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An unconditionally-positive finite difference (UPFD) and the standard explicit finite difference schemes are compared to the analytical solution of the advection–diffusion reaction equation which describes the exponential traveling wave in heat and mass transfer processes. It is found that although the unconditional positivity of the UPFD scheme, this scheme is less accurate than the standard explicit finite difference scheme. This is because the UPFD scheme contains additional truncation-error terms in the approximations of the first and second derivatives with respect to x, which are evaluated at different moments in time. While these terms tend to zero as the mesh is refined, the UPFD scheme nevertheless remains less accurate than its standard explicit finite difference counterpart. The presented results are important when modeling a heat and mass transfer processes using the investigated advection–diffusion reaction equation. Furthermore, current and future developers of coupled multi-species transport models may draw on the ideas of solutions methods employed in this study to further develop numerical models for various types of coupled multi-species transport problems.

Mathematically, the complex reactions can be simplified through different linear/non-linear algebraic techniques that are the easiest way to balance the complex models. Furthermore, the complex chemical equations can be balanced with the linear algebraic technique based on matrix-inversion applications. That is the other way for the researchers and chemists to balance the complex chemical equation. In this article, we provide information about coefficients that balance the chemical equations through null-space methodology and the steady-state approximation by applying linear algebra technique. The purpose is not only to identify the undetermined coefficients but also to compute the kernel matrix or null spaces. Furthermore, the steady state for a two-step reaction mechanism is established.

By using improved F-expansion method (IFEM), we study density-dependent space–time-fractional diffusion–reaction equation with quadratic nonlinearity (DDFDRE), which arises in mathematical biology. The fractional derivative is described in the sense of the CF derivative. Exact traveling wave solutions for DDFDRE are derived and expressed in terms of hyperbolic functions and rational functions. The IFEM is brief, efficient and easy to apply. It also can be used to solve many other fractional nonlinear evolution equations.

This study aimed to investigate the degradation kinetics of the total organic carbon (TOC) in real pharmaceutical wastewater. Because of the complexity of real wastewaters and the lack of knowledge on all reactions involved in their chemical degradation, the use of a kinetic approach becomes difficult. To overcome this difficulty, we adopted a kinetic model based on chemical equations validated for mechanisms of free radical generation. Photo-Fenton reactions were performed in a 1-L tubular reactor. The developed model satisfactorily described the degradation kinetics of TOC in wastewater, with an R2 of 0.9997. The TOC degradation rate constant (kd) was estimated at 1 × 1011 L2 mol2 s−1, in agreement with literature values for the degradation kinetics of organic compounds. UV radiation had a positive effect on degradation, contributing to HO• formation. The adopted mathematical approach was highly sensitive to the formation rate constant of HO• from H2O2 (k4). By assuming the reaction to follow variable order kinetics (β = 1 and β = 2), it was possible to improve the goodness of fit of the model. Our results suggest, albeit in an incipient way, that the approach used in this study can be extended to the kinetic understanding of other complex wastewaters.

The computational treatment of detailed kinetic reaction mechanisms for combustion is expensive, especially in the case of biodiesel fuels. In this way, great efforts in the search of techniques for the development of reduced kinetic mechanisms have been observed. As Methyl Butanoate (MB, $$C_3H_7COOCH_3$$ ) is an essential model frequently used to represent the ester group of reactions in saturated methyl esters of large chain, this paper proposes a reduction strategy and uses it to obtain a reduced kinetic mechanism for the MB. The reduction strategy consists in the use of artificial intelligence to define the main chain and produce a skeletal mechanism, apply the traditional hypotheses of steady-state and partial equilibrium, and justify these assumptions through an asymptotic analysis. The main advantage of the strategy employed here is to reduce the work required to solve the system of chemical equations by two orders of magnitude for MB, since the number of reactions is decreased in the same order.

The paper presents a new thermodynamic approach to studying polynuclear complex formation reactions in multicomponent two-phase systems, solid phase plus saturated aqueous solution, under real conditions. The quintessence of this developed approach consists in the thermodynamic analysis of concurrent reactions on the basis of the introduced notion of a generalized reaction equation. The formation of complex polynuclear species is characterized by certain peculiarities in the behavior of studied two-phase heterogeneous systems. The formation of polynuclear complexes of the metal ion with a complexing agent leads to expressions for the enthalpy ∆H and heat capacity ∆Cp as certain concentration functions, which are normative multipliers, referring these thermochemical quantities to one mole of the metal ion. It is shown that one of these functions, Φ, characterizes the degree of formation of polynuclear complexes. In the case of concurrent formation of mono- and polynuclear complexes, the derived expressions are useful in distinguishing separate contributions of individual complex formation reactions to the ∆H and ∆Cp of the overall process.

