Real gases
Real gases–– as opposed to a perfect oras opposed to a perfect or ideal gasideal gas–– exhibit properties that exhibit properties that
cannot be explained entirely using the
cannot be explained entirely using the ideal gas lawideal gas law.. To understand theTo understand the behaviour of real gases, the following must be taken into account: behaviour of real gases, the following must be taken into account: compressib
compressibility ility effects;effects; variable
variable specific heat capacityspecific heat capacity;; van der Waals forces
van der Waals forces;; non-equilibr
non-equilibrium ium thermodynamithermodynamic c effects;effects; issues with molecular dissociation and
issues with molecular dissociation and elementary reactions with variableelementary reactions with variable composition.
composition.
For most applications, such a detailed analysis is unnecessary, and the ideal gas For most applications, such a detailed analysis is unnecessary, and the ideal gas approximation can be used with reasonable accuracy. On the other hand, approximation can be used with reasonable accuracy. On the other hand, real-gas models have to be used near the
gas models have to be used near the condensationcondensationpoint of gases, nearpoint of gases, near criticalcritical
points
points,, at very high pressures, and in other less usual cases.at very high pressures, and in other less usual cases.
van der Waals model van der Waals model
Main article
Main article:: van der Waals equationvan der Waals equation
Real gases are often modeled by taking into account their molar weight Real gases are often modeled by taking into account their molar weight andand molar volume
molar volume
Where P is the pressure, T is the temperature, R the ideal
Where P is the pressure, T is the temperature, R the ideal gas constant, and Vgas constant, and Vmm
the
the molar volumemolar volume.. a and b are a and b are parameters that are determined empiriparameters that are determined empirically forcally for each gas, but
each gas, but are sometimes estimated from theirare sometimes estimated from their critical temperaturecritical temperature(T(Tcc) and) and
critical pressure
critical pressure(P(Pcc) using these relations:) using these relations:
Redlich
Redlich––Kwong modelKwong model
The
The RedlichRedlich––Kwong equationKwong equationiis another two-parameters equation that is useds another two-parameters equation that is used to model real gases. It is almost always more accurate than the
to model real gases. It is almost always more accurate than the van der Waalsvan der Waals
equation
equation,, and often more accurate and often more accurate than some equations with more than twothan some equations with more than two parameters. The equation is
parameters. The equation is
where a and b two empirical parameters that are
where a and b two empirical parameters that arenot not the same parameters as inthe same parameters as in
the van der Waals equation. These parameters can be determined: the van der Waals equation. These parameters can be determined:
Berthelot and modified Berthelot model Berthelot and modified Berthelot model
The Berthelot equation (named after D.
The Berthelot equation (named after D. BertheloBerthelot t [1][1]is very rarely used,is very rarely used,
but the modified version is somewhat more accurate but the modified version is somewhat more accurate
Dieterici model Dieterici model
This model (named after C.
This model (named after C. DietericDietericii[2][2])) fell out of usage in recent yearsfell out of usage in recent years
..
Clausius model Clausius model
The Clausius equation (named after
The Clausius equation (named after Rudolf ClausiusRudolf Clausius)) is a very simple three-is a very simple three-parameter equation used to model gases.
parameter equation used to model gases.
where where
where V
where Vccis critical volume.is critical volume. Virial model
Virial model
The
The VirialVirialequation derives from a perturbative treatment equation derives from aperturbative treatment of statisticalof statistical mechanics.
mechanics.
or alternatively or alternatively
where A, B, C, A
where A, B, C, A′′, B, B′′, and C, and C′′are temperature dependent constants.are temperature dependent constants.
Peng
Peng––Robinson modelRobinson model
This two parameter equation (named after D.-Y. Peng and D. B.
This two parameter equation (named after D.-Y. Peng and D. B. RobinsoRobinsonn[3][3])) hashas
the interesting property being useful in modeling some liquids as well as real the interesting property being useful in modeling some liquids as well as real gases.
gases.
Wohl model Wohl model
The Wohl equation (named after A. Woh
The Wohl equation (named after A. Wohll[4][4])) is formulated in terms of cis formulated in terms of c riticalritical
values, making it useful when real gas constants are not a
values, making it useful when real gas constants are not a vailable.vailable.
where where
..
Beattie
Beattie––Bridgeman modelBridgeman model [5]
[5]
this equation is based on
this equation is based on five experimentally determined constants. it isfive experimentally determined constants. it is expressed as
expressed as
where where
this equation is known to
this equation is known to be reasonably accurate for densities up to be reasonably accurate for densities up to about about 0.8ρcr where ρcr is the density of the substance at the critical point. the 0.8ρcr where ρcr is the density of the substance at the critical point. the
constants appearing in the above equation are available in following table when constants appearing in the above equation are available in following table when P is in KPa, v is
P is in KPa, v is in \frac{m^3}{Kmol} , T is in K andin \frac{m^3}{Kmol} , T is in K and R=8.314frac{kPa.m^3}{Kmol.K
R=8.314frac{kPa.m^3}{Kmol.K}}[6][6]
Gas
Gas A_0 A_0 a a B_0 B_0 b b cc
Air
Air 131.8441 131.8441 0.01931 0.01931 0.04611 0.04611 -0.001101 -0.001101 4.34×10^44.34×10^4 Argon,Ar
Argon,Ar 130.7802 130.7802 0.02328 0.02328 0.03931 0.03931 0.0 0.0 5.99×10^45.99×10^4 Carbon
Carbon Dioxide, Dioxide, Co_2 Co_2 507.2836 507.2836 0.07132 0.07132 0.10476 0.10476 0.07235 0.07235 6.60×10^56.60×10^5 Helium,He Helium,He 2.1886 2.1886 0.05984 0.05984 0.01400 0.01400 0.0 0.0 4040 Hydrogen,H_2 Hydrogen,H_2 20.0117 20.0117 -0.00506 -0.00506 0.02096 0.02096 -0.04359 -0.04359 504504 Nitrogen,N_2 Nitrogen,N_2 136.2315 136.2315 0.02617 0.02617 0.05046 0.05046 -0.00691 -0.00691 4.20×10^44.20×10^4 Oxygen,O_2 Oxygen,O_2 151.0857 151.0857 0.02562 0.02562 0.04624 0.04624 0.004208 0.004208 4.80×10^44.80×10^4 Benedict
Benedict ––WebbWebb––Rubin modelRubin model
Main article:
Main article: Benedict Benedict ––WebbWebb––Rubin equationRubin equation
The BWR equation, sometimes referred to as
The BWR equation, sometimes referred to as the BWRS equation,the BWRS equation,
where d is the molal density and where a,
where d is the molal density and where a, b, c, A, B, b, c, A, B, C, α, and γ C, α, and γ are empiricalare empirical constants. Note that the γ constant is a
constants. Note that the γ constant is a derivative of constant α and thereforederivative of constant α and therefore almost identical to 1.
almost identical to 1. The
Thecompressibilitcompressibility y factorfactor((ZZ), also known as the), also known as thecompression factorcompression factor, is a, is a
useful thermodynamic property for modifying the
useful thermodynamic property for modifying theideal gas lawideal gas lawto account forto account for
the
thereal gasreal gasbehaviobehavior.r.[1][1]In general, deviation from ideal In general, deviation from ideal behavior becomesbehavior becomes
more significant the closer a gas is to a phase c
more significant the closer a gas is to a phase c hange, the lower the temperaturehange, the lower the temperature or the larger the
or the larger the pressure. Comprespressure. Compressibility factor values are usually obtained bysibility factor values are usually obtained by calculation from
calculation fromequations of stateequations of state(EOS), such as the(EOS), such as thevirial equationvirial equationwhichwhich
take compound specific empirical constants as input. For a gas that is a mixture take compound specific empirical constants as input. For a gas that is a mixture of two or more pure gases (air or natural gas, for example), a
of two or more pure gases (air or natural gas, for example), agas compositiongas composition
is required before compressibili
is required before compressibility can ty can be calculated.be calculated.
