Standard state arbitrariness

In some cases, the temperature of the system may be larger than the critical temperature of one (or more) of the components, i.e., system temperature T may exceed T. . In that event, component i is a supercritical component, one that cannot exist as a pure liquid at temperature T. For this component, it is still possible to use symmetric normalization of the activity coefficient (y - 1 as x - 1) provided that some method of extrapolation is used to evaluate the standard-state fugacity which, in this case, is the fugacity of pure liquid i at system temperature T. For highly supercritical components (T Tj,.), such extrapolation is extremely arbitrary as a result, we have no assurance that when experimental data are reduced, the activity coefficient tends to obey the necessary boundary condition 1... 

Equation 20.180 shows that AG is dependent on K and on any arbitrary activity of reactants and products that may be introduced into the equation. When the reactants and products are in their standard states... 

If Uo is the absolute amount of intrinsic energy contained in a system (with reference to a state of absolute zero of energy) in an arbitrary standard state, and if in any change from a state [1] to a state [2] the total amounts of heat absorbed and work done are 2Q and 2A respectively, we have ... 

Just as the intrinsic energy of a body is defined only up to an arbitrary constant, so also the entropy of the body cannot, from the considerations of pure thermodynamics, be specified in absolute amount. We therefore select any convenient arbitrary standard state a, in which the entropy is taken as zero, and estimate the entropy in another state /3 as follows The change of entropy being the same along all reversible paths linking the states a and /3, and equal to the difference of the entropies of the two states, we may imagine the process conducted in the following two steps ... 

The integral does not furnish the absolute value of, the entropy, because the lower limit is undetermined. If this is regarded as fixed, the integral with various upper limits gives the values of the entropies referred to this arbitrary standard state, and the differences between these values and any one of them referred to this arbitrary standard state will be the values of the entropies referred to the new standard state (cf. 42). 

The standard state given by the unsymmetric convention for normalization has one very important advantage it avoids all arbitrariness about/2°, which is an experimentally accessible quantity the definition off2° given by Eq. (37) assures that the activity coefficient of component 2 is unambiguously defined as well as unambiguously normalized. There is no fundamental arbitrariness about f2° because Hl2p(M) can be determined from experimental measurements. 

Activity can be thought of as the quantity that corrects the chemical potential at some pressure and/or composition condition" to a standard or reference state. The concept of a standard state is an important one in thermodynamics. The choice of the pressure and composition conditions for the standard state are completely arbitrary, and unusual choices are sometimes made. The common choices are those of convenience. In the next section, we will describe and summarize the usual choices of standard states. But, first, we want to describe the effect of pressure and temperature on a,. 

For electrolytes where dissociation is extensive, but not complete, the classification is somewhat arbitrary, and the electrolyte can be considered to be either strong or weak. Thermodynamics does not prevent us from treating an electrolyte either way, but we must be careful to designate our assignment because the choice of standard state is different for a strong electrolyte and a weak electrolyte. Assuming that an electrolyte is weak requires that we have some nonthermodynamic procedure for distinguishing clearly between the dissociated and undissociated species. For example, Raman spectroscopy... 

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... 

Values for free energy are usually referred to the standard free energy G°. The standard state is arbitrary and designates the datum level. A gas is considered here to be at a standard state if it is at a pressure of 1 atm or 1 bar for the designated temperature of an isothermal process. Thus, integrating Equation 6.9 from standard pressure P< > to pressure P gives ... 

The choice of the standard state is largely arbitrary and is based primarily on experimental convenience and reproducibility. The temperature of the standard state is the same as that of the system under investigation. In some cases, the standard state may represent a hypothetical condition that cannot be achieved experimentally, but that is susceptible to calculations giving reproducible results. Although different standard states may be chosen for various species, throughout any set of calculations it is important that the standard state of a component be kept the same so as to minimize possibilities for error. 

If an arbitrary standard state is marked with, a formal definition of a Raoultian standard state for component A of a solution is... 

In Equation (15.11), the choice of is entirely arbitrary. However, it is conventional to choose m2 =1 mol kg that is, the standard state of the solute is a hypothetical one molal state that is the point of extrapolation of Henry s law behavior to a molality of 1 mol kg In a figure analogous to Figure 15.1, but with ni2 along the horizontal axis, the standard state would be a point on the Henry s law dotted line directly above m2=l mol kg ... 

