The use of recurrent equations in chromatography

Recurrent equations A(x + k) = aA(x) + b were shown to be applicable to the approximation not only of virtually arbitrary properties of organic compounds (A) in homologous series (A = n _C, k = 1 or 2) but also of the dependences of chromatographic retention parameters on the number of carbon atoms in homologue molecules (A = t _R). The same equations described the temperature dependences of retention times of arbitrary compounds under isothermal separation conditions in gas chromatography (x = T, k = ΔT = const) and the dependences of retention times on the concentration of an organic solvent as an eluent component (x = C, k = ΔC = const) under isocratic separation conditions in high performance liquid chromatography.     

Thermodynamics of the terpolymer of carbon monoxide,ethylene, and 1-butene

The heat capacity and the temperatures and enthalpies of physical transformations of the alternating terpolymer of carbon monoxide, ethylene, and 1-butene (the content of butene units is 10.7 mol.%) were studied by adiabatic and differential scanning calorimetry in the temperature range from 6 to 520 K. The energy of terpolymer combustion was measured at 298.15 K on an calorimeter with an isothermal shell and static bomb. The standard thermodynamic functions C°_p(T), H°(T)–H°(0), S°(T)–S°(0), and G°(T)–H°(0) for the range from Т → 0 to 400 K, the standard enthalpy of combustion, and the thermodynamic parameters of formation of the partially crystalline CO—ethylene—1-butene terpolymer at 298.15 K, as well as the thermodynamic characteristics of its synthesis in the range from T → 0 to 400 K were calculated.     

Heat capacity and thermodynamic functions of new cobalt manganites NdM<Subscript>2</Subscript><Superscript>I</Superscript> CoMnO<Subscript>5</Subscript> (M<Superscript>I</Superscript> = Li,Na, and K) in the range of 298.15–673 K

Temperature dependences of the heat capacity of cobalt manganites NdM₂ ^I CoMnO₅ (M^I = Li, Na, and K) are studied by means of dynamic calorimetry in the range of 298.15?673 K. It is found that λ-shaped effects are observed on the C _p ^° ~ f (T) curve of cobalt manganites, due probably to second order phase transitions. Based on the experimental data, equations for the temperature dependences of the heat capacity of cobalt manganite are derived with allowance for the temperatures of phase transitions. The values of thermodynamic functions Н°(T)–Н°(298.15), S°(T), and Ф^хх(T) are calculated.     

The <Emphasis Type="Italic">pVT</Emphasis> properties and phase equilibria in the <Emphasis Type="Italic">n</Emphasis>-pentane-water binary system

The pVT properties of the 0.075 n-C₅H₁₂ + 0.925H₂O binary system were studied experimentally. Thermal properties were measured along nine different isochores over the density and temperature ranges 87–698 kg/m³ and 303–684 K at pressures up to 60 MPa. Measurements were performed using a constant-volume piezometer. Inflection and break points of the p(T) dependences were used to construct phase equilibrium curves, which were described by scaling equations.     

Thermodynamic Properties of Zincate-Manganites of LaM<Subscript>2</Subscript> <Superscript>II</Superscript>ZnMnO<Subscript>6</Subscript> (М<Superscript>II</Superscript> = Mg,Ca, Sr,Ba) Composition

Temperature dependences of the heat capacity of new zincate-manganites of LaM₂^IIZnMnO₆ (M^II = Mg, Ca, Sr, Ba) composition are studied via experimental calorimetry in the interval of 298.15–673 K. It is found that all compounds have λ-shape effects on the curve of dependence C_p° ~ ?(T) with respect to phase transitions of the second kind. Equations for the temperature dependence of the heat capacity are derived with allowance for phase transition temperatures, and thermodynamic functions H°(T) ? H°(298.15), S°(T) and Φ^xx(T) are calculated on the basis of experimental data on C_p°(T) and the calculated S°(298.15) value.     

Thermodynamic properties of a liquid crystal carbosilane dendrimer

The temperature dependence of the heat capacity of a first-generation liquid crystal carbosilane dendrimer with methoxyphenyl benzoate end groups is studied for the first time in the region of 6–370 K by means of precision adiabatic vacuum calorimetry. Physical transformations are observed in this interval of temperatures, and their standard thermodynamic characteristics are determined and discussed. Standard thermodynamic functions C_p^°(T), H°(T) ? H°(0), S°(T) ? S°(0), and G°(T) ? H°(0) are calculated from the obtained experimental data for the region of Т → 0 to 370 K. The standard entropy of formation of the dendrimer in the partially crystalline state at Т = 298.15 K is calculated, and the standard entropy of the hypothetic reaction of its synthesis at this temperature is estimated. The thermodynamic properties of the studied dendrimer are compared to those of second- and fourth-generation liquid crystal carbosilane dendrimers with the same end groups studied earlier.     