The literature of the kinetics in thermal analysis deals mainly with models that consist of a single reaction equation. However, most samples with practical importance are too complex for such an oversimplified description. There is no universal way to overcome the difficulties, though there are well-established models that can express the complexity of the studied reactions for several important types of samples. The assumption of more than one reaction increases the number of unknown parameters. Their reliable estimation requests the evaluation of a series of experiments. The various linearization techniques cannot be employed in such cases, while the method of least squares can be carried out at any complexity of the models by proper numerical methods. It is advantageous to evaluate simultaneously experiments with linear and nonlinear temperature programs because a set of constant heating rate experiments is frequently not sufficient to distinguish between different models or model variants. It is well worth including modulated and constant reaction rate temperature programs into the evaluated series whenever they are obtainable. Sometimes different samples share some common features. In such cases one can try to describe their reactions by assuming parts of the kinetic parameters to be common for the samples. One should base the obtained models and parameter values on a sufficiently large amount of experimental information, in a reliable way. This article is based on the authors’ experience in the indicated directions from 1979 till the present. Though the examples shown are taken from biomass research, the models and methods shown in the article are also hoped to be relevant for other materials that have complicated structure or exhibit complicated thermal reactions, or both.

The metastable M2C carbide in a multi-component semi-high-speed steel shows an in situ decomposition during heat treatment. M2C carbides transform themselves into an aggregate of three secondary carbides (M6C, MC, M7C3) at the M2C carbide/γ-Fe matrix interface through the diffusion of various alloying elements. The content of the decomposition products decreases in the order of M6C, MC, and M7C3. The Fe-rich M6C is considered to be the η 2-type carbides of (Cr, V, Fe)4(Mo, W)2C, the MC consists primarily of V as the crystallized carbide does, and the M7C3 with 70 at. pct of Cr and Fe in total is formed only during heat treatment but not during solidification. At the heat treatment temperature of 1273 K (1000 °C), a new approximate reaction equation of M2C + 0.052γ-Fe → 0.687M6C + 0.044M7C3 + 0.320MC is proposed based on the measurement of the composition of carbides; moreover, a modified prediction model of transformation ratio of M2C carbide is obtained according to the DSC experiments.

UDC 519.63; 004.75; 536.252 The accuracy and convergence of the numerical solutions of a stationary one-dimensional linear convection-diffusion-reaction equation (with Dirichlet boundary conditions) by the Petrov–Galerkin finiteelement method with piecewise-linear basis functions and piecewise-quadratic weighting functions are analyzed and the accuracy estimates of the method are obtained in certain norms depending on the choice of the collection of stabilization parameters of weight functions.

This interdisciplinary study sketches the ways in which proportional reasoning is involved in the solution of chemistry problems, more specifically, problems involving quantities in chemical reactions (commonly referred to as stoichiometry problems). By building on the expertise of both mathematics and chemistry education research, the present paper shows how the theoretical constructs in proportional reasoning in mathematics education offer rich explanatory accounts of the complexities involved in solving stoichiometry problems. Using Vergnaud's concept of measure spaces, the theoretical analysis shows that proportionality situations are relatively more intricate, involving various layers of complexity in chemistry as compared to those in the mathematics curriculum. Knowledge of proportionality and chemistry are simultaneously required to provide solutions to chemical reactions. Our analysis of a range of stoichiometry situations led us to propose a problem analysis framework involving five levels of difficulty. Further, the specificity of proportionality in stoichiometry is that it can only be established when quantities are interpreted in the unit “mole,” a unit which does not have any physical embodiment in terms of a measure of quantity unlike mass and volume. Our analysis of student-teachers' solution to the stoichiometry problems, shows that they tend to incorrectly (probably intuitively) set proportional relationships when two quantities in a reaction are expressed in non-molar quantities such as mass. The data also bring to the fore the primarily formulaic approach that student-teachers use in setting inherent proportionality relationships. An important finding is the interpretation of a chemical equation as a mathematical equation, rather than a statement of proportionality.

