Table of Contents
- Monte Carlo: adsorption of binary mixture of CO₂ and methane in IRMOF-1
- Monte Carlo: NPT CO₂
- Monte Carlo: NPT propane
- Monte Carlo: Gibbs CO₂
- Molecular Dynamics: benzene in IRMOF-1
- Monte Carlo: adsorption CO₂ in LTA-4A sodium
- Molecular Dynamics: diffusion of CO₂ in LTA-4A sodium
- Monte Carlo: adsorption of C6-isomers in MFI
Monte Carlo: adsorption of binary mixture of CO₂ and methane in IRMOF-1
Adsorption of a binary mixture is specified at a total pressure and individual mol-fractions for the components.
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"ExternalTemperature" : 300.0,
"ExternalPressure" : 1e6
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"Name" : "methane",
"MolFraction" : 0.75,
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The fugacity \(f_i\) of component \(i\) in a gas-mixture is given by $$f_i = \gamma_i p_i$$ where \(\gamma_i\) is the fugacity coefficient and \(p_i\) the partial pressure of component \(i\).
The use of fugacity is prefered over using chemical potentials. For mixtures, the chemical potential diverges when the molar fraction of a component goes to zero. In contrast, the fugacity goes to zero. In addition, there is a very natural reference state for fugacity, namely an ideal gas. The fugacity and pressure converge at low pressure $$\lim_{p\rightarrow0}\frac{f_i}{p_i} = 1$$
Using the Peng-Robinson equation of the state, the fugacity coefficients and the number of excess molecules are computed:
Component 0 [CO2]
Mol-fraction: 0.25 [-]
Fugacity: 237587.6177345715 [Pa]
Fugacity coefficient: 0.9503504709382861 [-]
Compressibility: 0.9714389724700717 [-]
Excess molecules: 0.8675190741449798 [-]
Component 1 [methane]
Mol-fraction: 0.75 [-]
Fugacity: 734258.9620174329 [Pa]
Fugacity coefficient: 0.9790119493565772 [-]
Compressibility: 0.9714389724700717 [-]
Excess molecules: 2.6025572224349394 [-]
Appreciable adsorption in MOF materials occurs at higher pressure than zeolites, usually in the range up to 10 bar. At these high pressures absolute and excess adsorption are different, and excess adsorption eventually even goes down. This is due to the fact that excess adsorption is relative to what would have been in the free pore volume at these conditions.
To compute the excess adsorption the void fraction of a structure needs to be specified using ‘HeliumVoidFraction’. RASPA automatically uses an equation of state (default: Peng-Robinson) to compute the fugacities from the pressure and mol-fraction as is done here for a mixture of CO₂ and methane. It also computes the amount of excess molecules from this equation of state. It also computes the amount of excess molecules from this equation of state.
Component 0 (CO2)
Block[ 0] 1.363929e+01
Block[ 1] 1.355632e+01
Block[ 2] 1.360550e+01
Block[ 3] 1.362871e+01
Block[ 4] 1.360001e+01
---------------------------------------------------------------------------
Abs. loading average 1.360596e+01 +/- 3.988351e-02 [molecules/cell]
Abs. loading average 1.360596e+01 +/- 3.988351e-02 [molecules/uc]
Abs. loading average 2.209264e+00 +/- 6.476074e-03 [mol/kg-framework]
Abs. loading average 9.720498e+01 +/- 2.849395e-01 [mg/g-framework]
Block[ 0] 1.277177e+01
Block[ 1] 1.268880e+01
Block[ 2] 1.273798e+01
Block[ 3] 1.276119e+01
