The use of gas-liquid chromatography to determine activity coefficients and second virial coefficients of mixtures I. Theory and verification of method of data analysis

1966 ◽  
Vol 295 (1442) ◽  
pp. 259-270 ◽  
Keyword(s):  
Numerical Integration ◽  
Retention Volume ◽  
Simple Extension ◽  
Gas Liquid Chromatography ◽  
Local Pressure ◽  
Integration Procedure ◽  
Carrier Gas ◽  
Wide Range ◽  
Analytical Integration ◽  
Numerical Integration Procedure

This paper reformulates the differential equation describing the local elution rate in a g. l. c. column in terms of the local pressure and the carrier gas outlet flow rate. Analytical integration for an ideal carrier gas suggests an accurate method for extrapolating a function of the retention volume linearly to zero pressure, where the intercept V ° N is simply related to the thermodynamic activity coefficient of the solute (1) in the stationary liquid (3) and the gradient β gives B 12 for the mixture solute + carrier gas (2). We argue that a simple extension of the method should apply also, with fair accuracy, to a non-ideal carrier gas. We support this argument with data obtained by a numerical integration procedure which gives retention volume in terms of specified V ° N and B for a range of inlet and outlet pressures. The reliability of the numerical integration procedure is established by comparing results for the ideal gas case with the results of analytical integration. The retention volumes obtained by numerical integration for a non-ideal carrier gas are then treated as ‘experimental’ observations, using in addition to our extrapolation procedure, two previously published procedures. Our procedures are consistently more successful than the others and recover accurately the V ° N originally specified over a wide range of flow conditions, even when the carrier gas shows large deviations from ideality. In the case of β , our method is significantly in error only when the carrier gas deviates largely from ideality in a low pressure column with large pressure drop. A simple refinement of our method is satisfactory for even this case.

A first solution, for LiH, of a molecular transcorrelated wave equation by means of restricted numerical integration

1969 ◽  
Vol 311 (1505) ◽  
pp. 309-329 ◽  
Keyword(s):  
Wave Equation ◽  
Numerical Integration ◽  
Correlation Factor ◽  
Successful Implementation ◽  
Wide Range ◽  
The Many ◽  
Reacting Systems ◽  
Set Of Points ◽  
Ordinary Integral ◽  
Numerical Integration Procedure

The first calculation of a molecular wavefunction and energy by the solution of the appropriate transcorrelated wave equation ( C -1 HC – W )ϕ = 0 has been made for LiH. The results are in accord with the high accuracy found for the Ne atom by Handy & Boys (1968 c ). The spherical symmetry of the Ne problem gave such advantages that the integrals could virtually be evaluated exactly. The new problem of evaluating these integrals for functions about the many nuclei has now been overcome by a particular numerical integration procedure which gives a much higher accuracy in the energy than corresponds to the accuracy of integration for an ordinary integral. Hence a set of points which is much more restricted in number than otherwise can be used. The error per electron is nearly as small as 1 % of a bond energy, and it is reasonable to expect that later applications of this method will easily surpass this accuracy for a wide range of molecules, and for reacting systems. The method depends on the direct introduction of functions of r ij into the correlation factor in a way applicable to any molecule and the successful implementation of this for the first molecule may provide a turning point in methods of theoretical chemical prediction. The preparation of the whole set of programs requires less than one man year and the general characteristics are those of a method which is much simpler than the previous methods which do not appear capable of this level of accuracy.

scholarly journals Numerical Integration of Weight Loss Curves for Kinetic Analysis

Thermo ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 32-44
Author(s):  
Juan A. Conesa
Keyword(s):  
Weight Loss ◽  
Kinetic Analysis ◽  
Kinetic Parameters ◽  
Numerical Integration ◽  
Integration Procedure ◽  
Actual Temperature ◽  
Complex Processes ◽  
Decomposition Curve ◽  
Numerical Integration Procedure

Research abounds in the literature on kinetic analyses using thermogravimetric (TG) runs. Many of these studies use approximations of integral or derivative forms of the kinetic law and all of them use programmed temperatures. In the present work, a numerical integration procedure was discussed and applied to different examples. We focused on materials presenting a single decomposition curve as well as other materials with more complex processes. Different examples were explored, and the methodology was applied to a number of wastes such as coffee husks, polystyrene and polyethylene. In all cases, the actual temperature measured by thermocouples close to the sample is used, and several runs are fitted using the same kinetic parameters, giving robustness to the results.

scholarly journals Evaluation of astrophysically useful parameters for strontium monohydride and deuteride

