Definition of activity

With our definition of activity as the ratio of the fugacity of the component in the solution to that in the standard state, we find that... 

In the limit of infinitely dilute solutions, where equation (6.112) holds, m2 —f2/k2. If we maintain this ratio as our definition of activity, a2. then a2 = m2 in these solutions. For solutions which are not in the limiting region, we writecc... 

We connected our earlier definition of activity to a standard state of 1.0 bar or 1.0 M or a mole fraction of unity. None of these make much sense for electrons, but we may define electron... 

As explained before, a chemical reaction can seldom be described by a single elementary step, and hence we need to adapt our definition of activation for an overall reaction. Since we are not particularly interested in the effects of thermodynamics we define the apparent activation energy as... 

Several descriptions of electrode reaction rates discussed on the preceding pages and the difficulty to standardize electrode potential scales with respect to different temperatures imply several definitions of activation energies of electrode reactions. The easiest way to determine this quantity, for example, for an irreversible cathodic process, employs Eqs (5.2.9), (5.2.10) and (5.2.12) at a constant electrode potential,... 

Active transport. The definition of active transport has been a subject of discussion for a number of years. Here, active transport is defined as a membrane transport process with a source of energy other than the electrochemical potential gradient of the transported substance. This source of energy can be either a metabolic reaction (primary active transport) or an electrochemical potential gradient of a substance different from that which is actively transported (secondary active transport). 

Figure 5.8 Definition of activation energy with and without a catalyst. (Adapted with permission from Fraser, 2002)...

The ratio of perceived to real concentrations is called the activity coefficient y (because, from Equation (7.25), y = a -=- c). Furthermore, from the definition of activity in Equation (7.20), y will have a value in the range zero to one. The diagram in Figure 7.9 shows the relationship between y and concentration c for a few ionic electrolytes. 

The exact structure of the equilibrium constant on the right-hand side of Equation (7.32) follows from the definition of activity a in Equation (7.25). The product of the... 

Equation 2.63 is valid for any homogeneous or heterogeneous reaction. The only difference is in the definition of activities. For a species in a perfect gas-phase mixture a = pi/p°, where pi is the partial pressure of species i andp° is the standard pressure (1 bar). For a real gas-phase mixture a =f/p°, where is the fugacity of i. The fugacity concept was developed for the same reason as the activity to extend to real gases the formalism used to describe perfect gas mixtures. In the low total pressure limit (p -> 0), fi = pi. 

Using the definition of activation volume as given by Eq. (21) for every individual rate constant, the binding volume derived from Km becomes... 

From the definition of activity and concentration given earlier in equation (3.12), we can say that the formal and standard electrode... 

From the definition of activity (equation (3.12)) of a = c x y, and substituting for each activity term, we obtain the following ... 

However, the choice of standard states makes no difference in the value of AG for the transfer of solute between two solutions of different concentrations in the same solvent. We can observe this by applying the definition of activity to the equation for the free energy change in a transfer process ... 

For such solutions, the definition of activity is completed by the requirement that the activity approach the molality ratio in the limit of infinite dilution. That is. 

When activity data for a strong electrolyte such as HCl are plotted against 1712/m°), as illustrated in Figure 19.1, the initial slope is equal to zero. Thus, an extrapolation to the standard state yields a value of the activity in the standard state equal to zero, which is contrary to the definition of activity in Equations (16.1) and (16.3). Therefore, it is clear that the procedure for determining standard states must be modified for electrolytes. 

If followed in experimenrtally accessible dilute solutions, Henry s law would be manifested as a horizontal asymptote in a plot such as Figure 19.3 as the square of the molality ratio goes to zero. We do not observe such an asymptote. Thus, the modified form of Henry s law is not followed over the concentration range that has been examined. However, the ratio of activity to the square of the molality ratio does extrapolate to 1, so that the data does satisfy the definition of activity [Equations (16.1) and (16.2)]. Thus, the activity clearly becomes equal to the square of the molality ratio in the limit of infinite dilution. Henry s law is a limiting law, which is valid precisely at infinite dilution, as expressed in Equation (16.19). No reliable extrapolation of the curve in Figure 19.2 exists to a hypothetical unit molality ratio standard state, but as we have a finite limiting slope at = 0, we can use... 

In the case of the solvent (water, concentration = 55.5 mol dm-3), its standard activity is usually taken to be 1. For solvents, the definition of activity is the ratio of its vapour pressure when acting as a solvent, p, divided by the vapour pressure of the pure solvent, p°, taken as the standard state ... 

For solutions of ions, departures from ideality can be large even in quite dilute solutions because of the strong electrostatic attractions or repulsions between the ions. Furthermore, the simple definition of activity coefficient given in Eq. 2.3 fails for electrolytes because we can never measure the activity of, say, a cation Mm+ without anions Xx being present at the same time instead, we usually define a mean ionic activity a and coefficient /y as... 

It is important to note the constraints imposed by the definition of activity. These are (1) that the system be at equilibrium and (2) that the comparison be an isothermal compari-... 

Thus, firstly, the choice of the pure solvent as the reference state for the definition of activities of solutes in fact impairs a fair comparison of the activity of dilute solutes such as general adds to the activity of the solvent itself. Secondly, the observed first-order rate constants k or k0 for the reaction of a solute with the solvent water are usually converted to second-order rate constants by division through the concentration of water, h2o = oA iho, for a comparison with the second-order rate coefficients HA. Again, it is questionable whether the formal h2o coefficients so calculated may be compared with truly bimolecular rate constants kUA for the reactions with dilute general acids HA. It is then no surprise that the values for the rate coefficients determined for the catalytic activity of solvent-derived acids scatter rather widely, often by one or two orders of magnitude, from the regression lines of general adds.74... 

This simple experiment was important in that it clearly established the key notion that cellular extrusion of sodium ions by the sodium pump was coupled to metabolism. Because in this and subsequent experiments of the same sort the electrochemical gradient for sodium was known precisely, and since the fluxes of sodium (and later potassium) both into and out of the cell could be measured independently, this study also laid the groundwork for a theoretical definition of active transport, a theory worked out independently by Ussing in the flux ratio equation for transepithelial active transport of ions (see below). 

The activity of a component in a solution is essentially a relative quantity. From the definition of activity it follows that the numerical value of the activity of a particular component is dependent on the choice of the standard state. There is no fundamental reason for preferring one standard state over another. Convenience dictates the choice of the standard state. Up to now, we have chosen the pure state as the standard state. That is, a pure component in its stable state of existence at the specified temperature and latm pressure is chosen as the standard state. This particular choice is known as the Raoultian standard state. 

If the pure component exists in a physical state which is different from that of the solution at the temperature of interest (e.g., pure oxygen is a gas, but it is liquid when dissolved in water.), how can the pressure terms in the definition of activity be determined ... 

From the definition of activation energy in Eq. (8.3), we obtain the following expression for the difference in activation energies between the forward and the reverse direction of a reaction ... 

When the amount of material present is measured by either concentration, c or molality (= amount of solute/mass of solvent) i the definition of activity differs slightly from the case (equation (39.4)) where a dimensionless measure (like mole fraction, x) is used. Equations for chemical potential, fi, involving mole fractions, x apply quite well when one is examining equilibria in solution. However in other cases concentration, c or molalities, m are often used. Activity will then be defined in relation to a standard concentration, c° ... 

The above definitions of active/inactive subsystems is of course not restricted to the study of reactions but can be generalized to all static systems... 

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