Block[ 4] 1.273249e+01
---------------------------------------------------------------------------
Excess loading average 1.273844e+01 +/- 3.988351e-02 [molecules/cell]
Excess loading average 1.273844e+01 +/- 3.988351e-02 [molecules/uc]
Excess loading average 2.068401e+00 +/- 6.476074e-03 [mol/kg-framework]
Excess loading average 9.100717e+01 +/- 2.849395e-01 [mg/g-framework]
Component 1 (methane)
Block[ 0] 2.059514e+01
Block[ 1] 2.054633e+01
Block[ 2] 2.054283e+01
Block[ 3] 2.053066e+01
Block[ 4] 2.056720e+01
---------------------------------------------------------------------------
Abs. loading average 2.055643e+01 +/- 3.143894e-02 [molecules/cell]
Abs. loading average 2.055643e+01 +/- 3.143894e-02 [molecules/uc]
Abs. loading average 3.337845e+00 +/- 5.104890e-03 [mol/kg-framework]
Abs. loading average 5.354724e+01 +/- 8.189499e-02 [mg/g-framework]
Block[ 0] 1.799258e+01
Block[ 1] 1.794377e+01
Block[ 2] 1.794027e+01
Block[ 3] 1.792810e+01
Block[ 4] 1.796464e+01
---------------------------------------------------------------------------
Excess loading average 1.795387e+01 +/- 3.143894e-02 [molecules/cell]
Excess loading average 1.795387e+01 +/- 3.143894e-02 [molecules/uc]
Excess loading average 2.915255e+00 +/- 5.104890e-03 [mol/kg-framework]
Excess loading average 4.676786e+01 +/- 8.189499e-02 [mg/g-framework]
RASPA also computed the enthalpies of adsorption for the components in the mixture:
Component 0 [CO2]
-------------------------------------------------------------------------------
Block[ 0] -1.856496e+03
Block[ 1] -1.854692e+03
Block[ 2] -1.852168e+03
Block[ 3] -1.852296e+03
Block[ 4] -1.853725e+03
---------------------------------------------------------------------------
Enthalpy of adsorption: -1.853875e+03 +/- 2.235383e+00 [K]
-1.541398e+01 +/- 1.858601e-02 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Component 1 [methane]
-------------------------------------------------------------------------------
Block[ 0] -1.404500e+03
Block[ 1] -1.406274e+03
Block[ 2] -1.407237e+03
Block[ 3] -1.409140e+03
Block[ 4] -1.402377e+03
---------------------------------------------------------------------------
Enthalpy of adsorption: -1.405906e+03 +/- 3.213472e+00 [K]
-1.168935e+01 +/- 2.671830e-02 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Using Widom insertion we can verify that the measured fugacities are close to the imposed ones:
Component 0 [CO2]
Imposed fugacity: 2.375876e+05 [Pa]
Measured fugacity: 2.308130e+05 +/- 7.343665e+02 [Pa]
Component 1 [methane]
Imposed fugacity: 7.342590e+05 [Pa]
Measured fugacity: 7.143740e+05 +/- 1.022102e+03 [Pa]
Monte Carlo: NPT CO₂
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Block[ 0] 256
Block[ 1] 256
Block[ 2] 256
Block[ 3] 256
Block[ 4] 256
-----------------------------------------------------------------------
Density average 2.560000e+02 +/- 0.000000e+00 [molecules]
Density average 1.278004e-02 +/- 4.137859e-05 [molec/A^3]
Density average 9.337315e+02 +/- 3.023191e+00 [kg.m⁻³]
Average pressure tensor:
-------------------------------------------------------------------------------
9.9463e+01 -7.4457e-01 -2.1266e-01 +/- 3.2169e+00 2.6831e+00 3.3564e+00 [bar]
-7.4457e-01 9.9282e+01 -2.3006e-01 +/- 2.6831e+00 2.6834e+00 3.4452e+00 [bar]