Alternatively, the compressibility factor for specific gases can be
generalized compressibility chart
generalized compressibility chart ss[1][1]that that plot plot as as a a function function of of pressure pressure at at
constant temperature. The compressibility factor is defined as constant temperature. The compressibility factor is defined as
where
where is is thethe molar volumemolar volume,, is theis the molar
molar volume volume of of the the correspondincorresponding g ideal ideal gas, gas, is is thethe pressurepressure,, is theis the temperature
temperature,, and and is is thethe gas constant gas constant .. For engineering applications, it isFor engineering applications, it is frequently expressed as
frequently expressed as
where
where is is thethe densitydensityof of the the gas gas and and is is thethe specific gas constan
specific gas constant t ,,[2][2] being thebeing the molar massmolar mass..
For an
For an ideal gasideal gasthe the compressibicompressibility lity factor factor is is per per definition. definition. In In manymany real world applications requirements for accuracy demand that deviations from real world applications requirements for accuracy demand that deviations from ideal gas behaviour, i.e.
ideal gas behaviour, i.e.,, real gasreal gasbehaviour, is taken into account. The value of behaviour, is taken into account. The value of generally increases with pressure and decreases with temperature. At generally increases with pressure and decreases with temperature. At highhigh pressures molecules are colliding more often. This allows repulsive forces pressures molecules are colliding more often. This allows repulsive forces between molecules to have a noticeable effect, making the molar volume of the between molecules to have a noticeable effect, making the molar volume of the rea
real gal gas (s ( ) gr) greateater ter than han the the molmolar var voluolume ome of thf the coe correrrespspondonding ing ideideal gaal gas (s ( ),
), which which causes causes to to exceed exceed oneone..[3][3]WhenWhen
pressures are lower, the molecules are free to move. In this case attractive pressures are lower, the molecules are free to move. In this case attractive forces
forces dominate, dominate, making making . . The The closer closer the the gas gas is is to to itsits critical point critical point oror its
its boiling boiling point, point, the the more more deviates deviates from from the the ideal ideal case.case.
Generalized compressibility factor graphs for pure gases Generalized compressibility factor graphs for pure gases
Generalized compressibility factor diagram. Generalized compressibility factor diagram.
The unique relationship between the compressibility factor and the
The unique relationship between the compressibility factor and the reducedreduced
temperature
temperature,, , and the, and the reduced pressurereduced pressure,, , was first recognized by, was first recognized by Johannes Diderik van der Waals
Johannes Diderik van der Waalsin 1873 and is known as tin 1873 and is known as the two-parameterhe two-parameter principle of corresponding states
principle of corresponding states.. The principle of corresponding statesThe principle of corresponding states expresses the generalization that the properties of a
expresses the generalization that the properties of a gas which are dependent gas which are dependent on intermolecular forces are related to the critical properties of the gas in a on intermolecular forces are related to the critical properties of the gas in a universal way. That provides a most
universal way. That provides a most important basis for developing correlationsimportant basis for developing correlations of molecular
of molecular properties.properties. As for the
As for the compressibcompressibility of gases, the ility of gases, the principle of corresponding statesprinciple of corresponding states indicates
indicates that that any any pure pure gas gas at at the the same same reduced reduced temperature, temperature, , , and and reducedreduced pressure,
pressure, , , should should have have the the same same compressibilicompressibility ty factor.factor. The reduced temperature and pressure are defined
The reduced temperature and pressure are defined byby
and and Here
Here and and are are known known as as the the critical critical temperature temperature and and critical critical pressure pressure of of a
a gas. gas. They They are are characteristics characteristics of of each each specific specific gas gas with with being being thethe temperature
temperature above above which which it it is is not not possible possible to to liquify liquify a a given given gas gas and and is is thethe minimum pressure requir
minimum pressure required to liquify a ed to liquify a given gas at its given gas at its critical temperature.critical temperature.
Together they define the critical point of a fluid above which distinct liquid and Together they define the critical point of a fluid above which distinct liquid and gas phases of a given fluid do
gas phases of a given fluid do not exist.not exist. The pressure-volume-temp
The pressure-volume-temperature (PVT) data for erature (PVT) data for real gases varies from onereal gases varies from one pure gas to
pure gas to another. However, when the compressibility factors of variousanother. However, when the compressibility factors of various single-component gases are graphed versus pressure along with temperature single-component gases are graphed versus pressure along with temperature isotherms many of the
isotherms many of the graphs exhibit similar isotherm shapes.graphs exhibit similar isotherm shapes. In order to obtain a generalized graph that can
In order to obtain a generalized graph that can be used for many different gases,be used for many different gases, the
the reduced reduced pressure pressure and and temperature, temperature, and and , , are are used used to to normalize normalize thethe compressib
compressibility factor data. Figure 2 is an ility factor data. Figure 2 is an example of a generalizedexample of a generalized compressib
compressibility factor graph derived from hundreds of ility factor graph derived from hundreds of experimental PVT dataexperimental PVT data points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, points of 10 pure gases, namely methane, ethane, ethylene, propane, n-butane, i-pentane, n-hexane, nitrogen, carbon dioxide and
pentane, n-hexane, nitrogen, carbon dioxide and steam.steam.
There are more detailed generalized compressibility factor graphs based on There are more detailed generalized compressibility factor graphs based on asas many as 25 or more different pure gases, such as the Nelson-Obert graphs. Such many as 25 or more different pure gases, such as the Nelson-Obert graphs. Such graphs
graphs are are said said to to have have an aan accuracy ccuracy within within 1-2 p1-2 percent ercent for for values values greatergreater than
than 0.6 0.6 and and within within 4-6 4-6 percent percent for for values values of of 0.3-0.6.0.3-0.6.
The generalized compressibility factor graphs may be considerably in error for The generalized compressibility factor graphs may be considerably in error for strongly polar gases which are gases for which the centers of positive and strongly polar gases which are gases for which the centers of positive and negative
negative charge charge do do not not coincide. coincide. In sIn such cuch cases ases the esthe estimate timate for for may may be be inin error by as much as 15-20 percent.
error by as much as 15-20 percent.
