Partial molar volumes defined

These intensive variables are called the fatiial molar quantities associated with the extensive variable Y. As shown by (1.2.5), the chemical potential m is the partial molar quantity associated with the free energy G. Similarly to V corresponds the partial molar volume defined by... 

We define a partial molar volume Vi such that V = riiVi -I- U2V2... 

We may define, say, partial molar volume, enthalpy, or entropy by analogy with Eq. (8.5) ... 

The dependence of the rate constant on pressure provides another activation parameter of mechanistic utility. From thermodynamics we have (dGldP)T = V, where V is the molar volume (partial molar volume in solutions). We define the free energy of activation by AG = G — SGr. where SGr is the sum of the molar free energies of the reactants. Thus, we obtain... 

The volume of activation, AV is defined as the difference between the partial molar volumes of the transition state and the reactants. It is related to the pressure variation of the rate constant by Eq. (4) ... 

Because for a pure phase partial molar volume is equivalent to molar volume Vg, and since we have defined as standard state the condition of the pure component at P = 1 bar and T of interest ... 

For example, we may choose to as the average volume velocity, to = (c, V,) v,-. In more general terms, we may define to by X Pi vi > with Y Pi = 1 The s are weighting factors. If we formulate Eqn. (4.72) for two different reference velocities, to and to", and take into account the partial molar volumes (V,) which are not independent of each other (Gibbs-Duhem relation), we obtain after some algebraic rearrangements [H. Schmalzried (1981)] the quite general expression... 

Partial molar quantities can be defined as the change of an extensive variable with respect to the mole number of one component at constant temperature, pressure, electric field, and mole numbers of all other components. Then, with Equations (14.73) and (14.74), the change of the partial molar entropy and partial molar volume with the electric field is given by... 

This volume is called partial molar volume of A and denoted by symbol. V A. This partial volume is dependent on the composition of the solution. If the composition changes, this value also changes. As T, P and nB are kept constant during the process, we may define file partial molar volume as... 

The increase in (salt) for these onium salts can be understood in terms of mutual structure-forming or long-range hydrophobic interaction. The hydrocarbon side chains occupy voids in the solvent-lattice , which is stabilized by co-solvent but which collapses when x2 > xf. It is interesting therefore to note that the partial molar volume of R4N + ions decreases as x2 increases in ethyl alcohol + water mixtures, reaching a minimum near x2 0-1 which becomes more clearly defined with increase in size of the R-groups (Lee and Hyne, 1970). A similar trend is observed in acetone + water mixtures (Macdonald and Hyne, 1970). By way of contrast, the partial molar volumes, F (ion), for I-, Br , and Cl- ions decrease steadily to minima near x2 = 0-3 in ethyl alcohol + water mixtures where V2 (R4N+) is a maximum (Lee, 1972 1973). 

This equation defines the partial molar volume of species i in solution. It is simply the volumetric response of the system to the addition at constant T and P of a differential amount of species i A partial molar property may be defined in like fashion for each extensive thermodynamic property. Letting M represent the molar value of such a property, we write the general defining equation for a partial molar property as... 

Example 11.1 What physical interpretation can be given to the defining expression [Eq. (11.1)] for the partial molar volume ... 

The chemical potential is the partial molar Gibbs free energy. Partial molar quantities figure importantly in the theory of solutions and are defined at constant temperature and pressure thus, the Gibbs free energy is a natural state function for their derivation. As an example, the partial molar volume is found from the Maxwell relation... 

The partial molar volume V of component 1 in a mixture is defined as the increase in total volume V when 1 mol of the component is added to a sufficiently large amount of solution that the concentration does not change. For sodium chloride, the total volume V of solution is given by... 

The Kirkwood—Buff Integrals. The Kirkwood—Buff theory of solution relates the so-called Kirkwood—Buff integral (defined below) to macroscopic quantities, such as the compressibility, partial molar volumes, and the composition derivative of the activity coefficient. 

Partial Molar Quantities Equation (1.16) may be simplified by introducing the quantity Vi called the partial molar volume of the component i defined by ... 

R is the gas constant, Bn is the second virial coefficient for the probe, B13 is the mixed virial coefficient of the solute vapor and carrier gas, V is the partial molar volume of 1 at infinite dilution, P is the total pressure, and K is the equilibrium partition coefficient, defined as the ratio of concentration of solute in the stationary phase, q, to that in the gas phase, c, that is, K s q/c. 

Another asp>ect of partial molar volumes and heat contents, in particular, arises from the thermod3mamic requirement that for an ideal gas mixture or for an ideal liquid solution, as defined for example in 30a and 34a, respectively, there is no change of volume or of heat content upon mixing the components. This means that the partial molar volume and heat content of each substance in the mixture are equal to the respective molar values for the pure constituents. Any deviation of the partial molar quantity from the molar value then gives an indication of departure from ideal behavior this information is useful in connection with the study of solutions. 

On the basis of the above analysis it has been shown the partial molar quantities are easily obtained from intensive quantities like the molar volume when this quantity is plotted as a function of an intensive composition variable like the mole fraction. The plots in fig. 1.2 show that the molar volume is almost a linear function of the mole fraction of solute. If the curves in fig. 1.2 were actually perfect straight lines, the partial molar volumes would be constant independent of solution composition. Such a situation would arise if the solution were perfectly ideal. In reality, very few solutions are ideal, as will be seen from the discussion in the following section. In order to see more clearly the departure from ideality, one defines and calculates a quantity called the excess molar volume. This quantity is equal to the actual molar volume less the molar volume for the solution if it were ideal. The latter can be considered as the volume of the solution that would be found if the molecules of the two components form a solution without expansion or contraction. Thus, the ideal molar volume can be defined as... 

Equation (46) implies that the fraction of solute crystallized is proportional to the reduction in the partial molar volume of the solvent. Accordingly, the characteristic parameter for the antisolvent crystallization is the relative partial molar volume reduction (RPMVR), which is defined as... 

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