Equilibrium constant terms Links

This expression (2.3) follows on from another linear relation seen in thermodynamics linking the variation in free energy and the equilibrium constant K, (AG = —RT InK), for a homologous family of compounds in which each term differs from the preceding one by a supplementary CH2 unit. Since K= kj3 therefore fp = Ktfjf/fS, then log will increase as In K for the homologous family In fp = In K + (In fM//3). 

The databases associated with geochemical modeling programs consist of a list of the basis species, and a list of secondary or auxiliary species, minerals, and gases, each described in terms of the basis species, and with the equilibrium constant of the reaction linking the secondary species or mineral to the basis species. 

The link between Gibbs free energy and equilibrium constant can be appreciated qualitatively at this stage. The equilibrium constant in terms of pressures is given by ... 

Therefore, equilibrium binding constants such as or indeed fQ can be determined from a simple ratio of rate constants. Equations such as (7.57) are a form of Haldane relationship, although the term Haldane relationship is more usually applied to expressions in reaction kinetics that link rate constants with an equilibrium constant (see Chapter 8). If the receptor R is immobilised (imm) for any reason by covalent attachment to a solid phase, then the... 

Normally this consists of the stoichiometry of a secondary species in terms of basis species, plus the equilibrium constant of a reaction linking the two. Similarly, all minerals and gases included in the program must also be linked to the basis species by appropriate reactions and their equilibrium constants. 

This equation is the link between tables of thermodynamic data (such as Table 4.2.1), which allow the evaluation of ArG°, and the equilibrium constant Kr of the reaction (sometimes also denoted as reaction quotient Qr), which is a function of the composition of the system in terms of concentration, molar fractions, and so on. The value and definition of Kr depends on the choice of the standard state and the ideality of the system, as shown subsequently for ideal and real gases, liquids, and gas-solid systems. 

Thermodynamic properties of an assembly of many mutually independent molecules of the same kind are determined by one single quantity, the so called molecular partition function. The partition function is a statistical mechanics concept representing a link between microscopic and macroscopic thermodynamic properties. It enables the expression of the equilibrium constant in terms of energies of individual degrees of freedom for the molecules. The molecular partition function is defined as the sum over stationary quantum states of the molecules (numerated by i = 1,2,3,...) ... 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

Carrier facilitated transport involves a combination of chemical reaction and diffusion. One way to model the process is to calculate the equilibrium between the various species in the membrane phase and to link them by the appropriate rate expressions to the species in adjacent feed and permeate solutions. An expression for the concentration gradient of each species across the membrane is then calculated and can be solved to give the membrane flux in terms of the diffusion coefficients, the distribution coefficients, and the rate constants for all the species involved in the process [41,42], Unfortunately, the resulting expressions are too complex to be widely used. 

This equation shows the coupling effect between the metal ion [M] and the hydrogen ion [H] because both appear in the concentration term of the Fick s law expression linked by the equilibrium reaction constant K. Thus, there will be a positive uphill flux of metal ion from the downstream to the upstream solution (that is, in the direction l o) as long as... 

This holds provided that no stable intermediate is involved in the reaction. If the rate at which ML dissociates into M and L (defined by the rate constant, kb) is slower than the rate at which M and L assemble into ML (defined by the rate constant kf), then there will always at equilibrium be a larger amount of ML present than of M and L, which translates into a large thermodynamic stability constant K, which equates with the ratio of these two kinetic terms. In effect, for such simple steps, there is a thermodynamic-kinetic link. At this level, however, we shall not dwell on this too deeply. 

We stress again that the criteria for equilibrium as expressed in Equation 3.34 apply only if the corresponding terms in Equation 3.33 are independent of each other. Sometimes, however, these terms are not independent but coupled. An example of such coupling is the transport of ions. In an ion, matter and charge are linked to each other and, hence, transporting ions involves both chemical and electrical work, represented by the terms p d , and /dC- These terms are coupled since dQ = dn, (where z, is the valency of ion i and F is Faraday s constant). Then, the reasoning followed above leads to p + - pf + z,Fil) as the criterion for equilibrium... 

Nondiagonal terms in Equation [14] produce a macroscopic polarization Pjj(t) departs from zero at equilibrium and includes an oscillatory term at the frequency co y. The existence of a nondiagonal term Pij(t) 0 is linked to the creation of a coherent superposition of states i and / due to the perturbation. This coherence is broken through the interaction with the molecules of the bath, with a rate constant of the order of F y= I/T2. Usually this dephasing is much faster than the energy relaxation and T2 Ti. 

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