Concentration,temperature, and isotopy effects on the heat capacity of aqueous urea

Relations for the apparent molar heat capacity ?_c of urea in an aqueous solution depending on the molality m and temperature were obtained. A transition to the relations ?_c(m,T) for D₂O-(ND₂)₂CO and T₂O-(NT₂)₂CO systems was effected by temperature scaling. At low temperatures, the isotherms of the molar heat capacity C _p(m) of the protium and deuterium systems have minima shifted to more dilute solutions at elevated temperatures. At m = 1, C _p of a solution does not depend on temperature in both systems. The dependences C _p(T) also have minima at constant concentrations. The temperature of the minimum heat capacity is most effectively lowered by small additions of urea. For m = 0.25, T _min is 7.5 K lower than T _min of pure water, and its heat capacity is 0.08 J/(mol K) higher. A transition from m = 1.5 to m = 2 lowers the temperature of the minimum heat capacity by 3.6 K; thus, the heat capacity of solutions differs by 0.02 J/(mol K) only.     

Recurrent equations of physicochemical constants for homologs substantiated with numerical sequences

The applicability of the unified first order recurrent equations A(n + 1) = aA(n) + b to the approximation of the physicochemical constants of various organic compounds (not only members of homologous series), previously found for normal boiling points, was extended to the dielectric constant (ε). This tendency is equivalent to the general procedure for calculating the ε of any compound from the data for preceding homologs with an accuracy comparable to the average interlaboratory reproducibility of the results of ε measurements. To substantiate this universal character of recurrent relations we consider the analogy between the constants of successive homologs and the recursive numerical sequences (Fibonacci and Padovan sequences versus Lucas and Perrin sequences, respectively).     

Synthesis and study of high-temperature heat capacity of erbium orthovanadate

Orthovanadate ErVO₄ has been prepared by solid-phase synthesis from a stoichiometric mixture of high pure V₂O₅ and chemically pure Er₂O₃ by multistage calcination in air in the temperature range 873–1273 K. The effect of temperature (380–1000 K) on the heat capacity of orthovanadate ErVO₄ was studied by hightemperature calorimetry. Thermodynamic properties of erbium orthovanadate (enthalpy change H°(T)–H°(380 K), entropy change S°(T)–S°(380 K), and reduced Gibbs energy Φ°(T)) have been calculated from the experimental C_p = f(T) data. It has been shown that the specific heat varies in a row of oxides and orthovanadates of Gd-Lu naturally depending on the radius of the R³⁺ ion within the third and fourth tetrads.     

Exact treatment of generalized modifications of finite-dimensional systems by the LRM approach

LRM (Low Rank Modification) is a mathematical method that produces eigenvalues and eigenstates of generalized eigenvalue equations. It is similar to the perturbation expansion in that it assumes the knowledge of the eigenvalues and eigenstates of some related (unperturbed) system. However, unlike perturbation expansion, LRM produces correct results however large the modification of the original system. LRM of finite-dimensional systems is here generalized to the combined (external and internal) modifications. Parent n-dimensional system A _n containing n eigenvalues λ _i and n eigenstates \({| {\Phi_i}\rangle}\) is described by the generalized n × n eigenvalue equation. In an external modification system A _n interacts with another ρ-dimensional system B _ρ which is situated outside the system A _n . In an internal modification relatively small σ-dimensional subsystem of the parent system A _n is modified. Modified system C _n+ρ that contains external as well as internal modifications is described by the generalized (n + ρ) × (n + ρ) eigenvalue equation. This system has (n + ρ) eigenvalues \({\varepsilon_s}\) and (n + ρ) corresponding eigenstates \({| {\Psi_s}\rangle}\) . In LRM this generalized (ρ + n) × (ρ + n) eigenvalue equation is replaced with a (nonlinear) (ρ + σ) × (ρ + σ) equation which produces all eigenvalues \({\varepsilon_s \notin \left\{ {\lambda_i}\right\}}\) and all the corresponding eigenstates \({| {\Psi_s}\rangle }\) of C _{n + ρ}. Another equation produces remaining solutions (if any) that satisfy \({\varepsilon_s \in \left\{ {\lambda_i}\right\}}\) . Those two equations produce exact solution of the modified system C _{n + ρ}. If (ρ + σ) is small with respect to n, this approach is numerically much more efficient than a standard diagonalization of the original generalized eigenvalue equation. Unlike perturbation expansion, LRM produces exact results, however large modification of the parent system A _n .     