-2.1266e-01 -2.3006e-01 1.0180e+02 +/- 3.3564e+00 3.4452e+00 2.9063e+00 [bar]
Block[ 0] 4.424693e+07
Block[ 1] 4.406916e+07
Block[ 2] 4.396276e+07
Block[ 3] 4.420751e+07
Block[ 4] 4.407319e+07
---------------------------------------------------------------------------
Ideal gas pressure 4.411191e+07 +/- 8.601021e-03 [Pa]
4.411191e+02 +/- 1.428234e+00 [bar]
Block[ 0] -3.432308e+07
Block[ 1] -3.413620e+07
Block[ 2] -3.374362e+07
Block[ 3] -3.428751e+07
Block[ 4] -3.397906e+07
---------------------------------------------------------------------------
Excess pressure -3.409390e+07 +/- 2.961885e+05 [Pa]
-3.409390e+02 +/- 2.961885e+00 [bar]
Block[ 0] 9.923855e+06
Block[ 1] 9.932958e+06
Block[ 2] 1.021914e+07
Block[ 3] 9.919992e+06
Block[ 4] 1.009413e+07
---------------------------------------------------------------------------
Pressure average 1.001802e+07 +/- 1.665082e+05 [Pa]
1.001802e+02 +/- 1.665082e+00 [bar]
Monte-Carlo moves statistics
===============================================================================
Volume change all: 4197072
Volume change total: 4197072
Volume change constructed: 4197072
Volume change accepted: 2099085
Volume change fraction: 0.500131
Volume change max-change: 0.027143
Component 0 [CO2]
Reinsertion (CBMC) all: 83934567
Reinsertion (CBMC) total: 83934567
Reinsertion (CBMC) constructed: 81465120
Reinsertion (CBMC) accepted: 6662598
Reinsertion (CBMC) fraction: 0.079378
Reinsertion (CBMC) max-change: 0.000000
Translation all: 83927109
Translation total: 27977763 27975663 27973683
Translation constructed: 27970649 27968514 27966458
Translation accepted: 10966567 10967397 10961668
Translation fraction: 0.391974 0.392033 0.391856
Translation max-change: 0.010000 0.010000 0.010000
Rotation all: 83941252
Rotation total: 27986886 27978440 27975926
Rotation constructed: 27979687 27971583 27968722
Rotation accepted: 10969893 10970338 10971701
Rotation fraction: 0.391965 0.392100 0.392184
Rotation max-change: 0.010000 0.010000 0.010000
Monte Carlo: NPT propane
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Component 0 (propane)
Block[ 0] 256
Block[ 1] 256
Block[ 2] 256
Block[ 3] 256
Block[ 4] 256
-----------------------------------------------------------------------
Density average 2.560000e+02 +/- 0.000000e+00 [molecules]
Density average 7.761091e-03 +/- 3.026467e-06 [molec/A^3]
Density average 5.682864e+02 +/- 2.216054e-01 [kg.m⁻³]
Average pressure tensor:
-------------------------------------------------------------------------------
9.0757e+00 -8.8946e-02 2.4468e-02 +/- 7.4469e-01 1.8894e+00 9.6681e-01 [bar]
-8.8946e-02 9.6946e+00 -8.2002e-02 +/- 1.8894e+00 2.2443e+00 3.2412e+00 [bar]
2.4468e-02 -8.2002e-02 1.0125e+01 +/- 9.6681e-01 3.2412e+00 3.8172e+00 [bar]
Block[ 0] 2.678430e+07
Block[ 1] 2.678570e+07
Block[ 2] 2.678683e+07
Block[ 3] 2.680309e+07
Block[ 4] 2.678200e+07
---------------------------------------------------------------------------
Ideal gas pressure 2.678838e+07 +/- 6.290863e-04 [Pa]
2.678838e+02 +/- 1.044623e-01 [bar]
Block[ 0] -2.581568e+07
Block[ 1] -2.558996e+07
Block[ 2] -2.607333e+07
Block[ 3] -2.588163e+07
Block[ 4] -2.576545e+07
---------------------------------------------------------------------------
Excess pressure -2.582521e+07 +/- 2.183639e+05 [Pa]
-2.582521e+02 +/- 2.183639e+00 [bar]