In this paper, fractional derivatives in the sense of modified Riemann-Liouville derivative and first integral method are applied for constructing exact solutions of nonlinear fractional generalized reaction duffing model and nonlinear fractional diffusion reaction equation with quadratic and cubic nonlinearity. Our approach provides first integrals in polynomial form with a high accuracy for two-dimensional plane autonomous systems. Exact soliton solutions are constructed through established first integrals.

In this paper we report on a novel method for solving systems of nonlinear differential equations which is an extension of the spectral homotopy analysis method (SHAM). The proposed method extends the application of the SHAM to initial value problems that model the Michaelis-Menten enzymatic reaction equation. Results from the proposed method are compared with Runge-Kutta routines as a measure of accuracy and efficiency.

Bioethanol is a fuel additive or a fuel substitute that has the benefit of being cleaner and price competitive with gasoline. Therefore, we develop a reduced kinetic mechanism capable of modeling the ethanol combustion and the generation of the combustion products $$\text{H}_{2}\text{O},~\text{CO}_{2},~\text{CO},~ \text{H}_{2},~ \text{C}_{2}\text{H}_{4}$$ and OH. Based on a mechanism composed by 372 reversible elementary reactions among 56 reactive species, we propose a reduction strategy to obtain an eight-step mechanism for the ethanol. The reduction strategy consists in estimating the order of magnitude of the reaction rate coefficients, defining the main chain, applying the steady-state and partial equilibrium hypotheses, and justifying the assumptions through an asymptotic analysis. The main advantage of the obtained reduced mechanism is the decrease of the work needed to solve the system of chemical equations proportionally to the number of elementary reactions present in the complete mechanism. Numerical tests are carried out for a jet diffusion flame of ethanol and the results compare well with available data in the literature.

In the present work, the \((\frac{G^\prime}{G})\)-expansion method and its generalized version have been employed for obtaining a variety of exact traveling wave solutions of the nonlinear diffusion reaction equation with quadratic and cubic nonlinearities. We have examined the density independent nonlinear diffusion reaction equation with a convective flux term and successfully have obtained some new and more general solutions in terms of Jacobi elliptic functions. The work highlights the significant features of the employed methods and shows the variety in the obtained solutions.

The paper is aimed at defining reduction, oxidation, and redox reactions based both on the oxidation number and charge changes in reacting species. It is rationalized that the processes of oxidation and reduction, usually occurring simultaneously, can occur also as independent processes. It is explained that in balancing chemical equations of redox reactions the “gain” or “loss” of electrons should be understood as changes in oxidation number. A formal expressions “+n e−” and “−n e−” represent in reality a decrease and increase in oxidation number by n units, respectively.

Calculations are made using the equations ΔrG = ΔrH − TΔrS and ΔrX = ΔrH − ΔrQ where ΔrX represents the free energy change when the exchange of absorbed thermal energy with the environment is represented by ΔrQ. The symbol Q has traditionally represented absorbed heat. However, here it is used specifically to represent the enthalpy listed in tabulations of thermodynamic properties as (HT − H0) at T = 298.15 K, the reason being that for a given substance TS equals 2.0 Q for solid substances, with the difference being greater for liquids, and especially gases. Since ΔrH can be measured, and is tangibly the same no matter what thermodynamics are used to describe a reaction equation, a change in the absorbed heat of a biochemical growth process system as represented by either ΔrQ or TΔrS would be expected to result in a different calculated value for the free energy change. Calculations of changes in thermodynamic properties are made which accompany anabolism; the formation of anabolic, organic by-products; catabolism; metabolism; and their respective non-conservative reactions; for the growth of Saccharomyces cerevisiae using four growth process systems. The result is that there is only about a 1% difference in the average quantity of free energy conserved during growth using either Eq. 1 or 2. This is because although values of TΔrS and ΔrQ can be markedly different when compared to one another, these differences are small when compared to the value for ΔrG or ΔrX.

