Variable Temperature Standard States

The first thing to note about the standard state as used with activities is that the standard state and the state of interest are virtually always at the same temperature (T = T°). Because we are often interested in a series of equilibrium states at different temperatures, we therefore have a corresponding series of standard states, one for each temperature. This can be regarded rather as a single standard state having a variable temperature. 

The reason why T must generally equal T° can be seen by considering the integration of 

During integration, T is held constant, so that f, and f° necessarily refer to states at the same temperature. As temperature is often variable in a set of experimental or theoretical results, this naturally gives rise to a variable temperature standard state. This means that the comparison being made (in this case between /x of i in some state of interest, and n of i in its standard state) is always between two states having the same T. 

But not only are f, and f° necessarily at the same temperature, we must also note that G° (or /x°) and /° are independent of the system pressure. That is to say, they depend on P° but not on P. This can be seen by integrating (12.2) indefinitely rather than between limits as we have done. This gives 

It might be pointed out too, that there is no logical necessity to be restricted to using equation (12.1) and hence the variable temperature standard state. That is, if for some reason you preferred a fixed temperature standard state, you could modify 

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The second comment concerns the choice of standard states. Clearly, in defining the process of solvation, one must specify the thermodynamic variables under which the process is carried out. Here we used the temperature T, the pressure P, and the composition N1 ..., Nc of the system into which we added the solvaton. In the traditional definitions of solvation, one needs to specify, in addition to these variables, a standard state for the solute in both the ideal gas phase and in the liquid phase. In our definition, there is no need to specify any standard state for the solvaton. This is quite clear from the definition of the solvation process yet there exists some confusion in the literature regarding the standard state involved in the definition of the solvation process. The confusion arises from the fact that Ap is determined experimentally in a similar way as one of the conventional standard Gibbs energy of solvation. The latter does involve a choice of standard state, but the solvation process as defined in this section does not. For more details, see the next two sections. 

In considering the effect of pressure on activity, we must recall that the standard state pressure (P°) is not always the same as the system pressure (P), so that the differentiation with respect to pressure is not always completely analogous to differentiation with respect to temperature. First of all, for variable pressure standard states, those that do have P° = P, we have... 

Normally of course the expression for the variation of K with P is simpler than this, perhaps because all three states of matter may not be present, but also because it is quite unusual to use a variable pressure standard state for constituents whose fugacities are known or sought, (because this adds complexities rather than simplifying matters), and the In Qig) term is therefore essentially never required. To take a real example, let s consider the brucite-periclase reaction again. We have discussed the variation of the equilibrium constant for the brucite-periclase-water reaction with temperature at one bar, and showed that the equilibrium temperature for the reaction at one bar is about 265°C. Calculation of the equilibrium temperature of dehydration reactions such as this one at higher pressures was discussed briefly in 13.2.2. Here we will discuss the reaction in different terms to demonstrate the relationships between activities, standard states and equilibrium constants. 

Marcus has introduced a model for, S N 2 reactions of the ET type based on two interacting states which takes into account the relevant bond energies, standard electrode potentials, solvent contributions, and steric effects.87 The rate constant for intramolecular electron transfer between reduced and oxidized hydrazine units in the radical cation of the tetraazahexacyclotetradecane derivative (43) and its analogues has been determined by simulation of then variable temperature ESR spectra.88 The same researchers also reported then studies of the SET processes of other polycyclic dihydrazine systems.89,90... 

The state of a single-phase, one-component system may be defined in terms of the temperature, pressure, and the number of moles of the component as independent variables. The problem is to determine the difference between the values of the thermodynamic functions for any state of the system and those for the chosen standard state. Because the variables are not separable in the differential expressions for these functions, the integrations cannot be carried out directly to obtain general expressions for the thermodynamic functions without an adequate equation of state. However, each of the thermodynamic functions is a function of the state of the system, and the changes of these functions are independent of the path. The problem can be solved for specific cases by using the method outlined in Section 4.9 and illustrated in Figure 4.1. 

