Van’t Hoff relation

The enthalpy changes due to dimerization are determined from the van t Hoff relation. For a dimerization reaction between species i and j... 

Differentiation with respect to reciprocal temperature and use of the Arrhenius and van t Hoff relations gives... 

Osmotic pressure is often expressed by Equation 2.10, known as the Van t Hoff relation, but this is justified only in the limit of dilute ideal solutions. As we have already indicated, an ideal solution has ideal solutes dissolved in an ideal solvent. The first equality in Equation 2.9 assumes that yw is unity, so the subsequently derived expression (Eq. 2.10) strictly applies only when water acts as an ideal solvent (yw = 1.00). To emphasize that we are neglecting any factors that cause yw to deviate from 1 and thereby affect the measured osmotic pressure (such as the interaction between water and colloids that we will discuss later), riy instead of n has been used in Equation 2.10, and we will follow this convention throughout the book. The increase in osmotic pressure with solute concentration described by Equation 2.10 is... 

Figure 2-9. Relationship between concentration and osmotic pressure at 20°C for a nonelectrolyte (sucrose) and two readily dissociating salts (NaCl and CaCl2). The different initial slopes indicate the different degrees of dissociation for the three substances and are consistent with the Van t Hoff relation (Eq. 2.10). Data for osmotic pressure are based on the freezing point depression. (Data source Lide, 2008.)...

Hie concentration unit consistent with the conventional SI units for R is mol m-3. In that regard, the derivation of the Van t Hoff relation (Eq. 2.10) uses the volume of water, not the volume of the solution, so technically a unit based on molality is implied. However, the Van t Hoff relation is only an approximate representation of the osmotic pressure appropriate for dilute solutions, for which the numerical difference between molality and molarity is usually minor, as indicated in the text. Thus, molarity of osmotically active particles (= osmolarity) is suitable for most calculations and is generally more convenient (note that concentration in moles per m3 is numerically equal to ihm). 

In this chapter we will derive the Boyle-Van t Hoff relation using the chemical potential of water, and in Chapter 3 (Section 3.6B) we will extend the treatment to penetrating solutes by using irreversible thermodynamics. Although the Boyle-Van t Hoff expression will be used to interpret the osmotic responses only of chloroplasts, the equations that will be developed are general and can be applied equally well to mitochondria, whole cells, or other membrane-surrounded bodies. 

The Boyle-Van t Hoff relation applies to the equilibrium situation for which the water potential is the same on either side of the two membranes surrounding a chloroplast. When T1 equals T°, net water movement across the membranes ceases, and the volume of a chloroplast is constant. (The superscript i refers to the inside of the cell or organelle and the superscript o to the outside.) If we were to measure the chloroplast volume under such conditions, the external solution would generally be at atmospheric pressure (P° =0). By Equation 2.13a (T = P — H, when the gravitational term is ignored), the water potential in the external solution is then... 

To appreciate the refinements that this thermodynamic treatment introduces into the customary expression describing the osmotic responses of cells and organelles, we compare Equation 2.18 with Equation 2.15, the conventional Boyle-Van t Hoff relation. The volume of water inside the chloroplast is VM,n because n v is the number of moles of internal water and Vw is the volume per mole of water. This factor in Equation 2.18 can be identified with V — b in Equation 2.15. Instead of being designated the nonosmotic volume, b is more appropriately called the nonwater volume, as it includes the volume of the internal solutes, colloids, and membranes. In other words, the total volume (V) minus the nonwater volume (b) equals the volume of internal water (Ew ). We also note that the possible hydrostatic and matric contributions included in Equation 2.18 are neglected in the usual Boyle-Van t Hoff relation. In summary, although certain approximations and assumptions are incorporated into Equation 2.18 (e.g., that solutes do not cross the limiting membranes and that the... 

The intercept on the ordinate in Figure 2-11 is the chloroplast volume theoretically attained in an external solution of infinite osmotic pressure —a l/n° of zero is the same as a n° of infinity. For such an infinite 11°, all of the internal water would be removed = 0), and the volume, which is obtained by extrapolation, is that of the nonaqueous components of the chloroplasts. (Some water is tightly bound to proteins and other substances and presumably remains bound even at the hypothetical infinite osmotic pressure such water is not part of the internal water, Vwn v). Thus the intercept on the ordinate of a F-versus-l/n° plot corresponds to b in the conventional Boyle-Van t Hoff relation (Eq. 2.15). This intercept (indicated by an arrow in Fig. 2-11) equals 17 pm3 for chloroplasts both in the light and in the... 