Thermodynamic Properties of Polyphenylquinoxaline in the Temperature Range of <Emphasis Type="Italic">T</Emphasis> → 0 to 570 K

The thermodynamic properties of amorphous polyphenylquinoxaline in the temperature range of 6 to 570 K are studied via precision adiabatic vacuum calorimetry and differential scanning calorimetry. The thermodynamic characteristics of glass transition are determined. Standard thermodynamic functions C^°_p, H°(T) ? H°(0), S°(Т) ? S°(0), and G°(T) ? H°(0) in the range of T → 0 to 570 K and the standard entropy of formation at T = 298.15 K are calculated. The low-temperature (T ≤ 50 K) heat capacity is analyzed using a multifractal model for the processing of heat capacity, fractal dimension D values are determined, and conclusions on the topological structure of the compound are drawn.     

Dehydroxylation of high-energy ball-milled diasporic bauxite

The review consists of four parts. Part I is about the thermal transformation in clay minerals under firing and about the properties of fired clay. The transformations in the clay are expressed quantitatively with the extent of conversion, α, which is the function of two parameters of firing process, time t and temperature T: α?=?f(t, T). Part II is about the estimation of firing temperature after the analysis of the products of firing clays. All the estimations are speculative, without mathematical equations. It is impossible to derive the value of only one variable T from α?=?f(t,T), the function of two variables. Part III is about the long-term inverse transformation in fired ceramic paste under ambient conditions. The mass gain as a function of time is used for the evaluation of the time elapsed from the ancient firing, t. Part IV is about the visualization of the extent of conversion of the clay in the paste after all direct and reverse reactions, i.e., α.     

Thermodynamic properties of triphenylantimony dibenzoate

The temperature dependence of the heat capacity of triphenylantimony dibenzoate Ph₃Sb(OC(O)Ph)₂ is studied in the range of 6–480 K by means of precision adiabatic vacuum calorimetry and differential scanning calorimetry. The melting of the compound is observed in this temperature range, and its standard thermodynamic characteristics are identified and analyzed. Ph₃Sb(OC(O)Ph)₂ is obtained in a metastable amorphous state in a calorimeter. The standard thermodynamic functions of Ph₃Sb(OC(O)Ph)₂ in the crystalline and liquid states are calculated from the obtained experimental data: C_p^°(T), H°(T)–H°(0), S°(T), and G°(T)–H°(0) for the region from T → 0 to 480 K. The standard entropy of formation of the compound in the crystalline state at T = 298.15 K is determined. Multifractal processing of the low-temperature (T < 50 K) heat capacity of the compound is performed. It is concluded that the structure of the compound has a planar chain topology.     

Apparent Molar Volumes of Aqueous Solutions of Magnesium Tetraborate from 283.15 to 363.15 K and 0.1 MPa

Densities for aqueous solutions of magnesium tetraborate MgB₄O₇(aq) at the molalities of (0.00556–0.03341) mol·kg^?1 were measured with an Anton Paar Digital vibrating-tube densimeter at temperature intervals of 5 K from 283.15 to 363.15 K and 0.1 MPa. Apparent molar volumes were obtained based on the experimental density data, and the 3D diagrams of the apparent molar volume (V _? ) of MgB₄O₇(aq) against temperature (T) and molality (m) were plotted. On the basis of the Vogel–Tamman–Fulcher equation, the coefficients of the correlation equation for densities of MgB₄O₇(aq) against temperature and molality were parameterized. According to the Pitzer ion-interaction model of the apparent molar volume, the temperature correlation equations of Pitzer single-salt parameters F(i,p,T)?=?a₀?+?a₁?×?T?+?a₂?×?T ²?+?a₃/T?+?a₄?×?ln(T)?+?a₅?×?T ³ (where T is temperature in Kelvin, a _i are model parameters) for MgB₄O₇ were obtained for the first time.     