The quantum gases hydrogen, helium, and neon do not conform to the The quantum gases hydrogen, helium, and neon do not conform to the corresponding-s
corresponding-states behavior and ttates behavior and t he reduced pressure and he reduced pressure and temperature fortemperature for those three gases should be redefined in the following manner to improve the those three gases should be redefined in the following manner to improve the accuracy of predicting their compressibility factors when using the g
accuracy of predicting their compressibility factors when using the g eneralizedeneralized graphs:
graphs:
and and where the temperatures are in
where the temperatures are in kelvin and the kelvin and the pressures are in atmospheres.pressures are in atmospheres.[4][4]
Theoretical models Theoretical models
The virial equation is especially useful to describe the causes of non-ideality at a The virial equation is especially useful to describe the causes of non-ideality at a molecular level (very few
molecular level (very few gases are mono-atomic) as it is gases are mono-atomic) as it is derived directly fromderived directly from statistical mechanics:
statistical mechanics:
Where the coefficients in the numerator are known as virial coefficients and are Where the coefficients in the numerator are known as virial coefficients and are functions of temperature.
functions of temperature.
The virial coefficients account for interactions between successively larger The virial coefficients account for interactions between successively larger groups
groups of of molecules. molecules. For For example, example, accounts for accounts for interactions interactions between between pairs,pairs, for interactions between three gas molecules, and so on. Because
for interactions between three gas molecules, and so on. Because interactions between large numbers of molecules are rare,
interactions between large numbers of molecules are rare, the virial equation isthe virial equation is usually truncated after the third term
usually truncated after the third term..[5][5]
The
The Real gasReal gasarticle features more theoretical methods to computearticle features more theoretical methods to compute compressib
compressibility ility factorsfactors
Experimental values Experimental values
It is extremely difficult to
It is extremely difficult to generalize at what pressures or temperatures thegeneralize at what pressures or temperatures the deviation from the
deviation from the ideal gasideal gasbecomes imporbecomes important. As a rule of tant. As a rule of thumb, the idealthumb, the ideal gas law is reasonably accurate up to a
gas law is reasonably accurate up to a pressure of about 2pressure of about 2 atmatm,, and even higherand even higher for small non-associating molecules. For
for small non-associating molecules. For exampleexample methyl chloridemethyl chloride,, a highlya highly polarpolar
molecule
moleculeand therefore with and therefore with significant intermolecular forces, the experimentalsignificant intermolecular forces, the experimental value
value for for the the compressibilicompressibility ty factor factor is is at at a a pressure pressure of of 1010 atm and temperature of 100 °C
atm and temperature of 100 °C..[6][6]For air (small non-polar molecules) at For air (small non-polar molecules) at
approximately the same conditions, the
approximately the same conditions, the compressibilcompressibility factor is ity factor is onlyonly (see table below for 10
(see table below for 10 barsbars,, 400 K).400 K).
Compressibilit Compressibility of y of airair
Normal
Normal airaircomprises in crude numbers 80 percent comprises in crude numbers 80 percent nitrogennitrogenNN22and 20 percent and 20 percent
oxygen
oxygenOO22. Both molecules are small and. Both molecules are small and non-polarnon-polar(and therefore non-(and therefore
non-associating). We can therefore expect that
associating). We can therefore expect that the behaviour of air within broadthe behaviour of air within broad temperature and pressure ranges can be approximated as an ideal gas with temperature and pressure ranges can be approximated as an ideal gas with reasonable accuracy. Experimental values for the compressibility factor confirm reasonable accuracy. Experimental values for the compressibility factor confirm this.
this.
Z for air as function of pressure 1-500 bar Z for air as function of pressure 1-500 bar
75-200 K isotherms 75-200 K isotherms
250-1000 K isotherms 250-1000 K isotherms Compressi
Compressibility factor for bility factor for air (experimental values)air (experimental values) Pressure, bar (absolute)
Pressure, bar (absolute) Te Te mp, mp, K K 1 1 5 5 10 10 20 20 40 40 60 60 80 80 100 100 150 150 200 200 250 250 300 300 400 400 500500 75 75 520.00520.00 0.020.026060 0.0519190.05 0.100.103636 63630.200.20 0.300.308282 0.400.409494 990.50990.50 0.750.758181 1.011.012525 80 80 0.020.025050 0.040.049999 95950.090.09 0.190.198181 0.290.295858 270.39270.39 0.480.488787 0.720.725858 880.95880.95 1.191.193131 1.411.413939 90 90 640.97640.97 0.020.023636 0.040.045353 400.090.0940 0.180.186666 0.270.278181 86860.360.36 0.460.468181 0.670.677979 0.890.892929 981.10981.101.311.311010 1.711.716161 2.112.110505 100 100 970.97970.97 0.880.887272 0.040.045353 000.090.0900 0.170.178282 0.260.263535 98980.340.34 0.430.433737 0.630.638686 0.830.837777 951.03951.031.221.222727 1.591.593737 1.951.953636 120 120 800.98800.98 0.930.937373 0.880.886060 300.670.6730 0.170.177878 0.250.255757 71710.330.33 0.410.413232 0.590.596464 0.770.772020 300.95300.951.101.107676 1.501.509191 1.731.736666 140 140 270.99270.99 0.960.961414 0.920.920505 970.820.8297 0.580.585656 0.330.331313 37370.370.37 0.430.434040 0.590.590909 0.760.769999 140.91140.911.031.039393 1.321.320202 1.591.590303 160 160 510.99510.99 0.970.974848 0.940.948989 540.890.8954 0.780.780303 0.660.660303 96960.560.56 0.540.548989 0.630.634040 0.750.756464 400.88400.881.011.010505 1.251.258585 1.491.497070 180 180 670.99670.99 0.980.983232 0.960.966060 140.930.9314 0.860.862525 0.790.797777 32320.740.74 0.700.708484 0.710.718080 0.790.798686 000.90000.901.001.006868 1.221.223232 1.431.436161 200 200 780.99780.99 0.980.988686 0.970.976767 390.950.9539 0.910.910000 0.870.870101 74740.830.83 0.810.814242 0.800.806161 0.850.854949 110.93110.931.011.018585 1.201.205454 1.391.394444 250 250 920.99920.99 0.990.995757 0.990.991111 220.980.9822 0.960.967171 0.950.954949 63630.940.94 0.940.941111 0.940.945050 0.970.971313 521.01521.011.071.070202 1.191.199090 1.331.339292 300 300 990.99990.99 0.990.998787 0.990.997474 500.990.9950 0.990.991717 0.990.990101 03030.990.99 0.990.993030 1.001.007474 1.031.032626 691.06691.061.101.108989 1.201.207373 1.311.316363 350 350 001.00001.00 1.001.000202 1.001.000404 141.001.0014 1.001.003838 1.001.007575 21211.011.01 1.011.018383 1.031.037777 1.061.063535 471.09471.091.131.130303 1.211.211616 1.301.301515 400 400 021.00021.00 1.001.001212 1.001.002525 461.001.0046 1.011.010000 1.011.015959 29291.021.02 1.031.031212 1.051.053333 1.071.079595 871.10871.101.141.141111 1.211.211717 1.281.289090 450 450 031.00031.00 1.001.001616 1.001.003434 631.001.0063 1.011.013333 1.021.021010 87871.021.02 1.031.037474 1.061.061414 1.091.091313 831.11831.111.141.146363 1.201.209090 1.271.277878 500 500 031.00031.00 1.001.002020 1.001.003434 741.001.0074 1.011.015151 1.021.023434 23231.031.03 1.041.041010 1.061.065050 1.091.091313 831.11831.111.141.146363 1.201.205151 1.261.266767 600 600 041.00041.00 1.001.002222 1.001.003939 811.001.0081 1.011.016464 1.021.025353 40401.031.03 1.041.043434 1.061.067878 1.091.092020 721.11721.111.141.142727 1.191.194747 1.241.247575 800 800 041.00041.00 1.001.002020 1.001.003838 771.001.0077 1.011.015757 1.021.024040 21211.031.03 1.041.040808 1.061.062121 1.081.084444 611.10611.101.121.128383 1.171.172020 1.211.215050 100 100 00 041.00041.00 1.001.001818 1.001.003737 681.001.0068 1.011.014242 1.021.021515 90901.021.02 1.031.036565 1.051.055656 1.071.074444 481.09481.091.111.113131 1.151.151515 1.181.188989 Source:
Source:Perry's chemical engineers' handbook Perry's chemical engineers' handbook (6ed ed.). MCGraw-Hill. 1984.(6ed ed.). MCGraw-Hill. 1984.
