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THE DIFFERENTIAL EQUATIONS OF CONSECUTIVE REACTIONS
Canadian Journal of Chemistry
Abstract
The differential equations of consecutive first-order reactions are formulated in a general manner. The solutions of these equations are shown to fall into a pattern, according to which the solutions to all such systems can be written down without resort to formal integration. The pattern is also convenient for making numerical calculations.
... Murphy [33] studied consecutive first and second order reactions and determined their exact solution of rate equations. Westman and DeLury [34] discussed the ordinary differential equations of consecutive reactions. Chrastil [35] determined the rate constants of the first order consecutive reaction with the help of final product. ...
Mathematics Subject Classification: 44A05 35F40 97M10 97M60 a b s t r a c t The problem of motion of coupled harmonic oscillators can be modeled in the terms of system of ordinary differential equations (SODEs). In the present paper, authors used Sawi transformation for finding the solution of SODEs with application to determine the concentration of chemical reactants of chemical reaction in series. Authors have considered four numerical problems and determined their solutions step by step for explaining the complete method. After it, authors have taken a problem of physical chemistry for determining the concentration of chemical reactants (substances) of chemical reaction in series. Authors have obtained the solution of this problem by developing its mathematical model in terms of SODEs and then applying Sawi transformation on it. Results of this problem are visualized using tables and graphs. Results of the present paper show that the Sawi transformation provides us the exact solution of SODEs without large computational work. Ó 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Ain Shams University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/ by-nc-nd/4.0/).
We report high-spin aminyl triradicals with near-planar triphenylene backbones. Near-planarity of the fused aminyl radicals and the 2,6,10-triphenylene ferromagnetic coupling unit (FCU), magnetically equivalent to three fused 3,4'-biphenyl FCUs, assures an effective 2pπ-2pπ overlap within the cross-conjugated π-system, leading to an S = 3/2 (quartet) ground state that is well separated from low-spin excited doublet states. Thermal populations of the low-spin (S = 1/2) excited states are detectable both by SQUID magnetometry and electron paramagnetic resonance (EPR) spectroscopy, providing doublet-quartet energy gaps, ΔEDQ, corresponding to >85% population of the quartet ground states at room temperature. Notably, EPR-based determination of ΔEDQ relies on direct detection of the quartet ground state and doublet excited states. The ΔEDQ values are 1.0-1.1 kcal mol-1, with the more sterically shielded triradical having the larger value. The half-life of the more sterically shielded triradical in 2-methyltetrahydrofuran (2-MeTHF) is about 6 h at room temperature. The less sterically shielded triradical in 2-MeTHF decomposes at 158 K with a half-life of about 4 h, while at 195 K, the half-life is still about 2 h. The dominant products of the decay of triradicals are the corresponding triamines, suggesting hydrogen atom abstraction from the solvent as the primary mechanism. This study expands the frontier of the open-shell PAHs/nanographenes, of which the unique electronic, nonlinear optical, and magnetic properties could be useful in the development of novel organic electronics, photonics, and spintronics.
We developed a long-acting drug-delivery system that supports subcutaneous administration of the peptidic somatostatin agonist octreotide - a blockbuster drug used to treat acromegaly and neuroendocrine tumors. The current once-a-month polymer-encapsulated octreotide, Sandostatin LAR®, requires a painful intra-gluteal injection through a large needle by a health-care professional. To overcome such shortcomings, Tetra-PEG hydrogel microspheres were covalently attached to the α-amine of D-Phe1 or the ε-amine of Lys5 of octreotide by a self-cleaving β-eliminative linker; upon subcutaneous injection in the rat using a small-bore needle, octreotide was slowly released. The released drug from the ε-octreotide conjugate showed a remarkably long serum half-life that exceeded two months. The α-octreotide conjugate had a half-life of ~two weeks, and showed an excellent correlation of in vitro and in vivo drug release. Pharmacokinetic models indicate these microspheres should support once-weekly to once-monthly self-administered subcutaneous dosing in humans. The hydrogel-octreotide conjugate shows the favorable pharmacokinetics of Sandostatin LAR® without its drawbacks.
We have developed a unique long-acting drug-delivery system for the GLP-1 agonist exenatide. The peptide was covalently attached to Tetra-PEG hydrogel microspheres by a cleavable β-eliminative linker; upon s.c. injection, the exenatide is slowly released at a rate dictated by the linker. A second β-eliminative linker with a slower cleavage rate was incorporated in polymer crosslinks to trigger gel degradation after drug release. The uniform 40 micron microspheres were fabricated using a flow-focusing microfluidic device and in situ polymerization within droplets. The exenatide-laden microspheres were injected subcutaneously into the rat, and serum exenatide measured over a one-month period. Pharmacokinetic analysis showed a t1/2,β of released exenatide of about seven days which represents over a 300-fold half-life extension in the rat and exceeds the half-life of any currently approved long-acting GLP-1 agonist. Hydrogel-exenatide conjugates gave an excellent Level A in vitro-in vivo correlation of release rates of the peptide from the gel, and confirmed that linker cleavage was 3-fold faster in vivo than in vitro. Pharmacokinetic simulations indicate that the hydrogel-exenatide microspheres should support weekly or biweekly subcutaneous dosing in humans. The rare ability to modify in vivo pharmacokinetics by the chemical nature of the linker indicates that an even longer acting exenatide is feasible.
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