Standard states pressure dependent

This illustrates the statement made earlier that the most convenient choice of standard state may depend on the problem. For gas-phase problems involving A, it is convenient to choose the standard state for A as an ideal gas at 1 atm pressure. But, where the vapor of A is in equilibrium with a solution, it is sometimes convenient to choose the standard state as the pure liquid. Since /a is the same for the pure liquid and the vapor in equilibrium... 

The form of the equilibrium constant in Equation (10.21) is different from that presented in introductory courses. It has the advantages that 1) it is explicit that Kp is a dimensionless quantity 2) it is explicit that the numerical value of Kp depends on the choice of standard state but not on the units used to describe the standard state pressure the equilibrium constant has the same value whether P° is expressed as 750.062 Torr, 0.98692 atm, 0.1 MPa, or 1 bar. 

We see that Kp has an explicit temperature dependence. However, Kp is not pressure dependent. That is, from Eq. 9.43, Kp is seen to depend on the standard-state thermochemistry in other words, properties at the standard-state pressure p = p° alone. 

Note that the standard molar reaction free energy difference AG° depends only on temperature it is defined for the standard-state pressure of 1 bar. Therefore the partial derivative may be converted to an ordinary derivative... 

Let us recall the basic expression (5.53) for the P dependence of AG under isothermal conditions. For certain purposes, it is convenient to choose a standard state pressure, denoted P°, such that the free energy G = G(P) for any other pressure P is given by... 

The temperature dependence of the standard entropy change of reactionis developed similarly. Equation (6.21) is writtenfor the standard-state entropy of species i at the constant standard-state pressure P " ... 

The factor 1 in the equation is the standard-state pressure of 1 atm or 1 bar, and will be left out in subsequent equations for simplicity, with the understanding that/has the dimension of pressure in units of either atmospheres or bars, depending on the convention for g. It follows from the two equations above that, for an ideal-gas mixture, / = P , reducing for a pure ideal gas tof=p. 

Pa (1 bar). See the reference for information on the dependence of gas-phase entropy on the choice of standard state pressure. 

Similar results may be obtained also for very dilute liquid solutions, see (4.467), but because the corresponding standard state may depend beside T also on the pressure P, we confine ourselves usually to chemical kinetics with constant, say atmospheric pressure (its small variation may be neglected in liquids). 

One advantage offered by (5.5.11) is that it collects all the pressure effects into a single term. As we shall see in 5.6, many models for contain no pressure dependence hence, those models provide no pressure dependence for the activity coefficient, and such models are strictly valid only at the standard-state pressure P°. To include pressure in those models, we could use (5.5.11), if we have a reliable estimate for the partial molar volume—say, from a PvTx equation of state. 

Note that in FFF 5 neither the activity coefficient nor the standard-state fugacity depends on the mixture pressure. The Po)mting factor in FFF 5 can be computed, provided we can evaluate the partial molar volume for the real substance along the isotherm T from Pj to P. In contrast, the Po)mtmg factor appearing in FFF 3 applies to component i in its standard state and involves an integral over the partial molar volume of that standard-state substance. 

Otherwise, values of y, may be greater than one or less than one, but values less than one are more common. For a binary, the plot in Figure 10.5 applies to both a reference-solvent ideal solution and a solute-free ideal solution. This activity coefficient is particularly useful when we can choose the reference-solvent vapor pressure to be the standard-state pressure P , for then H,y is a function only of T, and we may place all pressure dependence either in an activity coefficient, as in FFF 4, or in a Poynting factor, as in FFF 5. 

Before a reaction-equilibrium calculation can be performed, we must select an appropriate standard state for each species. Moreover, we must clearly distinguish quantities, such as fugacities and activities, that depend on the final equilibrium state (T, P, x ), from those quantities, such as equilibrium constants, that depend only on the equilibrium temperature T, the standard-state pressures P , and the phase. Typically, the standard-state pressure and phase are chosen according to whether the real substance is gas, liquid, or solid at the equilibrium conditions. Those three possibilities are discussed, in turn, here, and each discussion culminates with a particular expression for the activity. Those expressions can be used either in the stoichiometric development, via (10.3.14), or in the nonstoichiometric development, via (10.3.38). We emphasize that when we use the stoichiometric approach, the standard states used for the fugacities must be consistent with those associated with the equilibrium constant. 

Recall that the standard state pressure here is 1 bar for all species, so the reaction pressure P in (10.4.26) must also be in bars. Also recall that Kj does not depend on P therefore, since Oy can be positive, negative, or zero, the product on the rhs may increase, decrease, or remain unchanged when P differs from P ... 

The molar enthalpy of any substance depends on its state. The standard state of a liquid or solid substance is specified to be the pure substance at a fixed pressure of exactly 1 bar (100,000 Pa), which we denote by P°. The standard state for a gas is defined to be the corresponding ideal gas at pressure P°. The difference between the molar enthalpy of a real gas at 1 bar pressure and the corresponding ideal gas at 1 bar is numerically very small, but we will discuss this difference in a later chapter. If substance number i is in its standard state, its molar enthalpy is denoted by H (i). A standard-state reaction is one in which all substances are in their standard states before and after the reaction. The enthalpy change for a standard-state reaction is denoted by A 71°. The standard-state pressure was at one time defined to equal 1 atm (101,325 Pa). The difference in numerical values is small, and the formulas involving P° are the same with either choice. For highly accurate work, one must determine which standard pressure was used for an older set of data. 

We can rewrite this expression in a slightly different but more general way by defining a = P/P° as the activity of an ideal gas. Thus, for an ideal gas, the activity is simply the partial pressure of the gas divided by P° = 1 bar, the standard state pressure. Although we will have more to say about activity in Section 13-8, for now we need only say that ultimately, the activity of a substance depends not only on the amount of substance but also on the form in which it appears in the system. The following rules summarize how the activity of various substances is defined (see also Table 13.5). It is beyond the scope of this discussion to explain the reasons for defining activities in these ways, so we will simply accept these definitions and use them. However, it is important to note that the activity of a substance is defined with respect to a specific reference state. 

From this equation, the temperature dependence of is known, and vice versa (21). The ideal-gas state at a pressure of 101.3 kPa (1 atm) is often regarded as a standard state, for which the heat capacities are denoted by CP and Real gases rarely depart significantly from ideaHty at near-ambient pressures (3) therefore, and usually represent good estimates of the heat capacities of real gases at low to moderate, eg, up to several hundred kPa, pressures. Otherwise thermodynamic excess functions are used to correct for deviations from ideal behavior when such situations occur (3). 

For liquid mixtures at low pressures, it is not important to specify with care the pressure of the standard state because at low pressures the thermodynamic properties of liquids, pure or mixed, are not sensitive to the pressure. However, at high pressures, liquid-phase properties are strong functions of pressure, and we cannot be careless about the pressure dependence of either the activity coefficient or the standard-state fugacity. 

An enthalpy of reaction also depends on the conditions (such as the pressure). All the tables in this book list data for reactions in which each reactant and product is in its standard state, its pure form at exactly 1 bar. The standard state of liquid water is pure water at 1 bar. The standard state of ice is pure ice at 1 bar. A solute in a liquid solution is in its standard state when its concentration is 1 mol-L". The standard value of a property X (that is, the value of X for the standard state of the substance) is denoted X°. 

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