ISBN
ISBN0-07-049479-70-07-049479-7.. (table (table 3-162). 3-162). values values are are calculated calculated from from values values of of pressure, volume (or density), and temperature in Vassernan, Kazavchinskii, pressure, volume (or density), and temperature in Vassernan, Kazavchinskii, and Rabinovich, "Thermophysical Properties of Air
and Rabinovich, "Thermophysical Properties of Air and Air and Air Components;'Components;' Moscow, Nauka, 1966, and NBS-NSF Trans.
Moscow, Nauka, 1966, and NBS-NSF Trans. TT 70-50095, 1971: and TT 70-50095, 1971: and VassernanVassernan and Rabinovich, "Thermophysical Propertie
and Rabinovich, "Thermophysical Properties of Liquid Air s of Liquid Air and Its Component,and Its Component, "Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.
"Moscow, 1968, and NBS-NSF Trans. 69-55092, 1970.
Compressibilit
Compressibility of y of ammonia gasammonia gas
Ammonia is small but highly polar
Ammonia is small but highly polar molecule with significant interactions. Valuesmolecule with significant interactions. Values can be obtained from Perry 4th
can be obtained from Perry 4th ed (awaits future library visit)ed (awaits future library visit) According to van der Waals, the
According to van der Waals, thetheorem of theorem of corresponcorresponding statesding states(or(or principle of corresponding states
principle of corresponding states) indicates that al) indicates that alll fluidsfluids,, when compared at when compared at
the same
the same reduced temperaturereduced temperatureandand reduced pressurereduced pressure,, have approximately thehave approximately the same
same compressibcompressibility ility factorfactorand all deviate from ideal gas behavior to about theand all deviate from ideal gas behavior to about the same degree
same degree..[1][2][1][2]
Material constants
Material constantsthat vary for each type of material are eliminated, in a recast that vary for each type of material are eliminated, in a recast reduced form of a
reduced form of a constitutive equationconstitutive equation.. The reduced variables are defined inThe reduced variables are defined in terms o
terms of f critical variablescritical variables.. It originated with the work of
It originated with the work of Johannes Diderik van der WaalsJohannes Diderik van der Waalsin about 187in about 18733[3][3]
when he used the
when he used the critical temperaturecritical temperatureandand critical pressurecritical pressureto characterize ato characterize a fluid.
fluid.
The most prominent example is the
The most prominent example is the van der Waals equationvan der Waals equationof state, theof state, the reduced form of which applies to all fluids.
reduced form of which applies to all fluids.
Compressibilit
Compressibility factor y factor at the critical at the critical point point
The compressibility factor at the critical point,
The compressibility factor at the critical point, which is defined aswhich is defined as
, , where where the the subscript subscript indicates indicates thethe critical point critical point isis predicted to be a constant independent of substance by many equations of state; predicted to be a constant independent of substance by many equations of state; the
the Van der Waals equationVan der Waals equatione.g. e.g. predicts predicts a a value value of of ..
Substance Value Substance Value H H22OO 0.2330.2 [4][4] 4 4HeHe 0.3110.3 [4][4] He He 0.3000.3 [5][5] H H22 0.3000.3 [5][5] Ne Ne 0.2990.2 [5][5] N N22 0.2990.2 [5][5] Ar Ar 0.2990.2 [5][5] In
Inphysicsphysicsandandthermodynamicsthermodynamics,, ananequation of stateequation of stateis a relation betweenis a relation between
state variables
state variables..[1][1]More specifically, an equation of state is aMore specifically, an equation of state is athermodynamicthermodynamic equation
equationdescribing the state of matter under a given set of physical conditions.describing the state of matter under a given set of physical conditions.
It is a
It is aconstitutive equationconstitutive equationwhich provides a which provides a mathematical relationshipmathematical relationship
between two or more
between two or morestate functionsstate functionsassociated with the matter, such as itsassociated with the matter, such as its
temperature
temperature,,pressurepressure,,volumevolume,, ororinternal energyinternal energy.. Equations of state areEquations of state are useful in describing the properties of
useful in describing the properties of fluidsfluids,, mixtures of fluidsmixtures of fluids,,solidssolids,, and evenand even
the interior of
the interior of starsstars.. OverviewOverview
The most prominent use of an equation of state is to correlate densities of gases The most prominent use of an equation of state is to correlate densities of gases and liquids to temperatures and pressures. One of
and liquids to temperatures and pressures. One of the simplest equations of the simplest equations of state for this purpose is the
state for this purpose is the ideal gas lawideal gas law,, which is roughly accurate for weaklywhich is roughly accurate for weakly polar gases at low
polar gases at low pressures and moderate temperatures. However, thispressures and moderate temperatures. However, this equation becomes increasingly inaccurate at higher pressures and lower equation becomes increasingly inaccurate at higher pressures and lower temperatures, and fails to predict condensation from a gas to a liquid. temperatures, and fails to predict condensation from a gas to a liquid. Therefore, a number of more accurate
Therefore, a number of more accurate equations of state have been equations of state have been developeddeveloped for gases and liquids. At present, there is no single equation of state that for gases and liquids. At present, there is no single equation of state that accurately predicts the properties of all
accurately predicts the properties of all substances under all conditions.substances under all conditions. In addition, there are
In addition, there are also equations of state describingalso equations of state describing solidssolids,, including theincluding the transition of solids from one crystalline state to
transition of solids from one crystalline state to another. There are equationsanother. There are equations that model the interior o
that model the interior of f starsstars,, includingincluding neutron starsneutron stars,, dense matterdense matter ((quark-
quark-gluon plasmas
gluon plasmas)) and radiation fields. A related concept is theand radiation fields. A related concept is the perfect fluidperfect fluid
equation of state used in
equation of state used in cosmologycosmology.. Historical
Historical
Boyle's law (1662) Boyle's law (1662)
Boyle's Law was perhaps the first expression of an equation of state. In 1662, Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and
the noted Irish physicist and chemist chemist Robert BoyleRobert Boyleperformed a series of performed a series of experiments employin
experiments employing a J-shaped glass tube, which was sealed on g a J-shaped glass tube, which was sealed on one end.one end. Mercury
Mercurywas added to the tube, trapping a was added to the tube, trapping a fixed quantity of air in the short,fixed quantity of air in the short, sealed end of the tube. Then the
sealed end of the tube. Then the volume of gas was carefully measured asvolume of gas was carefully measured as additional mercury was added to the tube. The pressure of the gas could be additional mercury was added to the tube. The pressure of the gas could be determined by the difference between the mercury level in the short end of the determined by the difference between the mercury level in the short end of the tube and that in the
tube and that in the long, open end. Through these experiments, Boyle notedlong, open end. Through these experiments, Boyle noted that the gas
that the gas volume varied inversely with the pressure. In volume varied inversely with the pressure. In mathematical form,mathematical form, this can be stated as:
this can be stated as:
The above relationship has also been attributed to
The above relationship has also been attributed to Edme MariotteEdme Mariotteand isand is sometimes referred to as
sometimes referred to as Mariotte'Mariotte's s lawlaw. However, Mariotte's work was not . However, Mariotte's work was not
published until 1676. published until 1676.