When data are available for the solute over the entire concentration range, from mole fraction 0 to 1, the choice of standard state, either the hypothetical unit mole fraction (Henry s law) or the actual unit mole fraction (Raoult s law), is arbitrary, but it is frequently easier to demonstrate Raoult s law as a limiting law than Heiuy s law. Figure 16.2 shows the relationships for activity and activity coefficient when Heiuy s law is used to define the standard state, and Figure 16.3 shows the same relationships when pure solute is chosen as the standard state. 

In calculating the lattice energy of a crystal, we adopt an arbitrary reference condition (two isolated ions in the gaseous state and at infinite distance) to which we assign a zero potential. It is worth stressing that this condition is not equivalent to the standard state commonly adopted in thermochemical calculations, which is normally that of element at stable state at reference P, T. 

While changes in internal energy and enthalpy (AC/ and Ai/) may be determined, it is not possible to measure either U or//absolutely. Consequently, an arbitrary datum is defined at which the enthalpy is zero. For this purpose, the enthalpy of all elements in their standard states is taken as zero at the stated reference temperature. The standard state of a pure substance at temperature T is defined as follows ... 

The allowance for polarization in the DH model obviates the need for separation of long-range and short-range attractive forces and for inclusion of additional repulsive interactions. Belief in the necessity to include some kind of covolume term stems from the confused analysis of Onsager (13), and is compounded by a misunderstanding of the standard state concept. Reference to a solvated standard state in which there are no interionic effects can in principle be made at any arbitrary concentration, and the only repulsive or exclusion term required is that described by the DH theory which puts limits on the ionic atmosphere size and hence on the lowering of electrical free energy. The present work therefore supports the view of Stokes (34) that the covolume term should not be included in the comparison of statistical-mechanical results with experimental ones. 

If the standard state is chosen consistently for all species, the arbitrary choice of P° will cancel out of AGrxn, AGf°[compound], and related free energy changes in chemical reactions. 

Such a statement reminds us that the standard state is a wholly arbitrary convention that need have no physical significance (except as a practical expedient), and may be chosen differently for different components of the solution. As indicated, the hypothetical standard state II is specifically adapted to the solute, i.e., to limiting Henry s law behavior as ji —> 1. 

The standard enthalpy of formation, A H (also represented by A or simply H ). of a substance at a given temperature is by definition, the enthalpy change when I mole of the substance in its standard state is formed, isothermally, at the indicated temperature from the elements, each in its standard stale. Usual units are kiloealories/mole. For all elements in their stable form at 25 C (298.15 K), the enthalpy of formation is zero If solid substances have more than one crystalline form, the most stable one is taken as the standard state, and the others have slightly different enthalpies. This convention about zero enthalpy is arbitrary hut universally accepted, and it may be compared to the arbitrary choice of zero lor terrestrial altitudes. The combination of enthalpies of formation, enthalpies of transition, and heat capacities makes possible the calculation of the enthalpy of a substance, in a given state at a given temperature, relative to a commonly accepted reference. 

To define an electrochemical "sea level," chemists have chosen a reference halfcell called the standard hydrogen electrode (S.H.E., shown in Figure 18.4). It consists of a platinum electrode in contact with H2 gas and aqueous H + ions at standard-state conditions [1 atm H2(g), 1 M H+ aq), 25°C]. The corresponding half-reaction, written in either direction, is assigned an arbitrary potential of exactly 0 V ... 

In this chapter, we continue the discussion begun in the last chapter of applying thermodynamics to chemical processes. We will focus our discussion here on two examples of biological interest. The principles are the same — all of our thermodynamic relationships work. But biochemists have their own vocabulary, and sometimes apply unique conditions to their systems. For example, as we shall see, unusual standard states are sometimes chosen. There is nothing wrong with this, since the choice of standard states is completely arbitrary as long as we keep track of what is done. Standard states are usually chosen in a way that makes the results most useful. That is true in this case. 

