Real gas
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Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
. To understand the behaviour of real gases, the following must be taken into account: *
compressibility In thermodynamics and fluid mechanics, the compressibility (also known as the coefficient of compressibility or, if the temperature is held constant, the isothermal compressibility) is a measure of the instantaneous relative volume change of a fl ...
effects; *variable
specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
; *
van der Waals force In molecular physics, the van der Waals force is a distance-dependent interaction between atoms or molecules. Unlike ionic or covalent bonds, these attractions do not result from a chemical electronic bond; they are comparatively weak and th ...
s; *non-equilibrium thermodynamic effects; *issues with molecular dissociation and elementary reactions with variable composition For most applications, such a detailed analysis is unnecessary, and the
ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...
approximation can be used with reasonable accuracy. On the other hand, real-gas models have to be used near the
condensation Condensation is the change of the state of matter from the gas phase into the liquid phase, and is the reverse of vaporization. The word most often refers to the water cycle. It can also be defined as the change in the state of water vapor to ...
point of gases, near critical points, at very high pressures, to explain the
Joule–Thomson effect In thermodynamics, the Joule–Thomson effect (also known as the Joule–Kelvin effect or Kelvin–Joule effect) describes the temperature change of a ''real'' gas or liquid (as differentiated from an ideal gas) when it is forced through a valv ...
, and in other less usual cases. The deviation from ideality can be described by the
compressibility factor In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to ...
Z.


Models


Van der Waals model

Real gases are often modeled by taking into account their molar weight and molar volume :RT = \left(p + \frac\right)\left(V_\text - b\right) or alternatively: :p = \frac - \frac Where ''p'' is the pressure, ''T'' is the temperature, ''R'' the ideal gas constant, and ''V''m the
molar volume In chemistry and related fields, the molar volume, symbol ''V''m, or \tilde V of a substance is the ratio of the volume occupied by a substance to the amount of substance, usually given at a given temperature and pressure. It is equal to the molar ...
. ''a'' and ''b'' are parameters that are determined empirically for each gas, but are sometimes estimated from their
critical temperature Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine *Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing in ...
(''T''c) and
critical pressure In thermodynamics, a critical point (or critical state) is the end point of a phase equilibrium curve. The most prominent example is the liquid–vapor critical point, the end point of the pressure–temperature curve that designates conditions ...
(''p''c) using these relations: :\begin a &= \frac \\ b &= \frac \end The constants at critical point can be expressed as functions of the parameters a, b: : p_c=\frac, \quad T_c=\frac, \qquad V_=3b, \qquad Z_c=\frac With the
reduced properties In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility facto ...
p_r = \frac,\ V_r = \frac,\ T_r = \frac\ the equation can be written in the ''reduced form'': :p_r = \frac\frac - \frac


Redlich–Kwong model

The Redlich–Kwong equation is another two-parameter equation that is used to model real gases. It is almost always more accurate than the
van der Waals equation In chemistry and thermodynamics, the Van der Waals equation (or Van der Waals equation of state) is an equation of state which extends the ideal gas law to include the effects of interaction between molecules of a gas, as well as accounting for ...
, and often more accurate than some equations with more than two parameters. The equation is :RT = \left(p + \frac\right)\left(V_\text - b\right) or alternatively: :p = \frac - \frac where ''a'' and ''b'' are two empirical parameters that are not the same parameters as in the van der Waals equation. These parameters can be determined: :\begin a &= 0.42748\, \frac \\ b &= 0.08664\, \frac \end The constants at critical point can be expressed as functions of the parameters a, b: : p_c=\fracR^\frac, \quad T_c=3^ (\sqrt 1)^ (\frac)^, \qquad V_=\frac, \qquad Z_c=\frac Using \ p_r = \frac,\ V_r = \frac,\ T_r = \frac\ the equation of state can be written in the ''reduced form'': : p_r = \frac - \frac with b' = \sqrt - 1 \approx 0.26


Berthelot and modified Berthelot model

The Berthelot equation (named after D. Berthelot) is very rarely used, :p = \frac - \frac but the modified version is somewhat more accurate :p = \frac\left + \frac \left(1 - \frac\right)\right/math>


Dieterici model

This model (named after C. Dieterici) fell out of usage in recent years :p = \frac \exp\left(-\frac\right) with parameters a, b, and : \exp\left(-\frac\right) = e^ = 1 - \frac + \dots


Clausius model

The Clausius equation (named after
Rudolf Clausius Rudolf Julius Emanuel Clausius (; 2 January 1822 – 24 August 1888) was a German physicist and mathematician and is considered one of the central founding fathers of the science of thermodynamics. By his restatement of Sadi Carnot's principle ...
) is a very simple three-parameter equation used to model gases. :RT = \left(p + \frac\right)\left(V_\text - b\right) or alternatively: :p = \frac - \frac where :\begin a &= \frac \\ b &= V_\text - \frac \\ c &= \frac - V_\text \end where ''V''c is critical volume.


