Van’t Hoff expression

V, is the molar volume of polymer or solvent, as appropriate, and the concentration is in mass per unit volume. It can be seen from Equation (2.42) that the interaction term changes with the square of the polymer concentration but more importantly for our discussion is the implications of the value of x- When x = 0.5 we are left with the van t Hoff expression which describes the osmotic pressure of an ideal polymer solution. A sol vent/temperature condition that yields this result is known as the 0-condition. For example, the 0-temperature for poly(styrene) in cyclohexane is 311.5 K. At this temperature, the poly(styrene) molecule is at its closest to a random coil configuration because its conformation is unperturbed by specific solvent effects. If x is greater than 0.5 we have a poor solvent for our polymer and the coil will collapse. At x values less than 0.5 we have the polymer in a good solvent and the conformation will be expanded in order to pack as many solvent molecules around each chain segment as possible. A 0-condition is often used when determining the molecular weight of a polymer by measurement of the concentration dependence of viscosity, for example, but solution polymers are invariably used in better than 0-conditions. 

Derive Equation (15.71), the van t Hoff expression for the elevation of the boiling point. 

For dilute polymer solutions, the osmotic pressure can be approximated by the limiting van t Hoff expression. A more general expression for the osmotic pressure of polymer solutions is given by (17)... 

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 error is not appreciable, however, so long as n is small compared with N, that is, so long as the solution is dilute Hence, van t Hoffs expression will apply to a very dilute solution even though this is not strictly ideal The order of magnitude may be estimated in the following way Consider a decmormal solution in water, and as an extreme case let (Va - bt + e) — io(Vx - b ) We then have njN = 1/550 and (N + n)jN = unity approx Hence,... 

It is evident that the Henry constant is simply the thermodynamic equilibrium constant for adsorption, and the temperature dependence should therefore follow a van t Hoff expression ... 

The first term of this equation is the van t Hoff expression (n/C q = RTIM ) for osmotic pressure at infinite dilution and the second term is related to the second virial coefficient (A2). Thus, and A2 can be, respectively, determined from the intercept and the slope from a k/C versus C plot (the third and higher virial coefficient terms are normally ignored) [90]. Compared to other experimental techniques (SEC and light scattering), MO is limited... 

This brings us to the conclusion that as far as the depletion interaction is concerned ideal polymer chains to a good approximation can be replaced by penetrable hard spheres with a diameter a = 2, where the depletion thickness bs now depends on the size ratio q = Rg/R. In dilute polymer solutions the ideal chain description suffices to describe depletion effects. In Chap. 4 we shall see that for polymers with excluded volume the depletion thickness not only depends on the size ration q but also on the polymer concentration, see also [36, 39-41]. Also the (osmotic) pressure is no longer given by the ideal (Van t Hoff) expression. Both features significantly affect depletion effects. 

The temperature dependence of solubility coefficients are typically described in terms of van t Hoff expressions shown below (39) ... 

Replacing AP by the Van t Hoff expression for the osmotic pressure, we find that the concentration of the solute in a droplet is equal to the initial number of moles present divided by the volume of the droplet at a time t. 

The number average molecular weight of a polysaccharide may be calculated from its osmotic pressure in solution, according to the Van t Hoff formula, but any association of the molecules in solution will give spurious high values. This may be overcome by using a limiting value of osmotic pressure/concentration extrapolated to zero concentration. Alternatively, a modified Van t Hoff expression ... 

The equilibrium between two eonformers A and B, where A B is dependent on temperature. The coneentration of the high-energy A conformer, Ca, relative to the concentration of the low-energy form, Cb, is given by the van t Hoff expression... 

Raoult s experimental work was unsurpassed, agreeing in many instances with modern measurements to within 0.0001 C. His thermometer was considered by many antediluvian (i.e. belonging to an era before the Flood), but as Van t Hoff expressed it "with this antediluvian thermometer the world was conquered."... 

For many years, it was thought that the macro solute forms a new phase near the membrane—that of a gel or gel-like layer. The model provided good correlations of experimental data and has been widely used. It does not fit known experimental facts. An explanation that fits the known data well is based on osmotic pressure. The van t Hoff equation [Eq. (22-75)] is hopelessly inadequate to predict the osmotic pressure of a macromolecular solution. Using the empirical expression... 

Historically, the identification of a linear correlation between log k and T 1 was empirical. First described by Hood [504], the relationship was given some theoretical significance by van t Hoff [505] who expressed the influence of temperature on equilibrium constants (Ke) by an equation of similar form, viz. 

The expression that we have just derived is a quantitative version of I.e Chate-lier s principle for the effect of temperature. It is normally rearranged (by multiplying through by —1 and then using In a — In h = In (alb)) into the van t Hoff equation ... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

There have been numerous approaches to describing the temperature dependence of the properties. For aqueous solubility, the most common expression is the van t Hoff equation of the form (Hildebrand et al. 1970) ... 

The van t Hoff equation also has been used to describe the temperature effect on Henry s law constant over a narrow range for volatile chlorinated organic chemicals (Ashworth et al. 1988) and chlorobenzenes, polychlorinated biphenyls, and polynuclear aromatic hydrocarbons (ten Hulscher et al. 1992, Alaee et al. 1996). Henry s law constant can be expressed as the ratio of vapor pressure to solubility, i.e., pic or plx for dilute solutions. Note that since H is expressed using a volumetric concentration, it is also affected by the effect of temperature on liquid density whereas kH using mole fraction is unaffected by liquid density (Tucker and Christian 1979), thus... 

The final rate expressions, which were used in the present work, were given by Hou and Hughes (2001). In these rate expressions all reaction rate and equilibrium constants were defined to be temperature-dependent through the Arrhenius and van t Hoff equations. The particular values for the activation energies, heats of adsorption, and pre-exponential constants are available in the original reference and were used in our work without alteration. 

Influenced by the form of the van t Hoff equation, Arrhenius (1889) proposed a similar expression for the rate constant kAin equation 3.1-2, to represent the dependence of (-rA) on T through the second factor on the right in equation 3.1-1 ... 

The correct use of the van t Hoff isotherm necessitates using the thermodynamic temperature (expressed in kelvin). 

Figure 5.17 The solubility s of a partially soluble salt is related to the equilibrium constant (partition) and obeys the van t Hoff isochore, so a plot of In s (as y ) against 1 IT (as x ) should be linear, with a slope of AF lution -t- R . Note how the temperature is expressed in kelvin a graph drawn with temperatures expressed in Celsius would have produced a curved plot. The label KIT on the x-axis comes from l/T -t- 1/K...

Words that can be used as topics in essays 5% rale buffer common ion effect equilibrium expression equivalence point Henderson-Hasselbalch equation heterogeneous equilibria homogeneous equilibria indicator ion product, P Ka Kb Kc Keq KP Ksp Kw law of mass action Le Chatelier s principle limiting reactant method of successive approximation net ionic equation percent dissociation pH P Ka P Kb pOH reaction quotient, Q reciprocal rule rule of multiple equilibria solubility spectator ions strong acid strong base van t Hoff equation weak acid weak base... 

This expression is analogous to Henry s Law for gas-liquid systems even to the extent that the proportionality constant obeys the van t Hoff equation and Ka = K0e AH/RT where AH is the enthalpy change per mole of adsorbate as it transfers from gaseous to adsorbed phase. At constant temperature, equation 17.1 becomes the simplest form of adsorption isotherm. Unfortunately, few systems are so simple. 

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