Variable Pressure Standard States

For cases where the standard state pressure for the various species is chosen as that of the system under investigation, changes in this variable will alter the values of AG° and AH0. In such cases thermodynamic analysis indicates that... 

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,... 

P, T, and also composition are the state variables most often used to characterize the state of the system, as they can be easily measured and controlled. As we show in Part II, Equations 2.5 and 2.14 are important to perform the thermodynamic analysis of a process ATS-m, which expresses the change in entropy of a reaction at 298 K and at standard pressure. The reaction is defined to take place between compounds in their standard state, that is, in the... 

Above all, thermodynamics is a useful subject. Its usefulness is largely dependent on the tabulation of thermodynamic quantities in an efficient and convenient form. Because it takes at least two variables to determine the state of a pure material, tables could get rather unwieldy. To avoid this, properties are tabulated at a standard (pressure) state and then converted to the pressure that is desired. For hquids and sohds, the standard state is just that of the pure material at 1.0 bar pressure. 

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. 

The foregoing is sufficiently complex that one should seek a simplified approach. This presentation applies if one uses solely the mole fraction xA as the composition variable and if all thermodynamic characterizations refer only to the standard state at a total pressure of P - 1 atm. In such circumstances the self-consistent equation (3.4.1) reduces to... 

Table 2.3 summarizes the essential relationships for pressure effects on chemical equilibrium for the variable-pressure standard-state convention. Note, that these relationships can apply to any consistent choice of standard part ial molar volumes, for example, one for which an ionic medium such as seawater is adopted as the solute reference state. For detailed discussion of applications to seawater see, for example, Millero (1969) and Whitfield (1975). A compie-... 

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 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. 

All the state variables and the changes in thermodynamic quantities during a process are measurable in principle. The value of A U is measurable, but the absolute values of U cannot be obtained. Thus, the thermodynamic data are reported with respect to certain internationally agreed standard or reference state values. Normally, a temperature of 25°C and a pressure of 1 bar = 105 pascal (Pa = N/m2) are taken as standard conditions, and for solutions, a molar concentration, c, of 1 mol/dm3 is used as a reference state. 

Now, the left-hand side of the above expression is only a function of T (the standard state of the pure solid is chosen as P = 1 bar or the vapor pressure, and so is only a function of T), while the right-hand side is only a function of T, p, and y (or yg). With the degrees of freedom set at two, any two of these intensive variables can be chosen without overspecifying the system. 

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. 

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. 

VARIABLE PRESSURE STANDARD STATES 12.4.1. Standard States Based on Raoult s Law... 

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)... 

Equations (12.10) and (12.14) are examples of expressions that use a variable pressure standard state. Although it is theoretically possible to keep P° fixed and to simply add BT In P) to the — term for each different P considered, in practice one usually considers that the pressure of the standard state (P°) and the pressure on the system or state of interest (P) are the same, so that p° is a function of the system pressure. 

Because the activities of solutes in dilute solutions can be more closely approximated with Henry s Law than with Raoult s Law, they are traditionally treated separately and use standard states different than those we have so far encountered. There are two variations of usage here, both of the variable pressure type, one required when using mole fractions, and another when using molalities. 

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... 

On the other hand, if even one of the reaction constituents has a standard state with a variable pressure, normally P° = P, then equation (13.27) is not true, and integration of equation (13.28) requires a knowledge of how ArG° varies with pressure. We will simplify the following discussion by assuming that reaction constituents having the same physical state (solid, liquid, gas, solute) will have the same kind of standard... 

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. 

Next, consider the case where the mineral standard states are of the variable pressure type, that is, the standard states for brucite and periclase are taken to be the pure phase at the system P and T, while water continues to have a standard state of ideal gaseous water at T and one bar. Because there is essentially no mutual solution between the three phases they are essentially pure when at mutual equilibrium, and the mineral activities are therefore 1.0 at all Ps and Ts. This is only an apparent simplification, because now the equilibrium constant varies with pressure. Its value at 2000 bars, 25°C can be calculated from equation (13.42), thus... 

The Ideal-gas equation, PV = iiRT, is the equation of state for an Ideal gas. The term R in this equation is the gas constant We can use the ideal-gas equation to calculate variations in one variable when one or more of the others are changed. Most gases at pressures less than 10 atm and temperatures near 273 K and above obey the ideal-gas equation reasonably well The conditions of 273 K (0 °C) and I atm are known as the standard temperature and pressure (STP). In all applications of the ideal-gas equation we must remember to convert temperatures to the absolute-temperature scale (the Kelvin scale). 

PRESSURE (SECTION 10.2) To describe the state or condition of a gas, we must specify four variables pressure (P), volume (V), temperature T), and quantity (n). Volume is usually measured in liters, temperature in kelvins, and quantity of gas in moles. Pressure is the force per unit area and is expressed in SI units as pascals. Pa (1 Pa = 1 N/m ). A related unit, the bar, equals 10 Pa. In chemistry, standard atmospheric pressure is used to define the atmosphere (atm) and the torr (also called the millimeter of mercury). One atmosphere of pressure equals 101.325 kPa, or 760 torr. A barometer is often used to measure the atmospheric pressure. A manometer can be used to measure the pressure of enclosed gases. 

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