A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Volumetric Properties of Binary Mixtures of Isomeric Butanols and C8 Solvents at 298.15 K

Excess molar volumes, V _m ^E, over the whole composition range for binary mixtures of 1-butanol, 2-butanol, and 2-methyl-2-propanol + 1-octanol, or 2-octanol, or di-n-butyl ether, or n-hexylacetate were determined at 298.15 K from density measurements carried out with a vibrating-tube densimeter. Small V _m ^E values, both positive and negative, are displayed by mixtures containing 1- or 2-octanol, whereas positive and larger values are always found for mixtures containing dibutyl ether and hexylacetate. These results can be justified in terms of H-bonding interactions and/or steric hindrance due to the branched alkyl chains. Partial molar volumes at infinite dilution of the isomeric butanols in the C8 compounds were also calculated from the apparent molar volumes in dilute solution. The solute-solvent interactions and the effects of the local organisation of the solvent around the butanol molecules were discussed using the void and cavity volumes as different estimates of the intrinsic volume of the molecules. The volumetric behavior of butanols seems to be determined by the solute-solvent interactions rather than packaging effects.     

Cation-crown ether complex formation in water. II. Alkali and alkaline earth cations and 12-crown-4, 15-crown-5, and 18-crown-6

The stability constants and the partial molal volume and isentropic partial molal compressibility changes of complex formation between cations and crown ethers in water at 25°C are presented. The cations involved are Na⁺, K⁺, Rb⁺, Cs⁺, Ca²⁺, and Ba²⁺, and the crown ethers are 12-crown-4, 15-crown-5, and 18-crown-6. Values of V of complex formation have been discussed in terms of two simple models, one based on the scaled particle theory, and the others on the Drude-Nernst continuum model. The results indicate that the charge of the potassium cation in 18-crown-6 is especially well screened from the water. On this basis hydration numbers of complexed cations have been calculated. This shows that the size of the cation compared to the crown ether hole is important for the contacts between complexed cations and water.     

Liquid-Phase Selective Oxidation of Propane on Silica-Supported Nafion Catalysts

The liquid-phase partial oxidation of propane in the presence of the Fen+/H2O2 Fenton system at 70℃ and 1.4 atm on silica supported Nafion catalysts has been investigated. The reaction proceeds via a radical reaction path the efficiency of which is improved by silica-supported Nafion catalysts. Because of the direct relationship between reaction rate and concentration of sulphonic acid sites of Nafion catalysts, it is inferred that the active phase enahnces the kinetics of propane conversion by promoting the rate of active radicals generation.     

The effect of high pressure on the ion pair equilibrium constant of alkali metal fluorides: A spectrophotometric study

Bromophenol blue indicator was used in UV-visible spectrophotometric measurements to study ion association constants of alkali metal fluorides. The equilibrium constants for the ion pair formation of the alkali metal fluorides were determined as a function of ionic strength at one atmosphere pressure and 25°C. The effect of pressure on these association constants was measured at a constant total ionic strength of 1.0 mol-kg^–1 over a pressure range of 1 to 2000 atmospheres at 25°C. The pressure dependences of the stoichiometric association constants of the alkali metal fluorides are given by: lnK _LiF ^* =0.77–2.47×10^–4P–2.12×10^–8P²; lnK _NaF ^* =0.53–1.08×10^–4P–1.66×10^–8P²; lnK _KF ^* =0.24–4.41×10^–5P–7.15×10^–8P²; lnK _RbF ^* =–0.17–8.65×10^–5P–4.51×10^–8P²; and lnK _CsF ^* = –0.37–1.14×10^–4P–6.82×10^–8P², where P is the pressure in atmospheres. The stoichiometric molar volume and compressibility changes for ion pair formation of the alkali metal fluorides were evaluated from the pressure dependence of K _MF ^* data. The thermodynamic association constants were also calculated making use of activity coefficient data from the Pitzer equations. The partial molal volume and compressibility changes for ion pair formation of each alkali metal fluoride are reported.     