CORE2D V4 is a finite element code for modeling partly or fully saturated water flow, heat transport, and multicomponent reactive solute transport under both local chemical equilibrium and kinetic conditions. It can handle coupled microbial processes and geochemical reactions such as acid–base, aqueous complexation, redox, mineral dissolution/precipitation, gas dissolution/exsolution, ion exchange, sorption via linear and nonlinear isotherms, and sorption via surface complexation. Hydraulic parameters may change due to mineral precipitation/dissolution reactions. Coupled transport and chemical equations are solved by using sequential iterative approaches. A sequential partly iterative approach (SPIA) is presented which improves the accuracy of the traditional sequential non-iterative approach (SNIA) and is more efficient than the general sequential iterative approach (SIA). While SNIA leads to a substantial saving of computing time, it introduces numerical errors which are especially large for cation exchange reactions. SPIA improves the efficiency of SIA because the iteration between transport and chemical equations is only performed in nodes with a large mass transfer between solid and liquid phases. The efficiency and accuracy of SPIA are compared to those of SIA and SNIA using synthetic examples and a case study of reactive transport through the Llobregat Delta aquitard in Spain. SPIA is found to be as accurate as SIA while requiring significantly less CPU time. In addition, SPIA is much more accurate than SNIA with only a minor increase in computing time. A further enhancement of the efficiency of SPIA is achieved by improving the efficiency of the Newton–Raphson method used for solving chemical equations. Such an improvement is obtained by working with increments of log concentrations and ignoring the terms of the Jacobian matrix containing derivatives of activity coefficients. A proof is given for the symmetry and non-singularity of the Jacobian matrix. Numerical analyses performed with synthetic examples confirm that these modifications improve the efficiency and convergence of the iterative algorithm.

Standard Galerkin finite element methods (GFEM) lack stability in solving advection-dominated solute transport in porous media. They usually require prohibitively fine grids and extremely small time steps to solve for advection-dominated problems. The algebraic subgrid-scale stabilized (ASGS) finite element method has been proved to overcome such problems for single-species reactive transport. Its potential for dealing with multicomponent reactive transport has not yet been explored. Here we present a numerical formulation of ASGS for steady and transient multicomponent reactive transport. Subgrid-scale transport equations are solved first by using an ASGS approximation and their solutions are substituted back into the grid-scale equations. A sequential iteration approach (SIA) is used to solve for coupled transport and chemical equations. Coupling of ASGS and SIA, ASGS+SIA, has been implemented in a reactive transport code, CORE2D V4, and verified for conservative solute transport. ASGS+SIA has been tested for a wide range of 1-D transient multicomponent reactive transport problems involving different types of chemical reactions such as: (1) Kinetically controlled aqueous species degradation, (2) Kinetic mineral dissolution, (3) Serial-parallel decay networks, and (4) Cation exchange and pyrite oxidation at local equilibrium. ASGS+SIA always provides accurate solutions and therefore offers an efficient option to solve for advection-dominated multicomponent reactive transport problems.

A definition of biological activity is proposed that is superficially analogous to the equation relating the thermodynamic activity of a solute to its concentration via an activity coefficient. The biological activity of a molecular entity is defined as A=cf, where A is the activity, c the amount-of-substance concentration, and f is a parameter designated as “inherent activity.” Units and dimensions are determined by the type of activity, catalytic (katal) or binding (mol−1 L). The measurand is described by a chemical equation that identifies the entity for which an activity is being monitored. This definition of biological activity has the advantage of separating the chemical characterization of the entity in terms of structure and amount from the assessment of biological activity. Ideally, a homogeneous entity is used for the measurement of f. In instances where impure materials are used or the chemical equation defining the activity is unknown, the evaluated parameter should be designated as f′ to denote its empirical nature. Any measurement of f or f′ should be qualified with an appropriate estimate of measurement uncertainty.

The theoretical isotherms (α=α(t), where α is the conversion degree and t is the time) and the theoretical thermogravimetric curves(α=α(T), where T is the temperature) were simulated for a system in which two consecutive reactions occur. A critical analysis of the use of an overall single rate reaction equation instead of the use of the true rate equations is presented. Both for isothermal and non-isothermal data and α≤0.50, the apparent reaction order depends on temperature. It is pointed out that the apparent reaction order for a given temperature can be evaluated if the parameters of the compensation effect are known.

Computer-based approaches to balancing chemical equations are discussed. Out of 13 methods surveyed, 6 are based on matrix, 2 are interactive programs, 1 is a stand alone system, 1 is developed in algorithm in Basic, 1 is based on design engineering, 1 is written in HyperCard, and 1 prepared for the World Wide Web. Features in addition to balancing chemical equations include the following: solve possible reactions from a list of molecules and ions, determine the number of independent chemical reactions, calculate yields, predict mass relationships, and balance rare reactions. Research and development issues are addressed.

