are substance-specific parameters.^[26] These parameters are expressed in terms of critical properties as follows:

a = 0.45724 \frac{R^2 T_c^2}{P_c},

b = 0.07780 \frac{R T_c}{P_c},

\alpha = \left[1 + m \left(1 - \sqrt{T_r}\right)\right]^2,

with m = 0.37464 + 1.54226\omega - 0.26992\omega^2 and T_r = T/T_c; here, T_c and P_c denote the critical temperature and pressure, while \omega is the acentric factor that adjusts for molecular shape deviations from sphericity.^[26] This structure enhances the original Redlich-Kwong model's handling of attractive forces, particularly for improved liquid-phase behavior.^[26] In chemical engineering applications, the Peng-Robinson equation is extensively employed in process simulation software such as Aspen Plus and HYSYS for vapor-liquid equilibrium calculations, adiabatic flash operations, and high-pressure chemical equilibria in systems like ammonia synthesis.^[27] It demonstrates particular superiority for non-polar and mildly polar compounds, including hydrocarbons, enabling reliable phase behavior predictions in natural gas processing and petrochemical separations across wide ranges of temperature and pressure.^[26]^[27]

Virial Expansion

The virial expansion represents the equation of state for real gases as a power series in density, offering a systematic way to account for non-ideal behavior at low densities where intermolecular interactions are weak. The compressibility factor Z = \frac{P V_m}{R T}, which measures deviations from ideal gas behavior, is expressed as

Z = 1 + \frac{B(T)}{V_m} + \frac{C(T)}{V_m^2} + \frac{D(T)}{V_m^3} + \cdots,

where V_m is the molar volume, and B(T), C(T), D(T), and higher-order terms are the virial coefficients that depend solely on temperature./02%3A_Gas_Laws/2.13%3A_Virial_Equations) The second virial coefficient B(T) captures pairwise interactions, the third C(T) includes three-body effects, and so on; for dilute gases, the series can often be truncated after the second or third term with good accuracy.^[28] From statistical mechanics, the virial coefficients derive from the cluster expansion of the partition function, providing a direct link to the underlying intermolecular potential. Specifically, the second virial coefficient arises from two-body interactions and is given by

B(T) = -2\pi N_A \int_0^\infty r^2 \left( e^{-u(r)/kT} - 1 \right) \, dr,

where N_A is Avogadro's number, u(r) is the pairwise intermolecular potential energy as a function of separation r, k is Boltzmann's constant, and T is temperature.^[28] This integral reflects the Mayer f-function f(r) = e^{-u(r)/kT} - 1, which is negative for repulsive forces (short-range hard-core effects) and positive for attractive forces (longer-range van der Waals interactions). The temperature dependence of B(T) is pronounced: at high temperatures, thermal energy overwhelms attractions, making B(T) > 0 due to excluded volume from repulsions; at lower temperatures, attractions dominate, yielding B(T) < 0. The temperature where B(T) = 0, known as the Boyle temperature, marks the transition between these regimes.^[29] A key advantage of the virial expansion lies in its firm theoretical foundation in statistical mechanics, enabling predictions from quantum or classical models of the potential u(r), such as the Lennard-Jones potential, without empirical fitting for low-density conditions.^[30] Unlike cubic equations of state, it excels in the dilute limit by allowing truncation of higher virials, which become negligible as density decreases, thus providing precise insights into interaction potentials. For practical use, virial coefficients are often determined experimentally via precise measurements of pressure-volume-temperature relations at low pressures, where the second virial dominates.^[31] As an example, for argon, experimental data reveal that B(T) is negative below its Boyle temperature of approximately 409 K, reflecting the prevalence of attractive intermolecular forces at moderate temperatures like room temperature (around 300 K).^[32] Measurements between 80 K and 125 K confirm this negative behavior, with values about 5–15% larger in magnitude than predictions from simple 12-6 Lennard-Jones potentials, highlighting the need for refined potential models.^[31] At room temperature, B for argon is approximately -21.7 cm³/mol, underscoring attractive deviations from ideality.^[33]