This equation gives the enthalpy of the system relative to the standard state, and the independent variable would now be (H — nH" ) rather than H itself. The quantities (H — H" ) and (ft — H ") are the changes of enthalpy when the state of aggregation of 1 mole of the component is changed from the triple-primed state to the primed state and to the double-primed state, respectively, at the temperature and pressure of the triple point. These quantities can be determined experimentally or from the Clapeyron equation, as discussed in Section 8.2. The three simultaneous, independent equations can now be solved, provided values that permit a physically realizable solution have been given to (H — nH "), V, and n. If such a solution is not obtained, the system cannot exist in three phases for the chosen set of independent variables. Actually, the standard state could be defined as one of the phases at any arbitrarily chosen temperature and pressure. The values of the enthalpy and entropy for the phase at the temperature and pressure... 

In the previous sections concerning reference and standard states we have developed expressions for the thermodynamic functions in terms of the components of the solution. The equations derived and the definitions of the reference and standard states for components are the same in terms of species when reactions take place in the system so that other species, in addition to the components, are present. Experimental studies of such systems and the thermodynamic treatment of the data in terms of the components yield the values of the excess thermodynamic quantities as functions of the temperature, pressure, and composition variables. However, no information is obtained concerning the equilibrium constants for the chemical reactions, and no correlations of the observed quantities with theoretical concepts are possible. Such information can be obtained and correlations made when the thermodynamic functions are expressed in terms of the species actually present or assumed to be present. The methods that are used are discussed in Chapter 11. Here, general relations concerning the expressions for the thermodynamic functions in terms of species and certain problems concerning the reference states are discussed. 

In order to evaluate each of the derivatives, such quantities as (V" — V-), (S l — Sj), and (dfi t/x t)T P need to be evaluated. The difference in the partial molar volumes of a component between the two phases presents no problem the dependence of the molar volume of a phase on the mole fraction must be known from experiment or from an equation of state for a gas phase. In order to determine the difference in the partial molar entropies, not only must the dependence of the molar entropy of a phase on the mole fraction be known, but also the difference in the molar entropy of the component in the two standard states must be known or calculable. If the two standard states are the same, there is no problem. If the two standard states are the pure component in the two phases at the temperature and pressure at which the derivative is to be evaluated, the difference can be calculated by methods similar to that discussed in Sections 10.10 and 10.12. In the case of vapor-liquid equilibria in which the reference state of a solute is taken as the infinitely dilute solution, the difference between the molar entropy of the solute in its two standard states may be determined from the temperature dependence of the Henry s law constant. Finally, the expression used for fii in evaluating (dx Jdx l)TtP must be appropriate for the particular phase of interest. This phase is dictated by the particular choice of the mole fraction variables. 

We recognize from our previous experience that pt is a function of the entropy, volume, temperature, or pressure in appropriate combinations and the composition variables. The splitting of into these two terms is not an operational definition, but its justification is obtained from experiment. The quantity pt is the quantity that is measured experimentally, relative to some standard state, whereas the electrical potential of a phase cannot be determined. Neither can the difference between the electrical potentials of two phases alone at the same temperature and pressure generally be measured. Only if the two phases have identical composition can this be done. If the two phases are designated by primes,... 

Because, at constant temperature, dGm = Vm dP and the molar volumes of condensed phases are very small, it is usually sufficiently accurate to take their molar free energy as pressure independent and the same as that at the 1.0-bar standard state. This is equivalent to setting the activity of pure, condensed phases equal to unity. (See Problem 9.) The activity of a condensed phase is also independent of just how much of the phase is present. As a result of these considerations, no variable describing the condensed phase appears in the equilibrium constant and the equilibrium is independent of just how much condensed phase is present. 

For estimation of thermodynamic properties of dissolved species, one can use the Entropy Correspondence Principle ( ), where the entropy of an ion at a given temperature is regarded as a function of the charge, the dielectric constant, mass, radius, and other variables. The function depends mainly upon the choice of the standard state, solvent, and temperature. The temperature dependency of entropy was derived based on the above principles and experimental data. By conducting the a square regression on Criss-Cobble s data ( ), we obtained the following eqtiation for calculating the entropies of species in aqueous solution. 