Block[ 0] 9.686217e+05
Block[ 1] 1.195745e+06
Block[ 2] 7.134989e+05
Block[ 3] 9.214610e+05
Block[ 4] 1.016545e+06
---------------------------------------------------------------------------
Pressure average 9.631744e+05 +/- 2.159625e+05 [Pa]
9.631744e+00 +/- 2.159625e+00 [bar]
Monte-Carlo moves statistics
===============================================================================
Volume change all: 2536350
Volume change total: 2536350
Volume change constructed: 2536350
Volume change accepted: 1266056
Volume change fraction: 0.499165
Volume change max-change: 0.020876
Component 0 [propane]
Reinsertion (CBMC) all: 50693568
Reinsertion (CBMC) total: 50693568
Reinsertion (CBMC) constructed: 38269580
Reinsertion (CBMC) accepted: 301948
Reinsertion (CBMC) fraction: 0.005956
Reinsertion (CBMC) max-change: 0.000000
Partial reinsertion (CBMC) all: 50699781
Partial reinsertion (CBMC) total: 50699781
Partial reinsertion (CBMC) constructed: 50689654
Partial reinsertion (CBMC) accepted: 25051762
Partial reinsertion (CBMC) fraction: 0.494120
Partial reinsertion (CBMC) max-change: 0.000000
Widom all: 50691571
Widom total: 50691571
Widom constructed: 37774707
Widom accepted: 0
Widom fraction: 0.000000
Widom max-change: 0.000000
Translation all: 50686797
Translation total: 16893938 16891675 16901184
Translation constructed: 16893938 16891675 16901184
Translation accepted: 8444975 8442754 8454909
Translation fraction: 0.499882 0.499817 0.500255
Translation max-change: 0.759807 0.749178 0.739907
Rotation all: 50691933
Rotation total: 16896569 16894845 16900519
Rotation constructed: 16896542 16894825 16900495
Rotation accepted: 8444825 8453701 8457582
Rotation fraction: 0.499795 0.500372 0.500433
Rotation max-change: 0.604939 0.601181 0.601781
Monte Carlo: Gibbs CO₂
We can compute Vapor-Liquid Equilibrium (VLE) using the Gibbs-ensemble.
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For each box we obtain a computed density (both the number of molecules and the volume of the boxes change), computing to a vapor phase and a liquid phase. For the gas-phase we obtain
Component 0 (CO2)
Block[ 0] 8.581800000000001
Block[ 1] 11.114
Block[ 2] 9.5334
Block[ 3] 10.2088
Block[ 4] 8.6978
-----------------------------------------------------------------------
Density average 9.627160e+00 +/- 1.318860e+00 [molecules]
Density average 4.605977e-04 +/- 5.480961e-05 [molec/A^3]
Density average 3.365205e+01 +/- 4.004484e+00 [kg.m⁻³]
and for the liquid phase we obtain
Block[ 0] 503.4182
Block[ 1] 500.886
Block[ 2] 502.4666
Block[ 3] 501.7912
Block[ 4] 503.3022
-----------------------------------------------------------------------
Density average 5.023728e+02 +/- 1.318860e+00 [molecules]
Density average 1.512513e-02 +/- 1.232200e-04 [molec/A^3]
Density average 1.105068e+03 +/- 9.002663e+00 [kg.m⁻³]
Uisng the NIST chemical database, we can compare to the experimental values of 33.295 and 1088.9 kg/m³. The experimental pressure from NIST is 12.825 bar. From the simulations we obtain \(13.08 \pm 1.3\) and \(12.27 \pm 9.5\) bar for the vapor and the liquid phases, respectively. Note that in equilibrium, the temperatures, chemical potentials, and pressures of the phases is equal, but the vapor-phase data has significantly lower error bars.