Effect of pressure on the solubility of naphthalene in water at 25°C

Solubility of naphthalene in water was measured at 25°C and pressures up to 200 MPa. The solubility decreased with increasing pressure. From the pressure coefficient of the solubility, the volume change V accompanying the dissolution was estimated as 13.8±0.4 cm ³ -mol ^–1 . Further we estimated the volume change V _CH accompanying hydrophobic hydration as –0.1±0.6 cm ³ -mol ^–1 using the V value, the molar volume of crystalline naphthalene, and the partial molar volume of naphthalene in n-heptane. This V _CH is much larger (i.e., less negative) than that for hydrophobic hydration of alkyl-chain compounds and suggests that the hydration structure of naphthalene differs from that of alkyl-chain compounds.     

The thermodynamics of mixing C6 and C7 compounds with a bicyclic compound at infinite dilution

The activity coefficients at infinite dilution have been measured at 25°C for cyclohexane, cyclohexene, 1,3-cyclohexadiene, 1,4-cyclohexadiene, benzene, n-hexane, 1-hexene, 1-hexyne, n-heptane, 1-heptene and 1-heptyne in decahydronaphthalene, bicyclohexyl, 1,2,3,4-tetrahydronaphthalene and cyclohexylbenzene. These results, together with previously determined H _m ^E and V _m ^E have been used to calculate the partial molar excess thermodynamic properties of mixing at infinite dilution.     

The Effect of Nb Loading on Catalytic Properties of Ni/Ce0.75Zr0.25O2 Catalyst for Methane Partial Oxidation

In this study,the effect of Nb loading on the catalytic activity of Ce_(0.75)Zr_(0.25)O_2-supported Ni catalysts was studied for methane partial oxidation.The catalysts were characterized by BET,H_2 chemisorption,XRD,TPR,TEM and tested for methane partial oxidation to syngas in the temperature range of 400-800℃at atmospheric pressure.The results showed that the activity of methane partial oxidation on the catalysts was apparently dependent on Nb loading.It seemed that the addition of Nb lowered the catalytic activity for methane partial oxidation and increased the extent of carbon deposition. This might be due to the strong interaction between NiO and Nb-modified support and reduction of surface oxygen reducibility.     

The effect of pressure on the dissociation constant of hydrofluoric acid and the association constant of the NaF ion pair at 25°C

The effect of pressure on the dissociation constant of hydrofluoric acid was determined by using the indicator technique at 25°C at an ionic strength of 0.1m over a pressure range of 1 to 2000 atm. A value of 3.14 for pK _a ^* at I =0 was obtained by extrapolation to zero ionic strength at 1 atm. The pressure dependence yielded a partial molar volume change of –9.6 cm³-mol^–1 and a compressibility change of — 35×10^–3 cm³-mol^–1 –atm^–1 for the dissociation. The dependence of ionic strength on the association constant K _A ^* of NaF was studied at 25°C and 1 atm. Extrapolation to I=0 yielded a pK _A ^* of –0.78. The pressure dependence of K _A ^* gave a change of volume of 3.26 cm³-mol^–1 and a change in compressibility of 6×10^–3 cm³-mol^–1-atm^–1 for the formation of the ion pair.     

Isentropic partial molal compressibilities of alcohols in propylene carbonate at 25°C

The isentropic coefficients of compressibility of the homologous series of alcohols and diols R _n CH₂OH (n=2–6), CH₃CHOHR _n (n=1–5), 1,2-propanediol, 1,3- 1,4- and 2,3-butanediol, 1,5-pentanediol, and 1,7-heptanediol dissolved in propylene carbonate have been measured at 25°C. Isentropic partial molal compressibilities and group partial molal compressibilities at infinite dilution have been evaluated. The isentropic partial molal compressibilities of these alcohols and diols have been compared with the corresponding values in water. This comparison shows that the values in propylene carbonate are higher than in water by a factor of 10 due to an increased compressibility of the solvation sheath around nonpolar groups in PC.