Integral molar volume

Here a suitable equation of state is required to provide a mathematical expression for the mixture molar volume, V. For some equations of state, it is better to use a form of equation 28 in which the integral is volume expHcit (3). Note also that for an ideal gas — Z — 1, and 0 = 1. 

The integral calculates the work done by the membrane resisting the pressure change in the cell. Assuming that the molar volume Vm is constant, the osmotic pressure is given by ... 

If the pressure dependence of the molar volume of the liquid is neglected, integration from a flat interface (r = oo) yields... 

The partial molar volume of the solvent can be obtained by the integration illustrated in Equation (18.37). To evaluate dy 2 we merely need to differentiate Equation (18.42) ... 

Let us now consider how these quantities are related to experimentally determined heats of adsorption. An essential factor is the condition under which the calorimetric experiment is carried out. Under constant volume conditions, AadU 1 is equal to the total heat of adsorption. In such an experiment a gas reservoir of constant volume is connected to a constant volume adsorbent reservoir (Fig. 9.3). Both are immersed in the same calorimetric cell. The total volume remains constant and there is no volume work. The heat exchanged equals the integral molar energy times the amount of gas adsorbed ... 

Sm is the molar entropy, Vm the molar volume of the gas at a distance x, and Um is the molar internal energy. Integrating the left side from infinite distance to a distance x leads to... 

In compressible gases, the molar volume changes with pressure. Using the ideal gas laws in integrating Equation (2.4) gives... 

In the case of limited solubility, for example an aqueous solution of sucrose, no difficulty arises in the use of Equation (6.65), if we use the subscript 1 to refer to the solvent and the subscript 2 to refer to the solute. Here V2 could be determined as a function of the mole fraction from the very dilute solution (pure solvent) to the saturation point. The lower limit of integration would then be V[ when = 1. With Equation (6.66), Vx could be determined as a function of the mole fraction from the very dilute solution to the saturation point. However, in order to determine the value of the integration constant, V2 would have to be known at some mole fraction. This is usually not the case, and so the integration cannot be completed. This problem is clarified by reference to Figure 6.3, where the molar volume of a system having limited solubility is plotted as a function of the mole fraction of the solvent. The value of the mole fraction at the saturation point is indicated by the vertical broken line is indicated by the solid curve from the saturation value to the pure solvent. The form of the curve of V2 can be obtained by integration of Equation (6.66), but its absolute value cannot be determined. The broken curves represent a family of curves that differ by a constant value. All of... 

In the case where the external pressure is not too high (< 100 atm.) and the compressibility remains independent of pressure, the second equation of 7.16 can be integrated with respect to pressure p to obtain Eq. 7.21 for the molar volume v(T, p) of a condensed substance ... 

Since Vf> the liquid-phase molar volume, is a very weak function of P at temperatures well below T , an excellent approximation is often obtained when evaluation of the integral is based on the assumption that Vt is constant at the value for saturated liquid, Vj ... 

The Clapeyron-Clausius equation as applied to liquid = vapour equilibrium, can be easily integrated. The molar volume of a substance in the vapour state is considerably greater than that in the liquid state. In the case of water, for example, the value of Vg at 100 C is 18 1670 = 30060 ml while that of V, is only a little more than 18 ml. Thus, V-Vt can be taken as Vwithout introducing any serious error. The Clapeyron-Clausius equation 1.51, therefore, may be written as... 

An adsorption step, during which an amount of liquid na is vaporized from the liquid phase (with a molar energy of vaporization Avap ) and adsorbed on the solid surface (with an integral molar adsorption energy A k, precisely defined for an adsorption process at constant volume (cf. Equation (2.59)), at temperature T and for the surface excess concentration r=n°/A. 

This quantity is called the Joule coefficient. It is the limit of -(A77AF) . corrected for the heat capacity of the containers as AUapproaches zero. With the van der Waals equation of state, we obtain p = ajy C - The eorrected temperature change when the two eontain-ers are of equal volume is found by integration to be AT = -a/2 FC , where V is the initial molar volume and C is the molar constant-volume heat capacity. It is instractive to calculate this AT for a gas such as CO2. In addition, the student may consider the relative heat capacities of 10 L of the gas at a pressure of 1 bar and that of the quantity of eopper required to constract two spheres of this volume with walls (say) 1 mm thiek and then eal-culate the AT expected to be observed with such an experimental arrangement. 

Experimental determination of excess molar quantities such as excess molar enthalpy and excess molar volume is very important for the discussion of solution properties of binary liquids. Recently, calculation of these thermodynamic quantities becomes possible by computer simulation of molecular dynamics (MD) and Monte Carlo (MC) methods. On the other hand, the integral equation theory has played an essential role in the statistical thermodynamics of solution. The simulation and the integral equation theory may be complementary but the integral equation theory has the great advantage over simulation that it is computationally easier to handle and it permits us to estimate the differential thermodynamic quantities. 

In principle this accomplishes the task one measures the (molar) volume of the nonideal gas as a function of the applied pressure at fixed temperature T, beginning at a very low, fixed value of the pressure, Pi, and ending at the pressure P for which the fugacity / is to be found. Alternatively, an equation of state may be employed for insertion in Eq. (3.1.10). If this is not convenient one may integrate by parts to obtain... 

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. 

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