Linear algebraic techniques based on minimization of thermodynamic functional and/or other constraints are illustrated for mass balance of chemical reactions that do not exhibit stoichiometrically unique solutions in the linear algebraic vector space. The techniques demonstrate elegant casting of chemical equations in terms of generalized linear Diophantine matrices and generalized elimination and variational schemes.

A critical analysis of the use of an overall single rate reaction equation instead of the true rate equation corresponding to a complex process consisting in two consecutive reactions is presented. In accordance with this approximation, often used in the kinetic analysis of the system in which several reactions take place, the overall process is described by the apparent activation parameters (the apparent activation energy, E ap, and the apparent pre-exponential factor, A ap) and the apparent conversion function. The theoretical isotherms (α=α(t), where a is the conversion degree and t is the time) have been simulated for a system in which two consecutive reactions occur. In this case, the apparent activation parameters depends on: (a) the considered range of the temperature; (b) the temperature, for a given conversion degree. It is shown that the apparent activation parameters are corrrelated by the compensation effect relationship: lnA ap=α*+β*E ap where α* and β* are the parameters of the linear regression. The possibility of using the apparent kinetic parameters to predict the isotherms α=α(t) for temperatures lower than those for which these parameters were evaluated, is discussed.

While the balanced chemical equations for a multireaction system are generally not unique, the minimum number of independent equations, R, is a characteristic property of the system. Deleting one nonspectator species from the system leads to a system with R reduced by one. In this way each system can be reduced to a single-reaction system and ultimately to a no-reaction system. The least number of chemical species that can be deleted to obtain a no-reaction system equals R. Every multireaction system, therefore, can be reduced to a number of single-reaction equations which can be balanced by any one of the standard techniques. Some examples are given where balancing by inspection is employed.

Algebraic equations were derived for plotting theoretical titration curves involving argentometric titration of binary halide mixtures. These generic equations are applied to three cases of binary halide mixtures in spreadsheet programs. The results are presented graphically as plots of half-cell potential versus volume of the titrant. This approach may serve as a more effective and logical method of showing students how to simulate a complex system involving multiple equilibria.

The disproportionation of pentavalent plutonium is traditionally represented by two inconsistent equations. The inconsistency puts the widely-accepted, fourth-power hydrogen-ion dependence of the reaction into question. The balanced reaction equation suggests that both the stoichiometry and the acid-dependence of Pu(V) disproportionation depend on the pH of the plutonium solution.

A simple procedure has been proposed for obtaining a combined chemical equation for a multiple reaction as defined by Temkin. The procedure is based on the concept of a “normal basis for chemical equations” which is introduced in the present paper. In some cases the proposed method makes it possible to obtain a form of combined chemical equation which is more convenient for practical use than that obtained by the familiar method.

While the balanced chemical equations for a multireaction system are generally not unique, the minimum number of independent equations, R, is a characteristic property of the system. Deleting one nonspectator species from the system leads to a system with R reduced by one. In this way each system can be reduced to a single-reaction system and ultimately to a no-reaction system. The least number of chemical species that can be deleted to obtain a no-reaction system equals R. Every multireaction system, therefore, can be reduced to a number of single-reaction equations which can be balanced by any one of the standard techniques. Some examples are given where balancing by inspection is employed.

An algorithm for parsing chemical equation directly input on computer terminal has been developed in APL programming language. By utilizing this algorithm, a user-oriented easy system for thermochemical parameter prediction for organic compounds has been constructed. This may be considered the first step to cognite a chemical equation for the ease of user interfacing

Thermodynamic charge is one of the central concepts in the thermokinetic theory of physico-chemical processes. This paper demonstrates that stoichiometric coefficients play the role of charges in elementary chemical reactions.

This paper describes an on-line interactive computer program that enables users to perform thermodynamic calculations by entering chemical equations and defining variables such as temperature and pressure. The data used in the calculations are automatically retrieved from a large data base. The scope of the program makes it possible to perform a very wide variety of calculations in which reactant and product compounds may be at different temperatures, pressures, and activities.