The free energy content of a system depends on temperature and pressure (and, for mixtures, on concentrations). The value of AG for a process depends on the states and the concentrations of the various substances involved. It also depends strongly on temperature, because the equation AG = A// — T AS includes temperature. Just as for other thermodynamic variables, we choose some set of conditions as a standard state reference. The standard state for AG is the same as for AH —1 atm and the specified temperature, usually 25°C (298 K). Values of standard molar free energy of formation, AG , for many substances are tabulated in Appendix K. For elements in their standard states, AG = 0. The values of AG may be used to calculate the standard free energy change of a reaction at 298 K by using the following relationship. 

The state of any pure component is uniquely determined by fixing any two variables. Consider first the case where pressure and temperature are taken to be the independent variables. The expressions for the thermodynamic functions are obtained by use of the path shown in Fig. 14-2. The standard state is taken to be a perfect gas at P0 and T0. The gas is taken from its standard state to its final state P and T by the following path. 

As with other thermodynamic variables, we usually compare entropy values for substances in their standard states at the temperature of interest 1 atm for gases, I M for solutions, and the pure substance in its most stable form for solids or liciuids. Because entropy is an extensive property, that is, one that depends on the amount of substance, we are interested in the standard molar entropy (S°) in units of J/moEK (or J mol -K ). The S° values at 298 K for many elements, compounds, and ions appear, with other thermodynamic variables, in Appendix B. 

The only mathematical restriction on /x° and f° in equation (12.1) is that they both refer to the same integration limit, or in physical terms, that they refer to the same equilibrium state. This state has been referred to in various places thus far as a reference state, which it is. We now consider it in more detail, with a more exact definition, and we refer to this more precise concept as a standard state. The exact nature of this state is completely a matter of definition, although a few definitions have themselves become standard because of their utility. We have used it in discussions of thermodynamic properties such as G°, H°, etc. to signify that the substance is in its pure state, and we have seen in the two conventions discussed in Chapter 7 that the pressure and temperature of the standard state could be different in different cases. With the introduction of the activity concept, standard states take on added importance because of their use in a wide variety of solutions, temperatures, and pressures, both fixed and variable, and we must now pay more attention to their definition than we have done so far. 

To see how standard states having variable pressure as well as variable temperature arise, we first go back to equation (12.1), which for a pure ideal gaseous component 1 having a standard state of ideal gas at T and 1 bar (so that f° = 1 and fi = P)... 

A5 a)° entropy change on adsorption at standard state, cal or kcal/mol-K 5 surface area per unit weight of adsorbent substrate eoneentration, mass/ volume culture scaled activity variable for deactivation s n) number of molecules adsorbed per weight of adsorbate T temperature, °C or K... 

We have seen how to use standard reduction potentials to calculate for cells. Real cells are usually not constructed at standard state conditions. In fact, it is almost impossible to make measurements at standard conditions because it is not reasonable to adjust concentrations and ionic strengths to give unit activity for solutes. We need to relate standard potentials to those measured for real cells. It has been found experimentally that certain variables affect the measured cell potential. These variables include the temperature, concentrations of the species in solution, and the number of electrons transferred. The relationship between these variables and the measured cell emf can be derived from simple thermodynamics (see any introductory general chemistry text). The relationship between the potential of an electrochemical cell and the concentration of reactants and products in a general redox reaction... 

AS°t = standard state entropy change for process x at temperature T (JK" T, = temperature of process x in K X = process subscript for state variables X = b normal boiling process... 

This chapter applies concepts introduced in earlier chapters to the simplest kind of system, one consisting of a pure substance or a single component in a single phase. The system has three independent variables if it is open, and two if it is closed. Relations among various properties of a single phase are derived, including temperature, pressure, and volume. The important concepts of standard states and chemical potential are introduced. 

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