Molar and molal scales

Molar volumes of solute and solvent, Vy and Vs, are determined from apparent molar volumes Oy (Section II.F) by density measurements. In terms of the measured densities of the solution and the pure solvent, d and d, the apparent molar volume is given by (molarity and molality scale, respectively)... 

It should be noted that there is sometimes an ambiguity in the concentration scales used because of the difference in densities between H2O and D2O the ratio K /K will have different values on the molarity and molality scales. It is probably most satisfactory to express all concentrations as moles per 55.51 moles of solvent this has been termed the aquamolality scale. Since H2O and D2O differ in molar volumes by only 0.36% the use of molarities will give virtually the same value for K /K. ... 

Figure 36. Calculated pH on the molar and molal scales for pure water as a function of temperature for a pressure of 400 bar. Reprinted from Ref. 5, Copyright (1997) with permission by Elsevier.

Concentration is expressed on the molar scale in terms of mol L of solvent or on the molal scale in terms of mol kg of solvent. The molal scale gives concentrations that are independent of temperature and pressure. In this chapter, the molar scale will be used on the basis that molarity and molality are almost identical for the low ionic strengths commonly associated with freshwaters. 

The values for are those obtained by Wesolowski and Palmer all values are given in unitless form but fundamentally are defined in terms of the products of the activities of ions on the molar or molal scales. 

Any activity can be written as the product of a concentration and activity coefficient. Here we usually express concentration in terms of mol liter" of solution (molarity, M). Concentration may also be expressed in terms of mol kg" of solvent (molality, M). The molal scale gives concentrations that are independent of temperature and pressure and is used in precise physicochemical calculations. The difference between molarity and molality is small in dilute solutions, especially in comparison to the uncertainties involved in determining... 

The standard potentials on the molal molar and rational scales E ) are related by... 

Another approach to non-ideality of aqueous solutions does not take into account the nature of interactions between constituents the simple solution concept allows one to determine thermodynamic properties of concentrated solutions of salts and also the density of various mixtures the latter parameter being required for the conversion of concentrations from the molar to molal scale [41]. 

Ionic Equilibria.. The ion product constant of D2O (see Table 3) is an order of magnitude less than the value for H2O (24,31,32). The relationship pD = pH + 0.41 (molar scale 0.45 molal scale) for pD ia the range 2—9 as measured by a glass electrode standardized ia H2O has been established (33). For many phenomena strongly dependent on hydrogen ion activity, as is the case ia many biological contexts, the difference between pH and pD may have a large effect on the iaterpretation of experiments. 

A concentration scale for solutes in aqueous solutions, equal to moles of solute/55.51 mol water. It is frequently used in studies of solvent isotope effects. As pointed out by Schowen and Schowen the choice of standard states can change the sign for the free energy of transfer of a species from one solvent to another, even from HOH and DOD. The commonly used concentration scales are molarity, mole fraction, aquamolality, and molality. Free energies tend to be nearly the same on all but the molality scale, on which they are about 63 cal mol more positive at 298 K than on the first three scales. The interested reader should consult Table I of Schowen and Schowen ... 

Here, p and m are the standard chemical potential and concentration (molal scale) of the /-component (z = 1 for solvent, z = 2 for biopolymer) A2 is the second virial coefficient (in molal scale units of cm /mol, i.e., taking the polymer molar mass into account) and m° is the standard-state molality for the polymer. 

Figure 6.10 Effect of CITREM on the molecular and thermodynamic parameters of maltodextrin SA-2 (DE = 2) in aqueous medium (phosphate buffer, pH = 7.2, ionic strength = 0.05 M 20 °C) (a) weight average molar mass, Mw (b) radius of gyration, Ra (c) structure sensitive parameter, p, characterizing die architecture of maltodextrin associates (d) second virial coefficient, A2 or A2, on the basis of the weight ( ) and molal (A) scales, respectively. The parameter R is defined as the molar ratio of surfactant to glucose monomer units in the polysaccharide. The indicated cmc value refers to the cmc of the pure CITREM solution. Reproduced from Anokhina et al. (2007) with permission.