Molecular Dynamics: benzene in IRMOF-1
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Monte Carlo: adsorption CO₂ in LTA-4A sodium
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[0.0, 0.5, 0.5, 4.0],
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[0.25, 0.0, 0.0, 1.0],
[0.75, 0.0, 0.0, 1.0],
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[0.75, 0.5, 0.0, 1.0],
[0.0, 0.25, 0.0, 1.0],
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[0.0, 0.25, 0.5, 1.0],
[0.0, 0.75, 0.5, 1.0],
[0.5, 0.25, 0.5, 1.0],
[0.5, 0.75, 0.5, 1.0],
[0.5, 0.0, 0.25, 1.0],
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[0.5, 0.5, 0.25, 1.0],
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Molecular Dynamics: diffusion of CO₂ in LTA-4A sodium
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[0.5, 0.5, 0.0, 4.0],
[0.0, 0.0, 0.5, 4.0],
[0.5, 0.0, 0.5, 4.0],
[0.0, 0.5, 0.5, 4.0],
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Monte Carlo: adsorption of C6-isomers in MFI
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],
"Components" : [
{
"Name" : "hexane",
"MolFraction" : 0.2,
"FugacityCoefficient" : 1.0,
"IdealGasRosenbluthWeight" : 8.099334e-3,
"TranslationProbability" : 0.5,
"RotationProbability" : 0.5,
"ReinsertionProbability" : 0.5,
"PartialReinsertionProbability" : 0.5,
"SwapProbability" : 1.0,
"WidomProbability" : 1.0,
"CreateNumberOfMolecules" : 0
},
{
"Name" : "2-methylpentane",
"MolFraction" : 0.2,
"FugacityCoefficient" : 1.0,
"IdealGasRosenbluthWeight" : 3.748358e-2,
"TranslationProbability" : 0.5,
"RotationProbability" : 0.5,
"ReinsertionProbability" : 0.5,
"PartialReinsertionProbability" : 0.5,
"SwapProbability" : 1.0,
"WidomProbability" : 1.0,
"CreateNumberOfMolecules" : 0
},
{
"Name" : "3-methylpentane",
"MolFraction" : 0.2,
"FugacityCoefficient" : 1.0,
"IdealGasRosenbluthWeight" : 5.348429e-2,
"TranslationProbability" : 0.5,
"RotationProbability" : 0.5,
"ReinsertionProbability" : 0.5,
"PartialReinsertionProbability" : 0.5,
"SwapProbability" : 1.0,
"WidomProbability" : 1.0,
"CreateNumberOfMolecules" : 0
},
{
"Name" : "22-methylbutane",
"MolFraction" : 0.2,
"FugacityCoefficient" : 1.0,
"IdealGasRosenbluthWeight" : 1.347700e-1,
"TranslationProbability" : 0.5,
"RotationProbability" : 0.5,
"ReinsertionProbability" : 0.5,
"PartialReinsertionProbability" : 0.5,
"SwapProbability" : 1.0,
"WidomProbability" : 1.0,
"CreateNumberOfMolecules" : 0
},
{
"Name" : "23-methylbutane",
"MolFraction" : 0.2,
"FugacityCoefficient" : 1.0,
"IdealGasRosenbluthWeight" : 5.814672e-2,
"TranslationProbability" : 0.5,
"RotationProbability" : 0.5,
"ReinsertionProbability" : 0.5,
"PartialReinsertionProbability" : 0.5,
"SwapProbability" : 1.0,
"WidomProbability" : 1.0,
"CreateNumberOfMolecules" : 0
}
]
}
Component 0 (hexane)
Block[ 0] 1.490829e+01
Block[ 1] 1.430523e+01
Block[ 2] 1.503197e+01
Block[ 3] 1.545259e+01
Block[ 4] 1.498112e+01
---------------------------------------------------------------------------
Abs. loading average 1.493584e+01 +/- 5.106443e-01 [molecules/cell]
Abs. loading average 1.866980e+00 +/- 6.383054e-02 [molecules/uc]
Abs. loading average 3.237103e-01 +/- 1.106740e-02 [mol/kg-framework]
Abs. loading average 2.792156e+01 +/- 9.546156e-01 [mg/g-framework]
Component 1 (2-methylpentane)
Block[ 0] 8.307880e+00
Block[ 1] 8.050410e+00
Block[ 2] 6.714440e+00
Block[ 3] 8.900490e+00
Block[ 4] 8.413380e+00
---------------------------------------------------------------------------
Abs. loading average 8.077320e+00 +/- 1.020321e+00 [molecules/cell]
Abs. loading average 1.009665e+00 +/- 1.275401e-01 [molecules/uc]
Abs. loading average 1.750629e-01 +/- 2.211381e-02 [mol/kg-framework]
Abs. loading average 1.510001e+01 +/- 1.907421e+00 [mg/g-framework]