Charles's law or Law of Charles and Gay-Lussac (1787) Charles's law or Law of Charles and Gay-Lussac (1787) In 1787 the French physicist
In 1787 the French physicist Jacques CharlesJacques Charlesfound that oxygen, nitrogen,found that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 hydrogen, carbon dioxide, and air expand to the same extent over the same 80 kelvin interval. Later, in 1802
kelvin interval. Later, in 1802,, Joseph Louis Gay-LussacJoseph Louis Gay-Lussacpublished results of published results of similar experiments, indicating a linear relationship between volume and similar experiments, indicating a linear relationship between volume and temperature:
temperature:
Dalton's law of
Dalton's law of partial pressures (1801)partial pressures (1801) Dalton's Law
Dalton's Lawof partial pressure states that the pressure of a mixture of gases isof partial pressure states that the pressure of a mixture of gases is equal to the sum of the pressures of all of
equal to the sum of the pressures of all of the constituent gases alone.the constituent gases alone. Mathematically, this can be represented for
The ideal gas law (1834) The ideal gas law (1834) In 1834
In 1834 Émile ClapeyronÉmile Clapeyroncombined Boyle's Law and Charles' law into the first combined Boyle's Law and Charles' law into the first statement of the
statement of theideal gas law ideal gas law .. Initially the law was formulated asInitially the law was formulated as
pV
pV mm==RR((T T C C + 267) (with temperature expressed in + 267) (with temperature expressed in degrees Celsius), wheredegrees Celsius), whereRRisis
the
the gas constant gas constant .. However, later work revealed that However, later work revealed that the number should actuallythe number should actually be closer to 273.2, and then the
be closer to 273.2, and then the Celsius scale was defined with 0 °C = 273.15 K,Celsius scale was defined with 0 °C = 273.15 K, giving:
giving:
Van der Waals equation of state (1873) Van der Waals equation of state (1873) In 1873,
In 1873, J. D. van der WaalsJ. D. van der Waalsintroduced the firsintroduced the first t equation of stateequation of statederived by thederived by the assumption of a finite volume occupied by
assumption of a finite volume occupied by the constituent moleculesthe constituent molecules..[2][2]His newHis new
formula revolutionized the study of equations of state, and
formula revolutionized the study of equations of state, and was most famouslywas most famously continued via the
continued via the RedlichRedlich––Kwong equation of stateKwong equation of stateand theand the Soave modificationSoave modification
of Redlich
of Redlich––KwongKwong..
Equation of state of an Ideal gas Equation of state of an Ideal gas::
Boyle’s law, Charles’ law and Avogadro’s law could be combined to give a Boyle’s law, Charles’ law and Avogadro’s law could be combined to give a general relation between the volume, pressure, temperature and the
general relation between the volume, pressure, temperature and the number of number of moles of a particular gas. The equation,
moles of a particular gas. The equation,PV = constant PV = constant , describes the variation, describes the variation
of
of PPwithwithVVat constant at constant TTand the equationand the equationV/TV/T, represents the variation of V, represents the variation of V
with T at constant P. On
with T at constant P. On combining these equations, we get:combining these equations, we get:
PV/T = constant
PV/T = constant and this relating the variables P. V and and this relating the variables P. V and T of an ideal T of an ideal gas isgas is known as the equation of state. The product ‘
known as the equation of state. The product ‘PVPV’ over’ overTTis always constant foris always constant for
all specified states of the gas. Hence, if we
all specified states of the gas. Hence, if we know these values for any one stateknow these values for any one state the constant can be calculated. In
the constant can be calculated. In standard state orstandard state orSTPSTP, with the pressure at 1, with the pressure at 1
atm and temperature at being 273.16 K , t
atm and temperature at being 273.16 K , t he volume occupied by a mole of anhe volume occupied by a mole of an ideal gas would be equal to 22.414 L.
ideal gas would be equal to 22.414 L. According to Avogadro’s law this volumeAccording to Avogadro’s law this volume is same for all ideal gases and if
is same for all ideal gases and if we consider ‘we consider ‘nn’ moles of an ideal gas ’ moles of an ideal gas at at STP,STP,
then the equation becomes then the equation becomes
PV/T = P
PV/T = PooVVoo/T/Too= nR= nR PV = nRT
PV = nRT, Where,, Where,RRis a universal gas constant per mole. The above equation isis a universal gas constant per mole. The above equation is known as the ideal gas equation and it
known as the ideal gas equation and it connects directly all the components andconnects directly all the components and permits all kinds of c
permits all kinds of c alculations.alculations. Major equations of state
Major equations of state For a given
For a given amount of substance contained in a amount of substance contained in a system, the temperature,system, the temperature, volume, and pressure are not independent quantities; they are connected by a volume, and pressure are not independent quantities; they are connected by a relationship of the general form:
relationship of the general form:
In the following equations the variables are defined as follows. Any consistent In the following equations the variables are defined as follows. Any consistent set of units may be used, although
set of units may be used, although SISIunits are preferredunits are preferred.. Absolute temperatureAbsolute temperature
refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero refers to use of the Kelvin (K) or Rankine (°R) temperature scales, with zero being absolute zero.
being absolute zero. = pressure (absolute) = pressure (absolute)
= volume = volume
= number of moles of a substance = number of moles of a substance
=
= ==molar volumemolar volume,, the volume of 1 mole of gas or the volume of 1 mole of gas or liquidliquid == absolute temperatureabsolute temperature
== ideal gas constant ideal gas constant (8.314472 J/(mol·K))(8.314472 J/(mol·K)) = pressure at the critical point
= pressure at the critical point = molar volume at the critical point = molar volume at the critical point = absolute temperature at the
= absolute temperature at the critical point critical point Classical ideal gas law
Classical ideal gas law The classical
The classical ideal gas lawideal gas lawmay be written:may be written: The ideal gas law may also be
The ideal gas law may also be expressed as followsexpressed as follows
where
where is is the the density, density, is is the the adiabatic adiabatic indexindex ((ratio of ratio of
specific heats
specific heats)), , is is the the internal internal energy energy per per unit unit mass mass (the(the "specific
"specific internal internal energy"), energy"), is the is the specific specific heat heat at at constant constant volume, volume, andand is the specific heat at constant pressure.
is the specific heat at constant pressure. Cubic equations of state
Cubic equations of state
Cubic equations of state are called such because they can be rewritten as a Cubic equations of state are called such because they can be rewritten as a cubiccubic
function
functionof Vof Vmm..