One of the most fundamental concepts of chemistry is the distinction between kinetic and thermodynamic factors nonetheless, such arguments are frequently ignored, or at best only tacitly considered, in wider discussions of reactivity. Chemical thermodynamics is concerned with the energetic relationships between chemical species. The most useful parameter is the Gibbs free energy, G, which, like all thermodynamic terms, is based on an arbitrary scale placing a value of zero upon pure elements in their stable standard states at 298 K and 1 atmosphere pressure. Differences between free energies are denoted by AG, as shown in Eq. (1.1). 

The thermodynamic functions have been defined in terms of the energy and the entropy. These, in turn, have been defined in terms of differential quantities. The absolute values of these functions for systems in given states are not known.1 However, differences in the values of the thermodynamic functions between two states of a system can be determined. We therefore may choose a certain state of a system as a standard state and consider the differences of the thermodynamic functions between any state of a system and the chosen standard state of the system. The choice of the standard state is arbitrary, and any state, physically realizable or not, may be chosen. The nature of the thermodynamic problem, experience, and convention dictate the choice. For gases the choice of standard state, defined in Chapter 7, is simple because equations of state are available and because, for mixtures, gases are generally miscible with each other. The question is more difficult for liquids and solids because, in addition to the lack of a common equation of state, limited ranges of solubility exist in many systems. The independent variables to which values must be assigned to fix the values of all of the... 

Although the choice of standard states is arbitrary, two choices have been established by convention and international agreement. For some systems, when convenient, the pure component is chosen as the substance in the standard state. For other systems, particularly dilute solutions of one or more solutes in a solvent, another state that is not a standard state is chosen as a reference state [19]. This choice determines the standard state, which may or may not be a physically realizable state. The reference state of a component or species is that state to which all measurements are referred. The standard state is that state used to determine and report the differences in the values of the thermodynamic functions for the components or species between some state and the chosen standard state. When pure substances are used in the definition of a standard state, the standard state and the reference state are identical. 

We may then choose the standard state of the system at the temperature T to be at some arbitrary pressure P0, so that the difference of the enthalpy of the system at the temperature T and pressure P and this standard state is given by... 

It may be convenient to define the standard state of the system as the state at an arbitrary temperature, T0, and an arbitrary pressure, P0. The enthalpy of the system in any state defined by the temperature T and the pressure P may then be calculated by a combination of Equations (8.5) and (8.8). Two alternate equations, depending on the path we choose, are obtained. These are... 

In Equation (8.22) fi°[T,P0] is the molar Gibbs energy of the substance in its standard state defined at the temperature T and the arbitrary pressure P0. The change of the Gibbs energy with temperature cannot be determined, because the absolute value of the entropy is not known, as stated before. 

With these standard states, comparison between the values of (dH/dn)T V at different temperatures cannot be made. In order to do so, we could choose the standard state to be the double-primed phase at some arbitrary temperature, T0, with the condition that this phase is saturated with respect to the primed phase the pressure of the standard state, P0, is thus determined by the temperature. We then define the enthalpy of the double-primed phase to be zero at T0. The molar enthalpy of the double-primed phase at any other temperature at which the two phases are at equilibrium may be calculated by the use of Equation (8.35), so... 

In this discussion the temperature and pressure of the reference state and of the standard state have been taken to be those of the solution this usage is consistent with the recommendations of the Commission on Symbols, Terminology, and Units of the Division of Physical Chemistry of the International Union of Pure and Applied Chemistry. For the standard state however, a fixed, arbitrary pressure (presumably 1 bar) might be chosen. If we define... 

In summary, a reference state or standard state must be defined for each component in the system. The definition may be quite arbitrary and may be defined for convenience for any thermodynamic system, but the two states cannot be defined independently. When the reference state is defined, the standard state is determined conversely, when the standard state is defined, the reference state is determined. There are certain conventions that have been developed through experience but, for any particular problem, it is not necessary to hold to these conventions. These conventions are discussed in the following sections. The general practice is to define the reference state. This state is then a physically realizable state and is the one to which experimental measurements are referred. The standard state may or may not be physically realizable, and in some cases it is convenient to speak of the standard state for the chemical potential, for the enthalpy, for the entropy,... 

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