2006 ◽  
pp. 13-16 ◽  
Author(s):  
Sri Ramachandran ◽  
N. Rajamanickam ◽  
S.P. Bagare
Keyword(s):  
Numerical Integration ◽  
Transition Probabilities ◽  
Integration Procedure ◽  
Giant Stars ◽  
Condon Factors ◽  
Astrophysical Molecules ◽  
Franck Condon ◽  
Numerical Integration Procedure

The Franck-Condon factors and r-centroids which are very closely related to transition probabilities, have been evaluated by the more reliable numerical integration procedure for the band systems B2?-X2?, C2?-X2? and F2?-X2? of astrophysical molecules strontium monohydride and strontium deuteride using an adequate potential. The Franck-Condon factors are more intense, particularly for the ?? = 0 bands, for all the systems examined here. Thus the bands of the molecules are expected to be present in sunspot spectra, SC-stars, cool M-giant stars and other prominent astrophysical sources.

Franck-Condon Factors and R-Centroids for the Band System a1Π-X1Σ+ of the InH Molecule

1993 ◽  
Vol 58 (7) ◽  
pp. 1491-1494 ◽  
Author(s):  
Narayanan Rajamanickam ◽  
Thangamariappan Murali ◽  
Thangasamy Sakthivel ◽  
Manuel Fernandez Gomez ◽  
Juan Jesus Lopez Gonzalez
Keyword(s):  
Numerical Integration ◽  
Transition Probabilities ◽  
Band System ◽  
Vibrational Transition ◽  
Integration Procedure ◽  
Condon Factors ◽  
Franck Condon ◽  
Suitable Potential ◽  
Numerical Integration Procedure

The Franck-Condon factors (vibrational transition probabilities) and r-centroids have been evaluated by a numerical integration procedure for the bands of the a3Π1-X1Σ+ system of the InH molecule using a suitable potential.

Studies of Polymer Properties via Inverse Gas Chromatography

1996 ◽  
Vol 69 (3) ◽  
pp. 347-376 ◽  
Author(s):  
Bincai(Pun Choi) Li
Keyword(s):  
Gas Chromatography ◽  
Stationary Phase ◽  
Mobile Phase ◽  
Inverse Gas Chromatography ◽  
Amorphous Polymer ◽  
Retention Volume ◽  
Polymer Surface ◽  
Gas Liquid Chromatography ◽  
Solvent Vapor ◽  
Carrier Gas

Abstract Gas Chromatosraphy (GC) using a polymer as the stationary phase to reveal the properties of the polymer — known as Inverse Gas Chromatography (IGC) — is in contrast to conventional GC where gaseous components in the mobile phase are separated and studied. Figure l(a) and l(b) are schematic diagrams showing the arrangement of apparatus in a gas Chromatograph for IGC. The column is filled with packings consisting of thin layer of polymer coated onto an inert support, typically Chromosorb W, Chromosorb G (70 ∼ 80 mesh, acid washed and dimethyldichlorosilane treated), or Teflon. The carrier gas, such as N2, H2, or He, acts as the mobile phase. The solvent, injected as a sharp pulse and vaporized immediately into the carrier gas stream at the entrance of the column, is called the probe. As the probe is carried forward, it is partitioned between the mobile gas phase and the stationary polymer phase. The time required to elute the probe through the column is called the retention time (elution is monitored in the detector and reflected on the recorder or integrator as a peak maximum). The corresponding amount of carrier gas needed is called the retention volume. The detector for the probe may be a thermal conductivity cell (TCD) or flame ionization detector (FID). When an FID is used, the flow of gas is diverted to the flow meter before it reaches the detector as shown in Figure l(b). Some notes on the experimental techniques will be discussed in Section IX. GC has been classified into Gas-Liquid Chromatography (GLC) and Gas-Solid Chromatography (GSC) according to whether the stationary phase is a liquid or a solid, respectively. In IGC, the process is GLC when the temperature of the polymer under investigation is far above its glass transition temperature Tg. The retention is due to absorption of the solvent vapor into the polymer bulk (an amorphous polymer above Tg is viewed as a liquid). When the temperature of the polymer is well below its Tg, the process is GSC and the retention mechanism becomes adsorption of the vapor onto the polymer surface. We shall initially discuss the GLC of polymers and then extend our discussions to GSC. Important applications of IGC to polymer research have been the studies of the thermodynamics of polymer-solvent and polymer-polymer interactions via GLC.

Sign in / Sign up

Export Citation Format

Share Document