Other Empirical Models

The Dieterici equation of state, proposed in 1899, modifies the ideal gas law to account for molecular volume and attractive forces through the form P e^{-a / ([R](/page/R) T [V_m](/page/Molar_volume))} (V_m - b) = [R](/page/R) T, where V_m is the molar volume, a and b are substance-specific constants, [R](/page/R) is the gas constant, and T is temperature; it predicts a critical compressibility factor of approximately 0.27, which is closer to the experimental value for many gases (around 0.27) than the van der Waals equation's 0.375.^[34]^[35]^[36] The Beattie-Bridgeman equation, introduced in 1927, is a five-parameter empirical expansion suitable for describing the behavior of gases such as air and helium at moderate pressures up to about two-thirds of the critical density, incorporating terms for both repulsive and attractive intermolecular interactions beyond simple cubic forms.^[34]^[37] The Benedict-Webb-Rubin equation, developed in 1940, employs an eighth-order polynomial in volume to model the thermodynamic properties of light hydrocarbons and natural gas mixtures, with parameters fitted directly from experimental pressure-volume-temperature data; it finds particular application in liquefied natural gas (LNG) processes for accurate phase equilibrium predictions.^[38]^[34] Other notable empirical models include the Berthelot equation, which resembles the van der Waals form but features a temperature-dependent attractive term a/T to better capture low-temperature deviations in gases like carbon dioxide./06%3A_Properties_of_Gases/6.03%3A_Van_der_Waals_and_Other_Gases) The Clausius equation represents an early modification of the van der Waals model by introducing temperature dependence in the attractive correction, aiding predictions for vapors near saturation. For mixtures, the Wohl equation extends empirical fitting to account for non-ideal interactions in ternary vapor-liquid equilibria, such as hydrocarbon blends.^[39]

Applications and Thermodynamic Calculations

Expansion Work

In thermodynamics, the work associated with the expansion of a real gas is fundamentally given by the integral W = \int_{V_1}^{V_2} P \, dV for a reversible process, where P is determined from an appropriate equation of state rather than the ideal gas law PV = nRT. This contrasts with ideal gases, where the pressure-volume relationship simplifies the integration. For real gases, intermolecular forces and finite molecular volume introduce deviations, requiring explicit integration using models like the van der Waals equation \left( P + \frac{an^2}{V^2} \right) (V - nb) = nRT, solved for P = \frac{nRT}{V - nb} - \frac{an^2}{V^2}.^[40]^[41] For isothermal expansion at constant temperature T, the work calculation incorporates these deviations directly through the equation of state. Substituting the van der Waals form yields W = nRT \ln \left( \frac{V_2 - nb}{V_1 - nb} \right) + an^2 \left( \frac{1}{V_1} - \frac{1}{V_2} \right). The first term modifies the ideal gas expression nRT \ln (V_2 / V_1) to account for excluded volume via b, while the second term arises from attractive forces parameterized by a. Unlike ideal gases, where internal energy U depends only on temperature (\Delta U = 0 for isothermal processes), real gases exhibit \Delta U \neq 0 due to volume-dependent interactions. This follows from the thermodynamic identity \left( \frac{\partial U}{\partial V} \right)_T = T \left( \frac{\partial P}{\partial T} \right)_V - P, which equals zero for ideal gases but for van der Waals gases simplifies to \frac{an^2}{V^2}, leading to \Delta U = -an^2 \left( \frac{1}{V_2} - \frac{1}{V_1} \right) via integration at constant T. Departure functions quantify these deviations: the internal energy departure (U - U^\text{ideal})_T = \int_{V}^\infty \left[ T \left( \frac{\partial P}{\partial T} \right)_V - P \right] dV', enabling computation of \Delta U between states using ideal gas values plus corrections.^[42]^[40]^[41] In adiabatic expansion, where no heat is exchanged (Q = 0), the first law gives \Delta U = -W (with W as work done by the system). For real gases, this requires solving coupled relations since temperature varies and C_V may depend on state, but assuming constant C_V, the van der Waals model yields T (V - nb)^{\gamma - 1} = \text{constant}, with \gamma = C_P / C_V = 1 + [R](/page/R) / C_V. Then, \Delta U = \int_{T_1}^{T_2} C_V \, dT - an^2 \left( \frac{1}{V_2} - \frac{1}{V_1} \right), where the first integral is the ideal contribution and the second is the real gas correction, so W = -\Delta U. This differs from ideal gases' \Delta U = C_V \Delta T and PV^\gamma = \text{constant}, as non-zero (\partial U / \partial V)_T introduces extra terms affecting efficiency in applications like engines. For instance, in supercritical CO₂ Brayton cycle engines, where CO₂ expands adiabatically in a turbine from high-pressure states near the critical point (using van der Waals constants a = 3.640 \, \text{L}^2 \cdot \text{bar} \cdot \text{mol}^{-2}, b = 0.0427 \, \text{L} \cdot \text{mol}^{-1}), real gas effects must be accounted for to accurately predict work output and enhance cycle efficiency for power generation.^[41]^[42]

