A mathematical approach to chemical equilibrium theory for gaseous systems—I: theory

Equilibrium theory occupies an important position in chemistry and it is traditionally based on thermodynamics. A novel mathematical approach to chemical equilibrium theory for gaseous systems at constant temperature and pressure is developed. Six theorems are presented logically which illustrate the power of mathematics to explain chemical observations and these are combined logically to create a coherent system. This mathematical treatment provides more insight into chemical equilibrium and creates more tools that can be used to investigate complex situations. Although some of the issues covered have previously been given in the literature, new mathematical representations are provided. Compared to traditional treatments, the new approach relies on straightforward mathematics and less on thermodynamics, thus, giving a new and complementary perspective on equilibrium theory. It provides a new theoretical basis for a thorough and deep presentation of traditional chemical equilibrium. This work demonstrates that new research in a traditional field such as equilibrium theory, generally thought to have been completed many years ago, can still offer new insights and that more efficient ways to present the contents can be established. The work presented here can be considered appropriate as part of a mathematical chemistry course at University level.

A mathematical approach to chemical equilibrium theory for gaseous systems—II: extensions and applications

Straightforward mathematical techniques are used innovatively to form a coherent theoretical system to deal with chemical equilibrium problems. For a systematic theory it is necessary to establish a system to connect different concepts. This paper shows the usefulness and consistence of the system by applications of the theorems introduced previously. Some theorems are shown somewhat unexpectedly to be mathematically correlated and relationships are obtained in a coherent manner. It has been shown that theorem 1 plays an important part in interconnecting most of the theorems. The usefulness of theorem 2 is illustrated by proving it to be consistent with theorem 3. A set of uniform mathematical expressions are associated with theorem 3. A variety of mathematical techniques based on theorems 1–3 are shown to establish the direction of equilibrium shift. The equilibrium properties expressed in initial and equilibrium conditions are shown to be connected via theorem 5. Theorem 6 is connected with theorem 4 through the mathematical representation of theorem 1.

EISCAT observations of ion composition and temperature anisotropy in the high-latitude F-region

The papers by Winser et al. [(1990) J. atmos. terr. Phys.52, 501] and Häggström and Collis [(1990) J. atmos. terr. Phys.52, 519] used plasma flows and ion temperatures, as measured by the EISCAT tristatic incoherent scatter radar, to investigate changes in the ion composition of the ionospheric F-layer at high latitudes, in response to increases in the speed of plasma convection. These studies reported that the ion composition rapidly changed from mainly O+ to almost completely (>90%) molecular ions, following rapid increases in ion drift speed by >1 km s−1. These changes appeared inconsisent with theoretical considerations of the ion chemistry, which could not account for the large fractions of molecular ions inferred from the obsevations. In this paper, we discuss two causes of this discrepancy. First, we reevaluate the theoretical calculations for chemical equilibrium and show that, if we correct the derived temperatures for the effect of the molecular ions, and if we employ more realistic dependences of the reaction rates on the ion temperature, the composition changes derived for the faster convection speeds can be explained. For the Winser et al. observations with the radar beam at an aspect angle of ϕ = 54.7° to the geomagnetic field, we now compute a change to 89% molecular ions in < 2 min, in response to the 3 km s−1 drift. This is broadly consistent with the observations. But for the two cases considered by Häggström and Collis, looking along the field line (ϕ = 0°), we compute the proportion of molecular ions to be only 4 and 16% for the observed plasma drifts of 1.2 and 1.6 km s−1, respectively. These computed proportions are much smaller than those derived experimentally (70 and 90%). We attribute the differences to the effects of non-Maxwellian, anisotropic ion velocity distribution functions. We also discuss the effect of ion composition changes on the various radar observations that report anisotropies of ion temperature.

The equilibrium structure of HCN

The equilibrium structure of HCN has been determined from the previously published ground state rotational constants of eight isotopomers by using (B0‐Be) values obtained from a variational calculation of the vibration–rotation spectrum. The results are re(CH)=1.065 01(8) Å, and re(CN)=1.153 24(2) Å.

Simulating the detailed chemical composition of secondary organic aerosol formed on a regional scale during the TORCH 2003 campaign in the southern UK

