Dilute solutions calculating densities/concentrations

Calculating Densities/Concentrations in Stoichiometric Compounds or Dilute Solutions... 

The terms in Eq. (6) include the gravitational constant, g, the tube radius, R, the fluid viscosity, p, the solute concentration in the donor phase, C0, and the penetration depth, The density difference between the solution and solvent (ps - p0) is critical to the calculation of a. Thus, this method is dependent upon accurate measurement of density values and close temperature control, particularly when C0 represents a dilute solution. This method has been shown to be sensitive to different diffusion coefficients for various ionic species of citrate and phosphate [5], The variability of this method in terms of the coefficient of variation ranged from 19% for glycine to 2.9% for ortho-aminobenzoic acid. 

A simple calculation reveals the limits of the numbers of water molecules that may be associated with an ion in a standard solution. A l mol dm-3 aqueous solution of sodium chloride has a density of 1038 kg m-3 at 25 °C, so 1 dm3 of such a solution has a mass of 1038 g. One mole of the salt has a mass of 58.44 g, so the water in the litre of solution has a mass of 1038-58.44 = 979.56 g. This amount of water contains 979.56/18.015 = 54.4 moles of the liquid. The molar ratio of water molecules to ions in the 1 mol dm-3 aqueous solution of Na h(aq) and Cl (aq) ions is therefore 54.4/2 = 27.2, assuming that the water molecules are shared equally between the cations and anions. This represents the theoretical upper limit of hydration of any one ion in a standard solution of 1 mol dm-3 concentration. The limit may be exceeded in more dilute solutions, but that depends upon the operation of forces over a relatively long range. Certainly, in more concentrated solutions, the limits of hydration of ions become more restricted as fewer water molecules are available to share out between the cation and anion assembly. 

For each series of measurements about 50 g of solvent was transferred quantitatively in the dry box to the cell by pouring it into the dilution bulb this was the minimum amount required to fill the cell bulb. The cell was removed from the dry box, placed in the oil bath, and connected to the bridge. Time was allowed for the attainment of thermal equilibrium then at least three resistance measurements were made at five-min intervals, and the average value was calculated. The cell was removed from the bath and returned to the dry box. Dilute stock solution was quantitatively added to the cell by means of a weighing buret. The contents of the cell were carefully mixed, and the resistance of the solution was measured as before. The procedure just described was repeated several times with the dilute stock solution and then with the concentrated stock solution. About ten concentrations with a hundredfold range were obtained. A portion of the final solution in the cell (the most concentrated solution) was removed, and the infrared spectrum taken no absorption band indicative of traces of water was observed at 3600 cm-1. It was necessary to obtain the densities of... 

Rates of mass transfer to the catalyst surface and pore diffusion can be calculated by the methods of Section 2.2.2 if the diffusion coefficients are known. However, the molecular theory of diffusion in liquids is relatively undeveloped and it is not yet possible to treat diffusion in liquids with the same rigour as diffusion in gases. The complicating factors are that the diffusion coefficient varies with concentration and that the mass density is usually more constant than the molar density of the solution. An empirical equation, due to Wilke and Chang, which applies in dilute solution, gives... 

A 2O-/1I homogenized blood sample is added to 5 ml diluting solution. Blood and diluent are mixed well. After 3 minutes the optical density is measured against water (X = 540 nm slit width = 0.02 mm I = 1.000 cm). The hemoglobin concentration is calculated by means of the equation... 

Problem Hydrogen peroxide is a powerful oxidizing agent used in concentrated solution in rocket fuels and in dilute solution as a hair bleach. An aqueous solution of H2O2 is 30.0% by mass and has a density of 1.11 g/mL. Calculate its... 

Experimentally, the dielectric constant and density of the substance are measured. Either the dielectric constant is measured at several temperatures or the refractive index must be measured the dipole moment is then calculated. The dielectric constant is measured in a conductance cell on the vapor or, more often, in dilute solution in a nonpolar solvent such as benzene.J For accurate work, the dielectric constant is measured at several concentrations to allow extrapolation to infinite dilution. ... 

Another frequent mistake among students is to try to apply the ideal gas law to calculate the concentrations of species in condensed-matter phases (e.g., liquid or solid phases). Do not make this mistake] The ideal gas law only applies to gases. To calculate concentrations for liquid or solid species, information about the density (pj) of the liquid or solid phase is required. Both mass densities and molar densities (concentrations) as well as molar and atomic volumes may be of interest. The complexity of calculating these quantities tends to increase with the complexity of the material under consideration. In this section, we will consider three levels of increasing complexity pure materials, simple compounds or dilute solutions, and more complex materials involving mixtures of multiple phases/compounds. 

When calculating the mass density, molar concentration, or molar volume of a specific individual species that is present in combination with other species (e.g., in a compound or solution), further work is needed. If the material s composition can be expressed in terms of a single stoichiometric compound or formula unit, the approach is still fairly straightforward— it just requires application of the compound stoichiometry. Similarly, dilute solutions, where the solute species is present in very low concentrations relative to the host solvent, can be handled in a relatively straightforward manner by assuming that the host material s density is not affected by the presence of the solute species. 

The degrees of dissociation and hydration numbers calculated from vapor pressures correlate quantitatively with the properties of dilute as well as concentrated solutions of strong electrolytes. Simple mathematical relations have been provided for the concentration dependences of vapor pressure, e.m.f. of concentration cells, solution density, equivalent conductivity and diffusion coefficient. Non-ideality has thus been shown to be mainly due to solvation and incomplete dissociation. The activity coefficient corrections are, therefore, no longer necessary in physico-chemical thermodynamics and analytical chemistry. 

To conclude this section on the DH theory, we would like to point out that these last two criticisms (neglecting short range repulsive interactions and linearizing the PBE) are the only valid criticisms. In fact the McMillan-Mayer theory (MMM) showed that, provided a correct definition of the "effective interaction potential" is given, the molecular structure of the solvent needs not to be considered explicitly(1) in calculating the thermodynamic properties of ionic solutions. This conclusion has very important consequences the first one is that, as the number density of ion in a typical electrolyte solutions is of the order of 10"3 ions/A, then the solution can be considered as a dilute ionic gas as a consequence the theories available for gases can be used for ionic fluids, provided the "effective potential" (more often called potential of the mean force at infinite dilution) takes the place ot the gas-gas interaction potential. Strictly this is true only in the limit of infinite dilution, but will hold also at finite concentrations, provided the chemical potential of the solvent in the given solution is the same as in the infinitely dilute solutions. This actually... 

Semi-empirical calculations carried out for the radical cation and its dimer revealed that frontier electron density involved in a radical substitution reaction is highest on the chlorine atom in the monomer form, whereas it is most important at the para-carbon atom in the dimer form [30]. In dilute solution, the hot radical cation would be rapidly trapped by NO present in large excess via the electronic interaction with the chlorine substituent, leading to the arenium ion intermediates. Such process would work favourably for o-substitution. In concentrated solution, the radical cation can be partially stabilised through coordination toward the chlorine atom of another chlorobenzene molecule to give a dimeric form, in which the frontier electron would... 

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