Van der Waals equation of state Van der Waals equation of state The
The Van der Waals equationVan der Waals equationof state may be written:of state may be written:
where
where is is molar molar volume, volume, and and and and are are substance-specifisubstance-specific c constants.constants. They can be calculated from the
They can be calculated from the critical propertiescritical properties and and (noting(noting that
that is is thethemolar molar volume at the critical point) as:volume at the critical point) as:
Also written as Also written as
Proposed in 1873, the van der Waals equation of state was
Proposed in 1873, the van der Waals equation of state was one of the first toone of the first to perform
perform markedly markedly better than better than the ideal the ideal gas law. gas law. In this In this landmark elandmark equation quation isis called th
called the attractie attraction paron parameter ameter and and the repulthe repulsion sion parameter parameter or or the effectivethe effective molecular volume. While the equation is definitely superi
molecular volume. While the equation is definitely superior to the ideal gas or to the ideal gas lawlaw and does predict the formation of a liquid phase, the agreement with
and does predict the formation of a liquid phase, the agreement with experimental data is limited for conditions where the
experimental data is limited for conditions where the liquid forms. While theliquid forms. While the van der Waals equation is
van der Waals equation is commonly referenced in text-books and papers commonly referenced in text-books and papers forfor historical reasons, it is now obsolete. Other modern equations of
historical reasons, it is now obsolete. Other modern equations of only slightlyonly slightly greater complexity are much more accurate.
greater complexity are much more accurate.
The van der Waals equation may be considered as the ideal gas
The van der Waals equation may be considered as the ideal gas law, “improved”law, “improved” due to two
due to two independent reasons:independent reasons: Molecules are thought as particles with
Molecules are thought as particles with volume, not material points. Thusvolume, not material points. Thus ca
cannnnoot bt be te tooo lo liittttlele, l, lesess ts thahan sn soomme ce cononsstatannt. t. SSo wo we ge get et (( ) i) insnstetead ad of of ..
While ideal gas molecules do
While ideal gas molecules do not interact, we consider molecules attractingnot interact, we consider molecules attracting others within a distance of several molecules' radii. It makes no effect inside the others within a distance of several molecules' radii. It makes no effect inside the material, but surface molecules are attracted into
material, but surface molecules are attracted into the material from the surface.the material from the surface. We see this as diminishing of pressure on the outer shell (which is used in the We see this as diminishing of pressure on the outer shell (which is used in the ide
ideal al gas gas lawlaw), ), so so we we wriwrite te ( ( somsomethethinging) ) insinsteatead d of of . . To To evaevalualuate te thithiss ‘something’, let's exa
‘something’, let's examine an additional force acting on an element of gasmine an additional force acting on an element of gas surf
surface. Whiace. While the forle the force actince acting on each sug on each surface morface molecullecule is ~e is ~ , the forc, the force actinge acting
oon n tthhe e wwhhoolle e eelleemmeennt t iis s ~~ ~~ .. With the reduced state variables, i.e. V
With the reduced state variables, i.e. Vrr=V=Vmm/V/Vcc, P, Prr=P/P=P/Pccand Tand Trr=T/T=T/Tcc, the reduced, the reduced
form of the Van der Waals equation can be
form of the Van der Waals equation can be formulated:formulated:
The benefit of this form is that for given T
The benefit of this form is that for given Trrand Pand Prr, the reduced volume of the, the reduced volume of the
liquid and gas can be calculated directly using
liquid and gas can be calculated directly using Cardano's methodCardano's methodfor thefor the reduced cubic form:
reduced cubic form:
For P
For Prr<1 and T<1 and Trr<1, the system is <1, the system is in a state in a state of vapor-liquid equilibrium. Theof vapor-liquid equilibrium. The
reduced cubic equation of state yields in that case 3 solutions. The largest and reduced cubic equation of state yields in that case 3 solutions. The largest and the lowest solution are the gas and liquid reduced volume.
the lowest solution are the gas and liquid reduced volume. Redlich
Redlich––Kwong equation of stateKwong equation of state
Introduced in 1949 the
Introduced in 1949 the RedlichRedlich––Kwong equation of stateKwong equation of statewas a considerablewas a considerable improvement over other equations of the time. It
improvement over other equations of the time. It is still of interest is still of interest primarilprimarilyy due to its relatively simple form. While superior to the van der Waals equation due to its relatively simple form. While superior to the van der Waals equation of state, it performs poorly with respect to the liquid phase and thus cannot be of state, it performs poorly with respect to the liquid phase and thus cannot be used for accurately calculating
used for accurately calculating vapor-liquid equilibriavapor-liquid equilibria.. However, it can be usedHowever, it can be used in conjunction with
in conjunction with separate liquid-phase correlations for this purpose.separate liquid-phase correlations for this purpose. The Redlich
The Redlich––Kwong equation is adequate for calculation of Kwong equation is adequate for calculation of gas phase propertiesgas phase properties when the ratio of the pressure to the
when the ratio of the pressure to the critical pressurecritical pressure(reduced pressure) is less(reduced pressure) is less than about one-half of the ratio of the temperature to the
than about one-half of the ratio of the temperature to the critical temperaturecritical temperature
(reduced temperature): (reduced temperature):
Soave modification of Redlich
Where ω is the
Where ω is the acentric factoracentric factorfor the species.for the species. This
This formulation formulation for for is due is due to to Graboski Graboski and Dauand Daubert. bert. The The originaloriginal formulation from Soave is:
formulation from Soave is: for hydrogen:
for hydrogen:
We can also write it in t
We can also write it in t hehepolynomial form, with A=aαP/R²T² and B=bP/RTpolynomial form, with A=aαP/R²T² and B=bP/RT then we have:
then we have: Z³-Z²+(A-B-B²)Z-AB=0Z³-Z²+(A-B-B²)Z-AB=0 In 1972 G. Soav
In 1972 G. Soavee[3][3]replaced the 1/√(replaced the 1/√(T T ) term of the Redlich) term of the Redlich––Kwong equationKwong equation
with a function α(T,ω
with a function α(T,ω) involving the temperature and the) involving the temperature and the acentric factoracentric factor(the(the resulting equation is also known as
resulting equation is also known as the Soavethe Soave––RedlichRedlich–Kwong equation). The α–Kwong equation). The α function was devised to fit the vapor pressure data of hydrocarbons and the function was devised to fit the vapor pressure data of hydrocarbons and the equation does fairly well for
equation does fairly well for these materials.these materials. Note especially that this replacement changes t
Note especially that this replacement changes t he definition of he definition of aaslightly, as theslightly, as the
is now to the second power. is now to the second power. Peng
Peng––Robinson equation of stateRobinson equation of state
In polynomial form: In polynomial form:
where,
where, is is thethe acentric factoracentric factorof of the the species, species, is is thethe universal gas constant universal gas constant
and Z=PV/(RT) is
and Z=PV/(RT) is compressibilcompressibility ity factorfactor.. The Peng
The Peng––Robinson equation was developed in 1976 in Robinson equation was developed in 1976 in order to satisfy theorder to satisfy the following goals
following goals::[4][4]
The parameters should be expressible in terms of t
The parameters should be expressible in terms of t hehe critical propertiescritical propertiesand theand the acentric factor
acentric factor..