Virial model

The Virial equation derives from a perturbative treatment of statistical mechanics. :pV_\text = RT\left + \frac + \frac + \frac + \ldots\right/math> or alternatively :pV_\text = RT\left + B'(T)p + C'(T)p^2 + D'(T)p^3 \ldots\right/math> where ''A'', ''B'', ''C'', ''A''′, ''B''′, and ''C''′ are temperature dependent constants.


Peng–Robinson model

Peng–Robinson equation of state Cubic equations of state are a specific class of thermodynamic models for modeling the pressure of a gas as a function of temperature and density and which can be rewritten as a cubic function of the molar volume. Equations of state are generally ...
(named after D.-Y. Peng and D. B. Robinson) has the interesting property being useful in modeling some liquids as well as real gases. :p = \frac - \frac


Wohl model

The Wohl equation (named after A. Wohl) is formulated in terms of critical values, making it useful when real gas constants are not available, but it cannot be used for high densities, as for example the critical isotherm shows a drastic ''decrease'' of pressure when the volume is contracted beyond the critical volume. :p = \frac - \frac + \frac\quad or: : \left(p - \frac\right)\left(V_\text - b\right) = RT - \frac or, alternatively: :RT = \left(p + \frac - \frac\right)\left(V_\text - b\right) where :a = 6p_\text T_\text V_\text^2 :b = \frac with V_\text = \frac\frac :c = 4p_\text T_\text^2 V_\text^3\ , where V_\text,\ p_\text,\ T_\text are (respectively) the molar volume, the pressure and the temperature at the critical point. And with the
reduced properties In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the fluid's state properties at its critical point. These dimensionless thermodynamic coordinates, taken together with a substance's compressibility facto ...
\ p_r = \frac,\ V_r = \frac,\ T_r = \frac\ one can write the first equation in the ''reduced form'': :p_r = \frac\frac - \frac + \frac


Beattie–Bridgeman model

This equation is based on five experimentally determined constants. It is expressed as :p = \frac\left(1 - \frac\right)(v + B) - \frac where :\begin A &= A_0 \left(1 - \frac\right) & B &= B_0 \left(1 - \frac\right) \end This equation is known to be reasonably accurate for densities up to about 0.8 ''ρ''cr, where ''ρ''cr is the density of the substance at its critical point. The constants appearing in the above equation are available in the following table when ''p'' is in kPa, ''v'' is in \frac, ''T'' is in K and ''R'' = 8.314\fracGordan J. Van Wylen and Richard E. Sonntage, ''Fundamental of Classical Thermodynamics'', 3rd ed, New York, John Wiley & Sons, 1986 P46 table 3.3


Benedict–Webb–Rubin model

The BWR equation, sometimes referred to as the BWRS equation, :p = RTd + d^2\left(RT(B + bd) - \left(A + ad - a\alpha d^4\right) - \frac\left - cd\left(1 + \gamma d^2\right) \exp\left(-\gamma d^2\right)\rightright) where ''d'' is the molar density and where ''a'', ''b'', ''c'', ''A'', ''B'', ''C'', ''α'', and ''γ'' are empirical constants. Note that the ''γ'' constant is a derivative of constant ''α'' and therefore almost identical to 1.


Thermodynamic expansion work

The expansion work of the real gas is different than that of the ideal gas by the quantity \int_^ (\frac-P_)dV .


See also

*
Compressibility factor In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to ...
*
Equation of state In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
*
Ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
:
Boyle's law Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as: The ...
and
Gay-Lussac's law Gay-Lussac's law usually refers to Joseph-Louis Gay-Lussac's law of combining volumes of gases, discovered in 1808 and published in 1809. It sometimes refers to the proportionality of the volume of a gas to its absolute temperature at constant pr ...


References


Further reading

* * * * * *


External links

*http://www.ccl.net/cca/documents/dyoung/topics-orig/eq_state.html Gases