Phase Behavior and Critical Phenomena

The critical point of a real gas marks the temperature T_c, pressure P_c, and volume V_c at which the distinction between the liquid and gaseous phases disappears, resulting in a single supercritical fluid phase with no distinct interface. At this point, the compressibility factor Z_c = P_c V_c / (R T_c) approaches approximately 0.3 for many real gases, reflecting significant deviations from ideal behavior.^[43] Real gas equations of state, such as the van der Waals model, predict specific values like Z_c = 3/8 at the critical point, providing a theoretical benchmark for phase vanishing. Phase diagrams for real gases illustrate the vapor pressure curve separating liquid and vapor regions, culminating at the critical point, with the binodal line defining coexistence boundaries. The Maxwell construction ensures thermodynamic consistency in these diagrams by applying the equal-area rule to isotherms below T_c, where the areas above and below the coexistence pressure on a pressure-volume plot are equal, representing the stable two-phase equilibrium.^[44] This construction resolves the unphysical oscillations in model isotherms, accurately depicting the flat coexistence plateau. A notable phenomenon in real gas phase behavior is retrograde condensation, observed in multicomponent gas mixtures like natural gas condensates, where liquid droplets form upon initial pressure reduction below the dew point, but further pressure decrease causes the liquid to re-vaporize, counter to typical condensation trends.^[45] This occurs due to the varying solubility of components at high pressures, impacting reservoir production.^[46] In 1873, Johannes Diderik van der Waals demonstrated the continuity of liquid and gaseous states through his equation, theoretically bridging the phases without abrupt transitions and laying the foundation for understanding critical phenomena.^[47] Applications of real gas phase behavior leverage supercritical fluids, such as carbon dioxide above its critical point at 31.1°C and 73.8 bar, for solvent-free extraction processes in industries like food decaffeination and pharmaceutical purification, exploiting tunable density and diffusivity.^[48] The law of corresponding states uses reduced variables—T_r = T / T_c, P_r = P / P_c, V_r = V / V_c—to reveal universal phase behavior across different gases, enabling predictive scaling from one substance to another without extensive experimentation.