The model should provide reasonable accuracy near the
The model should provide reasonable accuracy near the critical point,critical point, particularly for calculations of the
particularly for calculations of the compressibcompressibility ility factorfactorand liquid density.and liquid density. The mixing rules should not e
The mixing rules should not e mploy more than a single mploy more than a single binary interactionbinary interaction parameter, which should be independent of
parameter, which should be independent of temperature pressure andtemperature pressure and composition.
composition.
The equation should be applicable to all calculations of all fluid properties in The equation should be applicable to all calculations of all fluid properties in natural gas
natural gas processes.processes. For the most part the Peng
For the most part the Peng––Robinson equation exhibits performance similar toRobinson equation exhibits performance similar to the Soave equation, although it
the Soave equation, although it is generally superior in predicting the liquidis generally superior in predicting the liquid densities of many materials, especially nonpolar ones. The
densities of many materials, especially nonpolar ones. The departure functionsdeparture functions
of the Peng
of the Peng––Robinson equation are given on a Robinson equation are given on a separate article.separate article. Peng-Robinson-S
Peng-Robinson-Stryjek-Vera equations of tryjek-Vera equations of statestate PRSV1
PRSV1
A modification to the attraction term
A modification to the attraction term in the Peng-Robinson equation of statein the Peng-Robinson equation of state published by Stryjek and Vera
published by Stryjek and Vera in 1986 (PRSV) significantly improved thein 1986 (PRSV) significantly improved the
model's accuracy by introducing an adjustable pure
model's accuracy by introducing an adjustable pure component parameter andcomponent parameter and by modifying the polynomial fit of the
by modifying the polynomial fit of the acentric factoacentric factorr..[5][5]
The modification is: The modification is:
where
where is is an an adjustable adjustable pure pure component component parameter. parameter. Stryjek Stryjek and and VeraVera published pure component parameters for
published pure component parameters for many compounds of industrialmany compounds of industrial interest in their original journal article
interest in their original journal article..[5][5]
PRSV2 PRSV2
A subsequent modification published in 1986 (PRSV2) further improved the A subsequent modification published in 1986 (PRSV2) further improved the model's accuracy by introducing two additional pure component parameters to model's accuracy by introducing two additional pure component parameters to the previous attraction term
the previous attraction term modificationmodification..[6][6]
The modification is: The modification is:
where
where , , , , and and are are adjustable adjustable pure pure component component parameters.parameters. PRSV2 is particularly advantageous for
PRSV2 is particularly advantageous for VLEVLEcalculations. While PRSV1 doescalculations. While PRSV1 does offer an advantage over t
offer an advantage over t he Peng-Robinson model for describinghe Peng-Robinson model for describing thermodynami
thermodynamic behavior, it is c behavior, it is still not accurate enough, still not accurate enough, in general, for phasein general, for phase equilibrium calculations
equilibrium calculations..[5][5]The highly non-linear behavior of phase-equilibriumThe highly non-linear behavior of phase-equilibrium
calculation methods tends to amplify what would
calculation methods tends to amplify what would otherwise be acceptably smallotherwise be acceptably small errors. It is therefore recommended that PRSV2 be
errors. It is therefore recommended that PRSV2 be used for equilibriumused for equilibrium calculations when applying these models to a design. However, once the calculations when applying these models to a design. However, once the equilibrium state has been
equilibrium state has been determined, the phase specific determined, the phase specific thermodynamithermodynamicc values at equilibrium may be determined by one of several simpler models with values at equilibrium may be determined by one of several simpler models with a reasonable degree of accuracy
a reasonable degree of accuracy..[6][6]
One thing to note is that in
One thing to note is that in the PSRV equation, the parameter fit is done in athe PSRV equation, the parameter fit is done in a particular temperature range which is usually below the
particular temperature range which is usually below the critical temperature.critical temperature. Above the critical temperature, the PRSV alpha function tends to diverge and Above the critical temperature, the PRSV alpha function tends to diverge and become arbitrarily large instead of tending towards 0.
become arbitrarily large instead of tending towards 0. Because of this, alternateBecause of this, alternate equations for alpha should be employed above the
equations for alpha should be employed above the critical point. This iscritical point. This is especially important for systems containing hydrogen which is often found especially important for systems containing hydrogen which is often found at at temperatures far above its critical point. Several alternate
temperatures far above its critical point. Several alternate formulationformulations haves have been proposed. Some well known ones are by Twu et all
been proposed. Some well known ones are by Twu et all or by Mathias andor by Mathias and Copeman.
Copeman.
Elliott, Suresh, Donohue equation of state Elliott, Suresh, Donohue equation of state
The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in The Elliott, Suresh, and Donohue (ESD) equation of state was proposed in 1990
1990..[7][7]The equation seeks to correct a shortcoming in the PengThe equation seeks to correct a shortcoming in the Peng––Robinson EOSRobinson EOS
in that there was an inaccuracy in the
in that there was an inaccuracy in the van der Waals repulsive term. The EOSvan der Waals repulsive term. The EOS accounts for the effect of the shape of a
accounts for the effect of the shape of a non-polar molecule and can be extendednon-polar molecule and can be extended to polymers with the addition of an extra term (not shown). The EOS itself was to polymers with the addition of an extra term (not shown). The EOS itself was developed through modeling computer simulations and should capture the developed through modeling computer simulations and should capture the essential physics of the size, shape,
essential physics of the size, shape, and hydrogen bonding.and hydrogen bonding.
where: where:
and and is
is a a "shape "shape factor", factor", with with for for spherical spherical moleculesmolecules For non-spherical molecules, the following relation is suggested: For non-spherical molecules, the following relation is suggested:
where
where is is thethe acentric factoracentric factor
The
The reduced reduced number number density density is is defined defined asas where
where
is the characteristic size parameter is the characteristic size parameter is the number of
is the number of moleculesmolecules is the volume of the container is the volume of the container The char
The characteristic acteristic size psize parameter arameter is ris related to elated to the shthe shape parape parameter ameter throughthrough
where where
and is
and is Boltzmann's constant Boltzmann's constant ..
Noting the relationships between Boltzmann's constant and the
Noting the relationships between Boltzmann's constant and the Universal gasUniversal gas
constant
constant ,, and observing that the number of molecules can be expressed in termsand observing that the number of molecules can be expressed in terms of
of Avogadro's numberAvogadro's numberand theand the molar massmolar mass,, the the reduced reduced number number density density cancan be expressed in terms of the molar volume as
be expressed in terms of the molar volume as
The
The shape shape parameter parameter appearing appearing in in the the Attraction Attraction term term and and the the term term areare given by
given by
(and is hence also equal to 1
(and is hence also equal to 1 for sphericalfor spherical molecules).
molecules).
where
where is the is the depth of depth of the sqthe square-well uare-well potential and potential and is given is given byby
, , , , and and are are constants constants in in the the equation equation of of state:state: for spherical molecules (c=1)
for spherical molecules (c=1) for spherical molecules (c=1) for spherical molecules (c=1) for spherical molecules (c=1) for spherical molecules (c=1) The model can be
The model can be extended to associating components and mixtures of extended to associating components and mixtures of nonassociating compon
nonassociating components. Details are in the paper by ents. Details are in the paper by J.R. Elliott JrJ.R. Elliott Jret al.et al.