An existing computational technique was adapted for the derivation of ternary constitution diagrams in the sodium-carbon-oxygen system. The method involved the selection of appropriate chemical equations among seven condensed phases including sodium, carbon, Na2C2, Na2CO3, Na2O, Na2O2, NaO2 and the three gases CO, CO2 and sodium vapor. Using available thermochemical data, gas partial pressures were computed and ternary isotherms derived. At temperatures near 750°C and 1 atm pressure, carbon and Na2CO3 can coexist with liquid sodium at high carbon-oxygen ratios whereas carbon and Na2CO3 can coexist at low carbon-oxygen ratios. At somewhat higher temperatures and pressures, Na2C2 may be an important phase. Although solubility effects would alter slightly the computed data, the results of the study generally agree with the experimental observations of others.

A previously obtained condition on the boundary temperature for the existence of a unique steady state solution of the zeroth order Arrhenius reaction equation is shown to be sufficient to guarantee decay of the temperature in the corresponding transient problem. For the heat equation we obtain a bound on the size of the initial data. The bound depends upon domain size and geometry, which may be used to ascertain when disturbances decay to zero in time.

The present investigation is part of a program aimed at examining the mathematical basis of reaction rate theory (Bartholomay,Bull. Math. Biophys. 20, 1958) from the point of view of individual molecular events. Certain inconsistencies resulting from the classical procedure of using stoichiometric “chemical equations” to represent the kinds of mechanisms involved are pointed out and remedies are suggested by the introduction of set-theoretic notation within the framework of a so-calledMolecular Set Theory. It is concluded that whereas ordinary algebra is a suitable basis for stoichiometry, in discussing mechanisms of chemical reactions,molecular set theory is more appropriate. This is a theory of material transformations which is patterned afterabstract set theory: the individual molecules of a chemical species and chemical transformations from one species to another correspond to the abstract concepts of elements or points of a set and their mappings into other sets respectively. The new notions ofexact molecular sequences andequivalence classes of molecules arising from this study of chemical reactions may be of purely mathematical interest when referred back to the context of abstract set theory.

PbNb2O6 PbCrO4 PZT piezoelectric ceramics with low radial coupling coefficient has been developed. Its chemical equation is given. This kind of ceramics is characterized byk p =0.15,Q M =5000,f/f r =6×10−6. It has been used successfully to make low-frequency narrow-band ceramic filter.

THE general scheme of a chemical transformation can be reproduced by the equation: . A, B, C represent chemical elements or groups of elements, which are transferred as whole complexes from one side to the other of the chemical equation (i.e. NH4, SO4, NO2, etc.). We shall call these groups of elements for short the elementids of a chemical equation; the chemical elements are thus the simplest elementids. It is evident that in determining the number of elementids of a chemical equation the minimum rule must be observed—that is, the elements must be brought together into groups, so that the number of these groups (elementids) shall be the smallest possible. The composition of these groups must fulfil one condition: that their number taken as a whole and for each formula individually should be the same on the right and on the left side of the chemical equation. In most cases the problem of determining the elementids is simplified by the fact that the number of elementids is the same as the number of elements.

THE recent report of the Joint Committee of the Chemical, Faraday, and Physical Societies on Symbols for Thermodynamical and Physico-Chemical Quantities may be taken as an occasion for some remarks on the sign and symbol of heat of reaction and allied quantities. As is familiar, the heat of a reaction may be defined as the heat generated or lost by the system, when unit mass of reactants passes into resultants, the unit of mass being the number of grams represented by the sum of the molecular weights multiplied by their respective coefficients on either side of the chemical equation.

IT is well known that the Gibbs Phase Rule as applied to equilibrated macro-systems subject to changes of temperature, pressure and concentration may be written in the form where P is the number of phases, F the number of degrees of freedom and C the number of components. In applying the equation to physical and chemical equilibrium, the number of components C is taken as the minimum number of independently variable constituents in terms of which the equilibrium may be expressed in the form of a chemical equation. The term 'number of components' has thus to bear the brunt of the matter and assume a responsibility not shouldered by the equation itself. In fact, it is frequently necessary to write out a properly balanced chemical equation in order to decide upon the number of components in the system. Moreover, the equation cannot be applied without making allowance for certain restrictions when the phases approach identity, as in critical phenomena, or when one of the substances is kept in a definite weight -ratio to some other substance in the system, as in dissociation phenomena.