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Non-ideal behavior of gases (article) | Khan Academy
Compressibility: A measure of ideal behavior One way we can look at how accurately the ideal gas law describes our system is by comparing the molar volume of ...<|separator|>
[12]
Non-ideal gas - Van der Waal's Equation and Constants
The constants a and b are called van der Waals constants. They have positive values and are characteristic of the individual gas. If a gas behaves ideally, ...
[13]
Van der Waals Forces - Chemistry LibreTexts
Jan 29, 2023 · There are two kinds of Van der Waals forces: weak London Dispersion Forces and stronger dipole-dipole forces.Introduction · Van der Waals Equation
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5.10: Real Gases- The Effects of Size and Intermolecular Forces
Aug 14, 2020 · Deviations from ideal gas law behavior can be described by the van der Waals equation, which includes empirical constants to correct for the ...
[15]
Lennard-Jones Potential - Chemistry LibreTexts
Feb 18, 2025 · The Lennard-Jones potential describes the potential energy of interaction between two non-bonding atoms or molecules based on their distance of separation.
[16]
3.2 Real gas and compressibility factor
The compressibility factor is a dimensionless correction factor to account for the deviation of the real gas behaviour from the “ideal” gas model.
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Compressibility Chart - an overview | ScienceDirect Topics
Compressibility charts are graphical tools to determine the compressibility factor (Z) of gases, which accounts for deviations from ideal gas behavior.Missing: 300K | Show results with:300K
[18]
[PDF] The thermodynamic properties of nitrogen from 64 to 300* K ...
... Compressibility factor, . RT. -. Internal energy. -. Enthalpy. -. Entropy. -. Degree Kelvin. -. Degree Celsius. -. Tenaperature at saturation at 1 atm (77. 364* ...Missing: 300K | Show results with:300K
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Taking Another Look at the van der Waals Equation of State–Almost ...
Aug 2, 2019 · Van der Waals, J. D. On the Continuity of the Gaseous and Liquid States; Leiden, 1873. Google Scholar. There is no corresponding record for this ...Excess Properties from the van... · Mixing and Combining Rules... · References
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2.12: Van der Waals' Equation - Chemistry LibreTexts
Jul 7, 2024 · The pressure and volume appearing in van der Waals' equation are the pressure and volume of the real gas. We can relate the terms in van der ...
[21]
A8: van der Waal's Constants for Real Gases - Chemistry LibreTexts
Nov 13, 2024 · A8: van der Waal's Constants for Real Gases ; GeH · Germane, 5.743 ; He, Helium, 0.0346 ; HBr, Hydrogen bromide, 4.500 ; HCl, Hydrogen chloride ...Missing: source | Show results with:source
[22]
[PDF] Critical Constants of the van der Waals Gas
Jul 8, 2004 · One can test to see if an approximate equation of state gives a critical point by calculating these two derivatives for the equation of state ...
[23]
[PDF] Midterm Exam Problem 10 Example of using van der Waals ...
Nov 25, 2013 · Example of using van der Waals equation P = RT/(V-b) - a/V2 for n = 1 mole. At T = 273K, applying the ideal gas law to 1 mole of CO2 in V ...
[24]
The van der Waals Equation | PNG 520
One of the first things vdW recognized is that molecules must have a finite volume, and that volume must be subtracted from the volume of the container. At the ...
[25]
On the Thermodynamics of Solutions. V. An Equation of State ...
A classical thermodynamic model for dispersed nanophase stability and its application for investigating the stability of air nanobubbles in water.
[26]
Vapor-liquid equilibrium relations of binary systems propane-n ...
Application of L H C modification of Redlich-Kwong equation of state for saturated binary liquid mixtures. Chemical Engineering Science 1985, 40 (10) ...
[27]
A New Two-Constant Equation of State | Industrial & Engineering ...
Traditional Fossil Fuels February 1, 1976. A New Two-Constant Equation ... Peng–Robinson Equation of State. Industrial & Engineering Chemistry Research ...
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None
### Summary of Applications of Peng-Robinson EOS in Simulations and Advantages
[29]
[PDF] Virial Expansion – A Brief Introduction - The Schreiber Group
Note that. P = P(T,ρ) is entirely invertible in both variables; at every (P, T), a single phase with density ρ = P/kT is predicted. This EOS can be verified ...