Some pKSH values in non-aqueous solvents and water-organic solvent mixtures are listed in Table 6.6 [18, 19]. They are in molal scale but can easily be converted to molar scale by the relationship ... 

Other measures of the solubility of a gaseous solute are readily derived from its mole fraction. The Henry s law constant is KH lim(pjxj30) atp2 —> 0, which becomes on the molal scale Km lim(1000 p2/Mjc2) with Mj the molar mass of the solvent in g mob1 for use in Eq. (2.13) below. The Ostwald coefficient is the limit of y2 = RT(xJp lVl atp2 -> 0 and is related to the mass fraction w2 by ... 

Table 9.3. Standard chemical potential jj°, standard partial molar enthalpy h°, and standard partial molar entropy s,° for a few hydrated ions Standard state 101.3 kPa, 298 K, unit activity in molality scale.

At 25°C, paj, values on the molality scale are 0.043 unit higher than on the molarity scale,since the density of deuterium oxide is higher than that of water. The accuracy of measured pD values depends in part on the variability of the liquid-junction potential between the deuterium oxide and saturated potassium chloride in the reference electrode. 

Some further nomenclature is now necessary to describe absorption equilibria in ion exchange systems. For a species i, m and C, represent the molal and molar concentrations respectively, whilst A and Xi denote the mole fraction and equivalent ionic fraction of i respectively. Single ion activity coefficients are denoted yj and mean ionic activity coefficients by yj . Whether the latter quantities refer to the molar or molal concentration scales is decided by the choice of units defining concentration. Thermodynamic activities and activity coefficients for the resin phase using the equivalent or mole fraction concentration scale (rational scale) are sometimes defined differently and are discussed in a later section. Finally, the exchanger and external solution phases are differentiated by subscripts r and s respectively. 

A 0.1 m solution of MgCl2 in water has a density of 1.0047 gmL at 25°C. The mean ionic activity coefficient on the molal scale is 0.528. Calculate the mean activity and electrolyte activity on this scale. Repeat the calculations for the molarity scale. 

Here, cs denotes the concentration in mol/kg (molality scale), and [s] is the concentration in mol/liter (molarity scale). Both units are related in that [s] = pwcs where pw= 1 kg/dm3 is the density of water. Its variation with temperature causes the molarity scale to depend on temperature, whereas the molality scale does not. In the temperature range 0-25°C, however, the density of water differs from unity by less than 0.3%, so that [s] = cs with reasonable accuracy. Most Henry coefficients are less well known. From the definitions in Eqs. (8-7) and (8-8), the coefficients involved are related by... 

A great majority of studies on equilibria in aqueous solution have been carried out with the final result presented on the molar scale (M). In the presentation of the primary result this scale has been kept and no recalculation to the molal scale (m) performed. Even when not specifically mentioned in the text, the SIT procedure requires that all results be converted to the molal scale for extrapolation to the infinite dilution (/= 0) standard state. All pressures are expressed using the unit of bars and the necessary transformation has been made by the review. If this rule was not followed for any reason then the pressure unit used is explicitly mentioned. 

The word molar in this phrase refers to per mole, and has no connection with the molarity scale of concentration. An alternative name is partial moial quantity, which is synonymous, and has no connection with the molality scale of concentration. 

Figure A-52 Distribution coefficients of Th(IV) in the TBP extraction study of [2006NEC/ALT] (aqueous phase 0.05-3 M NaNOs + 0.02 M HNO3 organic phase TBP/w-dodecane). Experimental data and recalculation a) with the ion interaction model, b) with the nitrate complexation model. (All calculations are performed on the molal scale and back-transformed to the molar scale for direct comparison with the experimental data.)...

Constants were originally reported on the molality scale and have been converted to molarity units here by adding a correction to each log Kn value based on the density ps(W> of the water-saturated solvent phase, assuming the density of the solvent-saturated aqueous phase pw(S) remains unchanged. The solvent, corresponding value of ps(W) at 25 C, and resulting correction [equal to log(ps(W)/pwmolar volumes of water and l-hexanol), 0.2 DiPeMP, 0.96 g/cm [165], 0.03 TBP, 0,976 g/cm [250], -0.02. 

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