Van’t Hoff reaction isotherm

Here is the chemical potential of i in its standard state and a, is its molal activity. This equation can be recast in the form known as the van t Hoff reaction isotherm ... 

The Van t Hoff isotherm identifies the free energy relationship for bulk chemical reactions. 

Combining equation 6 and 7 with the Van t Hoff isotherm the Nemst equation for electrochemicA reactions is obtained ... 

Here, the a s refer to the activities in the chosen arbitrary state. The concept of activity is presented separately in a later section. For the present, the activity of a species in a system may just be considered to be a function of its concentration in the system, and when the species is in a pure form (or in its standard state), its activity is taken to be unity. The activities ac, aD, aA, aB given above correspond to the actual conditions of the reaction, and these may or may not correspond to the state of equilibrium. Two special situations can be considered. In the first, the arbitrary states are taken to correspond to those for the system at equilibrium. Q would then become identical to the equilibrium constant K and, according to the Van t Hoff isotherm, AG would then be zero. In the second situation, all the reactants and the products are considered to be present as pure species or in their standard states, and aA, aB, ac, and aD are all equal to 1. Then (7=1 and the free energy change is given by... 

The Van t Hoff isotherm establishes the relationship between the standard free energy change and the equilibrium constant. It is of interest to know how the equilibrium constant of a reaction varies with temperature. The Varft Hoff isochore allows one to calculate the effect of temperature on the equilibrium constant. It can be readily obtained by combining the Gibbs-Helmholtz equation with the Varft Hoffisotherm. The relationship that is obtained is... 

From the Van t Hoff isotherm it follows that for the oxidation reaction considered, the standard free energy change at temperature T is given by... 

Considering the phosphorus oxidation reaction given earlier, and applying the Van t Hoff isotherm for nonequilibrium conditions,... 

Q and K only have the same value when the reaction has reached equilibrium, i.e. when AGr = 0. At this extent of reaction, the relationship between and AG is given by the van t Hoff isotherm ... 

And we see a further point the equilibrium constant K of a reaction is a direct function of A Gr according to the van t Hoff isotherm (Equation (4.55)). If the overall energy of reaction remains unaltered by the catalyst, then the position of equilibrium will also remain unaltered. 

One of the most applicable terms in Van t Hoff s studies was the perturbation actions. In Van t Hoff s opinion, "normal transformations take place very rarely... . Reaction rate is subjected to various effects to such an extent that the investigation of the transformation process reduces mainly to that of perturbation effects. Van t Hoff did not concentrate on "perturbation effects as inhomogeneity, non-isothermality, and the occurrence of some secondary reaction. To his mind, the main thing that merits special consideration is the effect of the medium on the reaction rate ("primarily the effect of the media of obviously chemical nature ). 

Problem 4 Give the thermodynamic derivation of maximum work obtained from gaseous reactions or van t Hoff isotherm. 

Equations (4) and (5) are other forms of van t Hoff isotherm. The maximum work obtained is also given by equations (3), (4) and (5), and often - AG is called the affinity of the reaction. 

The enthalpy change of the reaction (6.4) was deduced to be -44.5 kj (mol H2) from the database indicated in Chen s report [6]. However, Chen et al. [6] and Kojima and Kawai [10] have carefully measured the PC isothermal curve for the Li-N-H system at different temperatures and evaluated AH from the van t Hoff plot to be —66kJ (mol Hz). Recently, Isobe et al. evaluated AH for hydrogen desorption on the Li-N-H system by the direct measurement of DSC, indicating AH to be -67kJ(mol Hz) [11]. 

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 ... 

The right-hand side is the complete expression for the equilibrium constant K in the case of a reaction between perfect gases In the case of a perfect gas, however, the internal energy and therefore C log T is independent of the volume and therefore of the pressure Hence K as defined above (m terms of volume concentrations) is independent of the pressure, a result which we have already obtained by a consideration of the van t Hoff isotherm K. is now seen to be explicitly a function of T If T be kept constant the terms on the right-hand side are constant, and hence for the system considered, the expression on the left is likewise constant, 1 e the equilibrium constant... 

U 0 represents the heat evolved by the reaction in the neighbourhood of the absolute zero At the actual zero the existence of a gaseous state is supposed to be impossible Now we have already seen that the affinity of a leaction such as the above occurring in a condensed system (liquid or solid) is given by a particular form of the van t Hoff isotherm, viz —... 

This is the general reaction isotherm, also known as the van t Hoff isotherm it is of prime importance. The logarithmic ratio is sometimes known as the activity quotient, and is written Q. As before, AG is a measure of the affinity of the process actually occurring, where the logarithmic term makes adjustment for non-unit activities. This equation would apply for example when it was required to determine the feasibility of a reaction for which all starting activities are known. 

Equations (10) and (11) can also be expressed in exponential form, as well as in forms which use AG as the dependent variable rather than pK (from the van t Hoff isotherm, AG = -RT In K). For equilibrium reactions in aqueous and other polar solutions, the ACp value is expected to have a finite value, due to the significant changes in solvent structure which occur when ionization takes place. For some compoimds, the ACp value may have a large uncertainty and be not statistically different to zero, depending on the precision of the raw data (e.g., 5,5-di-isopropylbarbituric acid) [89]. In these cases, the pRa temperature dependence is satisfactorily described by the integrated van t Hoff equation [Eq. (10) without the C log T term]. This equation will give a linear van t Hoff plot of pfCg versus 1/T. 

This is the Van t-Hoff equation or isotherm equation as it determines the extent of a systems nonequilibrium and direction of the reactions at constant temperature. It characterizes the maximum useful work which is necessary for the reaction j components to perform for achieving equilibrium, and helps to identify the process direction. 

FIGURE 1 Pressure-composition isotherms left) and van t Hoff plot right) tor the reaction of a metal or intermetallic compound with hydrogen. ph2. hydrogen pressure Ch. hydrogen concentration in the solid phase T, absolute temperature a, solid solution phase of hydrogen in the metal or Intermetallic compound metal hydride phase Peq, equilibrium pressure of metal hydride formation To, critical temperature AH, enthalpy of hydride formation. [From Schlapbach, L., Meli, R, and Zuttel, A. (1995). Intermetallic hydrides and their applications. In Intermetallic Compounds Principles and Practice (J. H. Westbrook and R. L. Fleischer, eds.), Vol. 2, pp. 475-488. Reproduced with permission from John Wiley Sons, New York.]... 

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