Component 2 (3-methylpentane)
Block[ 0] 1.430696e+01
Block[ 1] 1.429370e+01
Block[ 2] 1.391482e+01
Block[ 3] 1.212496e+01
Block[ 4] 1.425438e+01
---------------------------------------------------------------------------
Abs. loading average 1.377896e+01 +/- 1.165249e+00 [molecules/cell]
Abs. loading average 1.722371e+00 +/- 1.456561e-01 [molecules/uc]
Abs. loading average 2.986369e-01 +/- 2.525490e-02 [mol/kg-framework]
Abs. loading average 2.575886e+01 +/- 2.178355e+00 [mg/g-framework]
Component 3 (22-methylbutane)
Block[ 0] 5.939000e-01
Block[ 1] 5.042200e-01
Block[ 2] 4.890300e-01
Block[ 3] 6.582400e-01
Block[ 4] 5.171300e-01
---------------------------------------------------------------------------
Abs. loading average 5.525040e-01 +/- 8.888695e-02 [molecules/cell]
Abs. loading average 6.906300e-02 +/- 1.111087e-02 [molecules/uc]
Abs. loading average 1.197464e-02 +/- 1.926482e-03 [mol/kg-framework]
Abs. loading average 1.032869e+00 +/- 1.661682e-01 [mg/g-framework]
Component 4 (23-methylbutane)
Block[ 0] 1.731293e+01
Block[ 1] 1.816057e+01
Block[ 2] 1.947983e+01
Block[ 3] 1.617831e+01
Block[ 4] 1.640655e+01
---------------------------------------------------------------------------
Abs. loading average 1.750764e+01 +/- 1.680945e+00 [molecules/cell]
Abs. loading average 2.188455e+00 +/- 2.101182e-01 [molecules/uc]
Abs. loading average 3.794499e-01 +/- 3.643179e-02 [mol/kg-framework]
Abs. loading average 3.272936e+01 +/- 3.142416e+00 [mg/g-framework]
Component 0 [hexane]
-------------------------------------------------------------------------------
Block[ 0] 3.023371e+04
Block[ 1] 2.893068e+04
Block[ 2] 3.183916e+04
Block[ 3] 3.213888e+04
Block[ 4] 2.266373e+04
---------------------------------------------------------------------------
Enthalpy of adsorption: 2.916123e+04 +/- 4.786429e+03 [K]
2.424600e+02 +/- 3.979660e+01 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Component 1 [2-methylpentane]
-------------------------------------------------------------------------------
Block[ 0] 9.688533e+04
Block[ 1] 8.374968e+04
Block[ 2] 9.996776e+04
Block[ 3] 8.919909e+04
Block[ 4] 9.312475e+04
---------------------------------------------------------------------------
Enthalpy of adsorption: 9.258532e+04 +/- 7.920520e+03 [K]
7.697974e+02 +/- 6.585488e+01 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Component 2 [3-methylpentane]
-------------------------------------------------------------------------------
Block[ 0] 1.024919e+05
Block[ 1] 8.955039e+04
Block[ 2] 1.003299e+05
Block[ 3] 9.843426e+04
Block[ 4] 9.285472e+04
---------------------------------------------------------------------------
Enthalpy of adsorption: 9.673223e+04 +/- 6.674752e+03 [K]
8.042768e+02 +/- 5.549699e+01 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Component 3 [22-methylbutane]
-------------------------------------------------------------------------------
Block[ 0] 1.596507e+05
Block[ 1] 1.368162e+05
Block[ 2] 1.453298e+05
Block[ 3] 1.517618e+05
Block[ 4] 1.402389e+05
---------------------------------------------------------------------------
Enthalpy of adsorption: 1.467595e+05 +/- 1.135511e+04 [K]
1.220226e+03 +/- 9.441164e+01 [kJ/mol]
Note: need to subtract the ideal-gas energy.
Component 4 [23-methylbutane]
-------------------------------------------------------------------------------
Block[ 0] 1.775168e+05
Block[ 1] 1.672175e+05
Block[ 2] 1.729822e+05
Block[ 3] 1.777353e+05
Block[ 4] 1.721485e+05
---------------------------------------------------------------------------
Enthalpy of adsorption: 1.735201e+05 +/- 5.398403e+03 [K]
1.442727e+03 +/- 4.488483e+01 [kJ/mol]
Note: need to subtract the ideal-gas energy.
1.9.8