Local thermodynamics and the generalized Gibbs-Duhem equation in systems with long-range interactions

The local thermodynamics of a system with long-range interactions in d dimensions is studied using the mean-field approximation. Long-range interactions are introduced through pair interaction potentials that decay as a power law in the interparticle distance. We compute the local entropy, Helmholtz free energy, and grand potential per particle in the microcanonical, canonical, and grand canonical ensembles, respectively. From the local entropy per particle we obtain the local equation of state of the system by using the condition of local thermodynamic equilibrium. This local equation of state has the form of the ideal gas equation of state, but with the density depending on the potential characterizing long-range interactions. By volume integration of the relation between the different thermodynamic potentials at the local level, we find the corresponding equation satisfied by the potentials at the global level. It is shown that the potential energy enters as a thermodynamic variable that modifies the global thermodynamic potentials. As a result, we find a generalized Gibbs-Duhem equation that relates the potential energy to the temperature, pressure, and chemical potential. For the marginal case where the power of the decaying interaction potential is equal to the dimension of the space, the usual Gibbs-Duhem equation is recovered. As examples of the application of this equation, we consider spatially uniform interaction potentials and the self-gravitating gas. We also point out a close relationship with the thermodynamics of small systems.

Filtration of effluents for microirrigation systems

Clogging, measured through head loss across filters, and the filtration quality of different filters using different effluents were studied. The filters used were: 115, 130, and 200 m disc filters; 98, 115, 130, and 178 m screen filters; and a sand filter filled with a single layer of sand with an effective diameter of 0.65 mm. The filters were used with a meat industry effluent and secondary and tertiary effluents of two wastewater treatment plants. It was observed that clogging depended on the type of effluent. With the meat industry effluent, the poorest quality effluent, disc filters clogged more than the other filter types. When the wastewater treatment plant effluents were used, the disc filters showed less frequent clogging. Several physical and chemical parameters, such as total suspended solids, chemical oxygen demand, turbidity, electrical conductivity, pH, and number of particles, were analyzed in the effluents at the entry and exit points of the filters. In general, filters did not reduce the values of the main clogging parameters to a great degree. It was found that the parameter that explained the clogging, expressed as Boucher’s filterability index, was different depending on the type of effluent and filter. The best quality of filtration was achieved with a sand filter when the meat industry effluent was used. No significant differences were observed between the quality of filtration of disc and screen filters when operating with the secondary and tertiary effluents

Development, physical-chemical stability and release studies of four alcohol-free spironolactone suspensions for use in pediatrics

Dissolution studies have become of great significance because, in most cases, drug dissolution is the rate-limiting step in the absorption process. As occurs with solid oral dosage forms, heterogeneous disperse systems (suspensions) could also have some problems with their in vitro dissolution. The objective of this study was to evaluate influence of the excipients on the release of spironolactone from four alcohol free suspensions (pharmaceutical compounding) of spironolactone 5 mg/mL suitable for pediatric use. Also the comparison of the physical and chemical stability of the suspensions stored at 4, 25 and 40 ºC over a 60- day period has been studied. Rheological behavior, particle size, a prediction of long-term physical stability, pH and assay of spironolactone by HPLC were assessed at prefixed times. The dissolution profile of each suspension was determined and compared with that of the commercial tablets. A microbiological study of the best formula was also performed. Chemically, the four spironolactone suspensions were stable for 60 days stored at three temperatures; Suspension IV had optimum pH values and the highest recovery percentage. In terms of physical stability, sedimentation occurred in Suspension IV and flotation of spironolactone in Suspensions I, II and III. Suspension III had the highest viscosity and the slowest drug release. Suspension IV was also microbiologically stable for 60 days. In conclusion, Suspension IV had the best properties and the least suitable form was Suspension III, as its high viscosity made it difficult to achieve homogeneous redispersion, and it had the slowest dissolution profile.

Full instability behavior of N-dimensional dynamical systems with a one-directional nonlinear vector field

We show how certain N-dimensional dynamical systems are able to exploit the full instability capabilities of their fixed points to do Hopf bifurcations and how such a behavior produces complex time evolutions based on the nonlinear combination of the oscillation modes that emerged from these bifurcations. For really different oscillation frequencies, the evolutions describe robust wave form structures, usually periodic, in which selfsimilarity with respect to both the time scale and system dimension is clearly appreciated. For closer frequencies, the evolution signals usually appear irregular but are still based on the repetition of complex wave form structures. The study is developed by considering vector fields with a scalar-valued nonlinear function of a single variable that is a linear combination of the N dynamical variables. In this case, the linear stability analysis can be used to design N-dimensional systems in which the fixed points of a saddle-node pair experience up to N21 Hopf bifurcations with preselected oscillation frequencies. The secondary processes occurring in the phase region where the variety of limit cycles appear may be rather complex and difficult to characterize, but they produce the nonlinear mixing of oscillation modes with relatively generic features