In Chapter 2 (Section 2.3B) we derived the Boyle-Van t Hoff relation assuming that the water potential was the same on both sides of the cellular or organelle membrane under consideration. Not only were... 

When molecules cross the membranes bounding cells or organelles, the reflection coefficients of both internal and external solutes should be included in the Boyle-Van t Hoff relation. Because less than 1 when the external solutes can penetrate, the effect of the external osmotic pressure on Jv is then reduced. Likewise, the reflection coefficients for solutes within the cell or organelle can lessen the contribution of the internal osmotic pressure of each solute. Replacing nj by RTn j / (VM ji v ) (Eq. 2.10) in Equation 3.41 and dividing by a° leads to the following Boyle-Van t Hoff relation for the stationary state condition (Jv = 0 in Eq. 3.40) ... 

As indicated in Chapter 2 (Section 2.3B), V — b in the conventional Boyle-Van t Hoff relation [H°(V — b) = RTY ffinj (Eq. 2.15)] can be identified with Vwn v, the volume of internal water. Comparing Equation 2.15 with Equation 3.44, the osmotic coefficient of species /, permeation properties of solutes, both internal and external. Indeed, failure to recognize the effect of reflection coefficients on

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Equation 3.48 indicates that not only does Js depend on An, as expected from classical thermodynamics, but also that the solute flux density can be affected by the overall volume flux density, Jv. In particular, the classical expression for Js for a neutral solute is P Ac (Eq. 1.8), which equals (Pj/RT)ATlj using the Van t Hoff relation (Eq. 2.10 II, = PT Cj). Thus to, is analogous to P/RT of the classical thermodynamic description (Fig. 3-19). The classical treatment indicates that Js is zero if An is zero. On the other hand, when An is zero, Equation 3.48 indicates that Js is then equal to c,(l - cr,)/y solute molecules are thus dragged across the membrane by the moving solvent, leading to a solute flux density proportional to the local solute concentration and to the deviation of the reflection coefficient from 1. Hence, Pj may not always be an adequate parameter by which to describe the flux of species , because the interdependence of forces and fluxes introduced by irreversible thermodynamics indicates that water and solute flow can interact with respect to solute movement across membranes. 

Temperature Effects. The temperature range for which this model was assumed to be valid was 0°C through 40°C, which is a range covering most natural surface water systems (28). Equilibrium constants were adjusted for temperature effects using the Van t Hoff relation whenever appropriate enthalpy data was available (23, 24, 25). Activity and osmotic coefficients were temperature corrected by empirical equations describing the temperature dependence of the Debye-Huckel parameters of equations 20 and 21. These equations, obtained by curve-fitting published data (13), were... 

Inclusion of the surface energy terms gives a new van t Hoff relation (3,4), showing that size reduction lowers the enthalpy of hydride formation (A// ) for the nanostructured hydride as long as is positive. As seen in Eq. (2),... 

The standard approach for the search of new hydrogen-storage materials is to synthesize bulk samples and to use gravimetric [1,2] or volumetric [1,3] techniques to follow their hydrogenation reaction and to record pressure-concentration isotherms (pcT). The equilibrium pressure of the metal-to-hydride transition is determined from the plateau of the pressure composition isotherms. The enthalpy of hydride formation is exttacted from the temperature dependence of the equilibrium pressure, by means of the Van t Hoff relation ... 

Adsorption isotherms were obtained for four amino acids in an investigation of their interaction with calcium montmorillonite and sodium and calcium illite. Linear isotherms were obtained in the study of their adsorption by the calcium clay. These isotherms were described in terms of a constant partition of solute between the solution and the adsorbent Stem layer. Free-energy values were calculated using the van t Hoff relation [16,27]. 

D" AF (c) have been derived from dissociation pressure dependence upon temperature (van t Hoff relation) the standard entropy for salts D+AF (j.) have been evaluated. Table I contains data from these investigations, which are also pertinent to Figure 2. Standard entropies (7), for a wide variety of these and other salts (all of which are close-packed solids), show a linear dependence on volume, as illustrated in Figure 2. As may be seen, the linear relationship does satisfy the expectation that S° should become zero at zero volume. The standard entropy for a close-packed salt D+AF is therefore assumed, from this experience, to be determined by its FUV, the numerical relationship being ... 

K, can be calculated from the free energy change of the reaction. Using the van t Hoff relation, we obtain the dependence of K, on temperature ... 

For ideal solutions, the van t Hoff relation of equation (17), and the Hildebrand relation of equation (24), state that the In(XB) term is linearly dependent on /T and on In (7j. The enthalpy of solution is equal to the enthalpy of melting... 

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