Characterization of neutron spectrum at Es-Salam Research Reactor using Høgdahl convention and Westcott formalism for the k<Subscript>0</Subscript>-based neutron activation analysis

For the general applicability of the k ₀-NAA method two formalisms were carried out to deal with “1/ν and non-l/ν ((n,γ)” reaction nuclides, respectively. In the Høgdahl-formalism the reactor neutron spectrum parameters, such as α and f were measured using three methods: Cd-ratio, Cdcovered and bare triple monitors. In addition, bare bi-isotopic method using Zr is also utilized for the calculation of f. According to the Westcottformalism the modified spectral index r(α)√T _n /T ₀ and g(T _n ) factor for monitoring neutron temperature T _n , were measured employing Lu as non “1/v” monitor and ¹⁹⁷Au, ⁹⁶Zr and ⁹⁴Zr as “1/v” monitors. The reduced resonance integral of lutetium s _0,Lu was also calculated. To evaluate the applicability of k ₀-NAA in our analytical system, the analysis of two kinds of SRMs was executed. The analytical results showed that the relative error of most of the elements was less than 10%.     

Thermodynamic properties of first- and third-generation carbosilane dendrimers with terminal phenyldioxolane groups

The heat capacities of first- and third-generation carbosilane dendrimers with terminal phenyldioxolane groups are studied as a function of temperature via vacuum and differential scanning calorimetry in the range of 6 to 520 K. Physical transformations that occur in the above temperature range are detected and their standard thermodynamic characteristics are determined and analyzed. Standard thermodynamic functions C_p^ο(T), [H°(T) ? H°(0)], [S°(T) ? S°(0)], and [G°(T) ? H°(0)] in the temperature range of T → 0 to 520 K for different physical states and the standard entropies of formation of the studied dendrimers at T = 298.15 K are calculated, based on the obtained experimental data.     

A Search for a Direct Relationship Between Chromatographic and Adsorption Data

Shifts of chromatographic peak maxima and centres of gravity have been investigated for different amounts of propane injected on to a chromatographic column in ideal, non-linear chromatography. Specific retention volumes (V _{g (273)}, corrected to the standard temperature, 273.15 K), propane adsorption isotherms, and the first and second derivatives of the isotherms, (da/dp) _T and (d²a/dp²) _T , were determined for samples of active carbon and for different amounts of propane injected on column. Relationships between specific retention volume and the molar differential work of adsorption, A, were calculated on the basis of the propane isotherms and using the retention times of the peak maxima and the centres of gravity of the peaks. The equations obtained, ln V _{g (273)}=f₁(A) and(dW/dA) _{T, F c} = f₂(ln V _{g (273)}), have been used to explain the relationships between (i) chromatographic peak profiles and (ii) the distribution function of pore volumes filled with propane and the molar differential work of adsorption at different column temperatures (303–318 K).     

Heat capacities and thermodynamic functions of new nanosized ferro-chromo-manganites LaM<Subscript>0.5</Subscript><Superscript>II</Superscript>FeCrMnO<Subscript>6.5</Subscript> (M<Superscript>II</Superscript>–Mg,Ca, Sr,Ba)

The heat capacities of nanosized ferro-chromo-manganites LaM_0.5 ^IIFeCrMnO_6.5 (M^II–Mg, Ca, Sr, Ba) are measured via dynamic calorimetry in the temperature range of 298.15–673 K using an IT-S-400 instrument. It is established that the C°_p~f(T) function of LaM_0.5 ^IIFeCrMnO_6.5 (M^II–Mg, Ca, Sr, Ba) has λ-type effects, due probably to phase transitions of the second order. Considering the temperatures of the phase transitions, equations of the heat capacity of ferro-chromo-manganites LaM_0.5 ^IIFeCrMnO_6.5 (M^II–Mg, Ca, Sr, Ba) as a function of temperature are derived on the basis of experimental data. Thermodynamic functions Н°(Т)–Н°(298.15), S°(Т), and Ф ^хх(Т) are calculated in the temperature range of 298.15–675 K.     

High-temperature heat capacity and thermodynamic properties of pyrovanadate Pb<Subscript>2</Subscript>V<Subscript>2</Subscript>O<Subscript>7</Subscript> and orthovanadate Pb<Subscript>3</Subscript>(VO<Subscript>4</Subscript>)<Subscript>2</Subscript>

The heat capacities of Pb₂V₂O₇ and Pb₃(VO₄)₂ as a function of temperature in the range 350–965 K have been studied by the differential scanning calorimetry method. The C_P = f(T) curve for Pb₂V₂O₇ is described by the equation C_p = (230.76 ± 0.51) + (73.60 ± 0.50)×10^-3T ? (18.38 ± 0.54)×10⁵T^-2 in the entire temperature range. For Pb₃(VO₄)₂, there is a well-pronounced extreme point in the C_P = f(T) curve at T = 371.5 K, which is caused by the existence of a structural phase transition. The thermodynamic properties of the oxide compounds have been calculated.