[30]
[PDF] Statistical Mechanics I - MIT OpenCourseWare
(a) Calculate the second virial coefficient B2(T ), and comment on its high and low tem perature behaviors. The second virial coefficient is obtained from.
[31]
Virial Equation - an overview | ScienceDirect Topics
The virial equation of state is a polynomial series in the density, and is explicit in pressure and can be derived from statistical mechanics.<|separator|>
[32]
Second Virial Coefficients of Argon, Krypton, and ... - AIP Publishing
Second virial coefficients of argon between 80–125°K, and krypton (105–140°K), have been measured. The experimental values are 5–15% larger in magnitude ...Missing: dependence | Show results with:dependence
[33]
Isotherms for the rigidity of fluid argon from the Boyle temperature...
Isotherms for the rigidity of fluid argon from the Boyle temperature (409 K) just above red isotherm at 400 K to the critical temperature (151 K) purple ...
[34]
[PDF] Measurements of virial coefficients of Helium, Argon and Nitrogen ...
At room temperature, the second virial coefficient is positive for helium and negative for argon and nitrogen. (Figure 3). Figure 2. The compressibility ...
[35]
[PDF] DEVELOPMENT OF AN EQUATION OF STATE FOR GASES
More than a hundred equations of state relating the pressure, vol- ume, and temperature of gases have been proposed according to.
[36]
[PDF] Precise Numerical Differentiation of Thermodynamic Functions with ...
Nov 29, 2021 · Such equations of state for fluids do indeed exist: • The equation of state of Dieterici [1] of 1899 was published in a pressure-explicit ...
[37]
Generalized Beattie-Bridgeman Equation of State for Real Gases1
This article is cited by 12 publications. Manuel F. Pérez-Polo, Manuel Pérez-Molina, Elena Fernández Varó, Javier Gil Chica. Analysis of the state equations ...Missing: seminal | Show results with:seminal
[38]
Predicting Phase and Thermodynamic Properties of Natural Gases ...
The Benedict-Webb-Rubin equation of state was used in digital computer programs to make rapid determinations of natural gas equilibrium phase compositions.Missing: parameters experiments
[39]
Application of the Wohl equation to ternary liquid-vapor equilibria
The use of the Wohl equation for activity coefficients is tested successfully for the most nonideal ternary systems for which experimental data could be ...
[40]
Work of van der Waals Gas - Collection of Solved Problems in Physics
Sep 18, 2020 · The expansion is isothermal, the temperature T is therefore constant. We determine it using initial volume V1 and pressure p1. We obtain: T=(p1+ ...
[41]
[PDF] Van der Waal's gas equation for an adiabatic process and its Carnot ...
Nov 8, 2017 · The above result is obtained by using the ideal gas equation of state for the isothermal process PV = constant and the ideal gas equation for ...
[42]
[PDF] Thermodynamics of Real Gases
Our task now boils down to evaluating the constants V0 and ueff,0. Let us start by the excluded volume, V0. The minimum allowed distance between two.Missing: hard primary source
[43]
A Modified Form of the van der Waals Equation of State
Nov 6, 2014 · ... 3/8 of the van der Waals fluid. Assuming an average value of the critical compressibility factor barZ_c=0.275, the predicted generalized ...<|control11|><|separator|>
[44]
[PDF] 5. Phase Transitions - DAMTP
This condition, known as the Maxwell construction, tells us the pressure at which gas and liquid can co-exist. I should confess that there's something slightly ...
[45]
retrograde condensation - Energy Glossary - SLB
Retrograde condensation is the formation of liquid hydrocarbons in a gas reservoir when pressure decreases below dewpoint, some gas condensing into liquid ...
[46]
Retrograde Phenomenon | PNG 520: Phase Behavior of Natural ...
Retrograde phenomenon is when, during compression, liquid vaporizes instead of condensing, contrary to normal behavior, and is typical of gas condensate ...
[47]
The Continuity of the Liquid and Gaseous States of Matter
Title: The Continuity of the Liquid and Gaseous States of Matter ; Author: Waals, J. D. van der (Johannes Diderik), 1837-1923 ; Translator: Threlfall, Richard.Missing: 1887 | Show results with:1887
[48]
Carbon Dioxide Supercritical Extraction - ScienceDirect.com
Carbon dioxide at temperature and pressure above supercritical values (T > 31.1 °C, P > 7.38 MPa) exhibits increased transport properties and ability to extract ...