Following on from the companion study (Johnson et al., 2006), a photochemical trajectory model (PTM) has been used to simulate the chemical composition of organic aerosol for selected events during the 2003 TORCH (Tropospheric Organic Chemistry Experiment) field campaign. The PTM incorporates the speciated emissions of 124 nonmethane anthropogenic volatile organic compounds (VOC) and three representative biogenic VOC, a highly-detailed representation of the atmospheric degradation of these VOC, the emission of primary organic aerosol (POA) material and the formation of secondary organic aerosol (SOA) material. SOA formation was represented by the transfer of semi and non-volatile oxidation products from the gas-phase to a condensed organic aerosol-phase, according to estimated thermodynamic equilibrium phase-partitioning characteristics for around 2000 reaction products. After significantly scaling all phase-partitioning coefficients, and assuming a persistent background organic aerosol (both required in order to match the observed organic aerosol loadings), the detailed chemical composition of the simulated SOA has been investigated in terms of intermediate oxygenated species in the Master Chemical Mechanism, version 3.1 ( MCM v3.1). For the various case studies considered, 90% of the simulated SOA mass comprises between ca. 70 and 100 multifunctional oxygenated species derived, in varying amounts, from the photooxidation of VOC of anthropogenic and biogenic origin. The anthropogenic contribution is dominated by aromatic hydrocarbons and the biogenic contribution by alpha-and beta-pinene (which also constitute surrogates for other emitted monoterpene species). Sensitivity in the simulated mass of SOA to changes in the emission rates of anthropogenic and biogenic VOC has also been investigated for 11 case study events, and the results have been compared to the detailed chemical composition data. The role of accretion chemistry in SOA formation, and its implications for the results of the present investigation, is discussed.

Insights into redox cycling of sulfur and iron in peatlands using high-resolution diffusive equilibrium thin film (DET) gel probe sampling

High spatial resolution vertical profiles of pore-water chemistry have been obtained for a peatland using diffusive equilibrium in thin films (DET) gel probes. Comparison of DET pore-water data with more traditional depth-specific sampling shows good agreement and the DET profiling method is less invasive and less likely to induce mixing of pore-waters. Chloride mass balances as water tables fell in the early summer indicate that evaporative concentration dominates and there is negligible lateral flow in the peat. Lack of lateral flow allows element budgets for the same site at different times to be compared. The high spatial resolution of sampling also enables gradients to be observed that permit calculations of vertical fluxes. Sulfate concentrations fall at two sites with net rates of 1.5 and 5.0nmol cm− 3 day− 1, likely due to a dominance of bacterial sulfate reduction, while a third site showed a net gain in sulfate due to oxidation of sulfur over the study period at an average rate of 3.4nmol cm− 3 day− 1. Behaviour of iron is closely coupled to that of sulfur; there is net removal of iron at the two sites where sulfate reduction dominates and addition of iron where oxidation dominates. The profiles demonstrate that, in addition to strong vertical redox related chemical changes, there is significant spatial heterogeneity. Whilst overall there is evidence for net reduction of sulfate within the peatland pore-waters, this can be reversed, at least temporarily, during periods of drought when sulfide oxidation with resulting acid production predominates.

Flux difference splitting for inviscid, real gases with non-equilibrium chemistry

A flux-difference splitting method is presented for the inviscid terms of the compressible flow equations for chemical non-equilibrium gases

An efficient finite difference scheme for three-dimensional, non-equilibrium flows using operator splitting

An efficient finite difference scheme is presented for the inviscid terms of the three-dimensional, compressible flow equations for chemical non-equilibrium gases. This scheme represents an extension and an improvement of one proposed by the author, and includes operator splitting.

Boundary condition effects in the simulation study of equilibrium properties of magnetic dipolar fluids

In this paper we investigate the equilibrium properties of magnetic dipolar (ferro-) fluids and discuss finite-size effects originating from the use of different boundary conditions in computer simulations. Both periodic boundary conditions and a finite spherical box are studied. We demonstrate that periodic boundary conditions and subsequent use of Ewald sum to account for the long-range dipolar interactions lead to a much faster convergence (in terms of the number of investigated dipolar particles) of the magnetization curve and the initial susceptibility to their thermodynamic limits. Another unwanted effect of the simulations in a finite spherical box geometry is a considerable sensitivity to the container size. We further investigate the influence of the surface term in the Ewald sum-that is, due to the surrounding continuum with magnetic permeability mu(BC)-on the convergence properties of our observables and on the final results. The two different ways of evaluating the initial susceptibility, i.e., (1) by the magnetization response of the system to an applied field and (2) by the zero-field fluctuation of the mean-square dipole moment of the system, are compared in terms of speed and accuracy.

Equilibrium behavior of symmetric ABA triblock copolymer melts

Melts of ABA triblock copolymer molecules with identical end blocks are examined using self-consistent field theory (SCFT). Phase diagrams are calculated and compared with those of homologous AB diblock copolymers formed by snipping the triblocks in half. This creates additional end segments which decreases the degree of segregation. Consequently, triblock melts remain ordered to higher temperatures than their diblock counterparts. We also find that middle-block domains are easier to stretch than end-block domains. As a result, domain spacings are slightly larger, the complex phase regions are shifted towards smaller A-segment compositions, and the perforated-lamellar phase becomes more metastable in triblock melts as compared to diblock melts. Although triblock and diblock melts exhibit very similar phase behavior, their mechanical properties can differ substantially due to triblock copolymers that bridge between otherwise disconnected A domains. We evaluate the bridging fraction for lamellar, cylindrical, and spherical morphologies to be about 40%–45%, 60%–65%, and 75%–80%, respectively. These fractions only depend weakly on the degree of segregation and the copolymer composition.