(1990) (1990)..[7][7]
Non-cubic equations of state Non-cubic equations of state Dieterici equation of state Dieterici equation of state where
whereaais associated with the interaction between molecules andis associated with the interaction between molecules andbbtakes intotakes into account the finite size of the molecules, similarly to the Van der Waals
account the finite size of the molecules, similarly to the Van der Waals equation.equation. The reduced coordinates are:
The reduced coordinates are:
Virial equations of state Virial equations of state Virial equation of state Virial equation of state Main article
Main article:: Virial expansionVirial expansion
Although usually not the
Although usually not the most convenient equation of state, the most convenient equation of state, the virial equationvirial equation is important because it can be derived directly from
is important because it can be derived directly from statistical mechanicsstatistical mechanics.. ThisThis equation is also called the
equation is also called the Kamerlingh OnnesKamerlingh Onnesequation. If appropriateequation. If appropriate assumptions are made about the mathematical form of intermolecular forces, assumptions are made about the mathematical form of intermolecular forces, theoretical expressi
theoretical expressions can be developed for each of ons can be developed for each of thethe coefficientscoefficients.. In this caseIn this case
B
Bcorresponds to interactions between pairs of corresponds to interactions between pairs of molecules,molecules,C C to triplets, and soto triplets, and so on. Accuracy can be
on. Accuracy can be increased indefinitely by considering higher order terms.increased indefinitely by considering higher order terms. The coefficients
The coefficientsBB,,C C ,,DD, etc. are functions of temperature only., etc. are functions of temperature only.
It can also be used to
It can also be used to work out the Boyle Temperature (the temperature at work out the Boyle Temperature (the temperature at which B = 0 and ideal gas laws apply) from a and b from the Van der Waals which B = 0 and ideal gas laws apply) from a and b from the Van der Waals equation of state, if you use the value for B
equation of state, if you use the value for B shown below:shown below:
The BWR equation of state The BWR equation of state Main article
Main article:: Benedict Benedict ––WebbWebb––Rubin equationRubin equation
where where
p
p= pressure= pressure
ρ
ρ= the molar density= the molar density
Values of the various parameters for 15 substances can be found in K.E. Starling Values of the various parameters for 15 substances can be found in K.E. Starling (1973).
(1973).Fluid Properties for Fluid Properties for Light Petroleum SystemsLight Petroleum Systems.. Gulf Publishing CompanyGulf Publishing Company.. Multiparameter equations of state
Multiparameter equations of state Helmholtz Function form
Helmholtz Function form Multiparameter equatio
Multiparameter equations of state ns of state (MEOS) can be used to (MEOS) can be used to represent pure fluidsrepresent pure fluids with high accuracy, in both the liquid and gaseous states. MEOS's represent the with high accuracy, in both the liquid and gaseous states. MEOS's represent the Helmholtz function of the fluid as the sum of ideal gas
Helmholtz function of the fluid as the sum of ideal gas and residual terms. Bothand residual terms. Both terms are explicit in reduced temperature and reduced density - thus:
terms are explicit in reduced temperature and reduced density - thus: Where:
Where:
The reduced density and temperature are typically, though not always, the The reduced density and temperature are typically, though not always, the critical values for the
critical values for the pure fluid. Other thermodynamic functions can be derivedpure fluid. Other thermodynamic functions can be derived from the MEOS by using
from the MEOS by using appropriate derivatives of the Helmholtz function;appropriate derivatives of the Helmholtz function; hence, because integration of the MEOS is not required, there are few hence, because integration of the MEOS is not required, there are few restrictions as to the functional form of the ideal or residual terms. Typical restrictions as to the functional form of the ideal or residual terms. Typical MEOS use upwards of 50 fluid specific parameters, but are able to represent the MEOS use upwards of 50 fluid specific parameters, but are able to represent the fluid's properties with high accuracy. MEOS are available c
fluid's properties with high accuracy. MEOS are available c urrently for about 50urrently for about 50 of the most common industrial fluids including refrigerants. Mixture models of the most common industrial fluids including refrigerants. Mixture models also exist.
also exist.
Other equations of state of
Other equations of state of interest interest Stiffened equation of state
Stiffened equation of state
When considering water under very high
When considering water under very high pressures (typical applications arepressures (typical applications are underwater nuclear explosions
underwater nuclear explosions,, sonic shock lithotripsysonic shock lithotripsy,, andand sonoluminescencesonoluminescence)) the stiffened equation of state is often used:
the stiffened equation of state is often used: where
where is is the the internal internal energy energy per per unit unit mass, mass, is is an an empirically empirically determineddetermined constant
constant typically typically taken taken to to be be about about 6.1, 6.1, and and is is another another constant,constant,
representing the molecular attraction between water molecules. The magnitude representing the molecular attraction between water molecules. The magnitude of the correction is about
of the correction is about 2 gigapascals (20,000 atmospheres).2 gigapascals (20,000 atmospheres).
The equation is stated in this form because the speed of sound in water The equation is stated in this form because the speed of sound in water is givenis given by
by ..
Thus water behaves as though it is an ideal gas t
Thus water behaves as though it is an ideal gas t hat ishat isalready already under about under about
20,000 atmospheres (2 GPa) pressure, and explains why
20,000 atmospheres (2 GPa) pressure, and explains why water is commonlywater is commonly assumed to be incompressible: when the external pressure changes from 1 assumed to be incompressible: when the external pressure changes from 1 atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an atmosphere to 2 atmospheres (100 kPa to 200 kPa), the water behaves as an ideal gas would when
ideal gas would when changing from 20,001 to changing from 20,001 to 20,002 atmospheres20,002 atmospheres (2000.1 MPa to 2000.2
(2000.1 MPa to 2000.2 MPa).MPa). This equation mispredicts the
This equation mispredicts the specific heat capacityspecific heat capacityof water but few simpleof water but few simple alternatives are available for severely nonisentropic processes such as strong alternatives are available for severely nonisentropic processes such as strong shocks.
shocks. Ultrarelativisti
Ultrarelativistic equation of c equation of statestate An
An ultrarelativistiultrarelativistic c fluidfluidhas equation of statehas equation of state where
where is is the the pressure, pressure, is is the the mass mass density, density, and and is is thethe speed of soundspeed of sound.. Ideal Bose equation of state
Ideal Bose equation of state The equation of state for an idea
The equation of state for an ideall Bose gasBose gasisis
where α is an exponent specific to the
where α is an exponent specific to the system (e.g. in the absence of a potentialsystem (e.g. in the absence of a potential field, α=3/2),
field, α=3/2), z z is exp(is exp( μ μ//kT kT ) where) where μ μis theis the chemical potentialchemical potential,, Li is theLi is the
polylogarithm
polylogarithm,, ζ is theζ is the Riemann zeta functionRiemann zeta function,, andandT T ccis the critical temperatureis the critical temperature
at which a
at which a BoseBose––Einstein condensateEinstein condensatebegins to form.begins to form. Equations of state for solids
Equations of state for solids Rose
Rose––Vinet equation of stateVinet equation of state
Birch
Birch––Murnaghan equation of stateMurnaghan equation of state
Johnson
Johnson––Holmquist damage modelHolmquist damage model
Mie-Gruneisen equation of state
Mie-Gruneisen equation of state
General Expressions General Expressions
General Expressions for the
General Expressions for the EnthalpyEnthalpyH H , the, the EntropyEntropyS S and theand the Gibbs EnergyGibbs EnergyGG
are given b are given byy[1][1]
