Pure ideal gases

However, the chemical potential is given by Eq. (4-341) for gas-phase reactions and standard states as the pure ideal gases at T°, this equation becomes... 

The thermodynamic functions for the gas phase are more easily developed than for the liquid or solid phases, because the temperature-pressure-volume relations can be expressed, at least for low pressures, by an algebraic equation of state. For this reason the thermodynamic functions for the gas phase are developed in this chapter before discussing those for the liquid and solid phases in Chapter 8. First the equation of state for pure ideal gases and for mixtures of ideal gases is discussed. Then various equations of state for real gases, both pure and mixed, are outlined. Finally, the more general thermodynamic functions for the gas phase are developed in terms of the experimentally observable quantities the pressure, the volume, the temperature, and the mole numbers. Emphasis is placed on the virial equation of state accurate to the second virial coefficient. However, the methods used are applicable to any equation of state, and the development of the thermodynamic functions for any given equation of state should present no difficulty. 

The equations just derived show that for ideal gases a mixture property depe only on the properties of the pure ideal gases which comprise the mixture, information about the mixture other than its composition is required, circumstance is not limited to ideal gases, but extends more generally to i solution wherein all molecules are of the same size and all forces betw molecules (like and unlike) are equal. Equations based on these characteristi provide a model of behavior known as the ideal solution. 

The above fact gives us a quantitative basis for measurement of thermodynamic activity of a constituent. As we would like to use this concept of activity in relation to a reference (standard) state, we define activity at the standard state to be unity. Thus pure ideal gases at 25 °C (298 °K) and at 1 atmosphere pressure are assumed to have unit activity. At this temperature, if they were kept at half atmospheric pressure, they would have an activity of 0.5. 

Consider now the process of forming an ideal gas mixture at temperature T and pressure P from a collection of pure ideal gases, all at that temperature and pressure. From the discussion here it is clear that for each species Vi(T, P, x) = V-(T, P) and U (T,x) = U iT). It then follows immediately from equations such as Eqs. 8.1-14... 

In Sec. 9.1 we considered the changes in thermodynamic properties on forming an ideal gas mixture from a collection of ideal gases at the same temperature and pressure. A second, less common way of forming an ideal gas mixture is to start with a collection of pure ideal gases, each at the temperature T and volume V, and mix and compre.ss the mixture to produce an ideal gas mixture at temperature T and volume V. 

Just because this is the simple mixture, the partial entropy s may be interpreted as specific entropy of pure (ideal) gas at a density equal to those in the mixture (see (4.426) and below), and the mixing entropy may be calculated as the sum of entropy changes at the expansion of pure (ideal) gases a (with masses Wa) from starting density (before mixing) to final density (as in the mixture). 

Further, the difference between the heat capacities for ideal-gas mixtures is the same as for pure ideal gases (4.1.4). In summary, all first-law properties of ideal-gas mixtures are rigorously obtained by mole-fraction averaging pure ideal-gas properties. For second-law properties, we substitute (4.1.35)-(4.1.37) into (3.4.4) to find... 

Entropy is frequently interpreted physically as a measure of the amount of "order" or "disorder" in a system. Specifically, statements are made to the effect that increases in the disorder of a system are reflected by increases in entropy. In this section we explore such claims. Mixing is one process in which substances can be considered to become less ordered, and so, if the conventional wisdom is correct, the mixing of pure substances should be accompanied by entropy increases. To test this, we consider two processes for mixing pure ideal gases (a) one at fixed T and P, (b) another at fixed T and V. 

First, the term A ix° refers to the difference in Gibbs energies of products and reactants when each product and each reactant, whether solid, liquid, gas, or solute, is in its pure reference state. This means the pure phase for solids and liquids [e.g., most minerals, H20( ), H20(/), alcohol, etc.], pure ideal gases at Ibar [e.g., 02( ), H20( ), etc.], and dissolved substances [solutes, e.g., NaCl(ag), Na+, etc.] in ideal solution at a concentration of Imolal. Although we do have at times fairly pure solid phases in our real systems (minerals such as quartz and calcite are often quite pure), we rarely have pure liquids or gases, and we never have ideal solutions as concentrated as 1 molal. 

When pure ideal gases mix at constant T and p to form an ideal gas mixture, the molar entropy change A5 (mix) = -R yt In (Eq. 11.1.9) is positive. 

From these considerations, one might conclude that the fundamental reason the entropy increases when pure ideal gases mix is that different substances become intermingled. This conclusion would be mistaken, as we will now see. 

From Eqs. 11.1.24 and 11.1.25, and the fact that the entropy of a mixture is given by the additivity rule S = Y ,i ntSi, we conclude that the entropy of an ideal gas mixture equals the sum of the entropies of the unmixed pure ideal gases, each pure gas having the same temperature and occupying the same volume as in the mixture. 

We can now understand why the entropy change is positive when ideal gases mix at constant T and p Each substance occupies a greater volume in the final state than initially. Exactly the same entropy increase would result if the volume of each of the pure ideal gases were increased isothermally without mixing. 

In making Washburn corrections, we must use a single standard state for each substance in order for Eq. 11.5.9 to correctly give the standard internal energy of combustion. In the present example we choose the following standard states pure solid or liquid for the reactant compound, pure liquid for the H2O, and pure ideal gases for the O2 and CO2, each at pressure p° = Ibar. 

Fugacity was invented to remedy the counterintuitive behavior of the chemical potential, which makes it approach minus infinity as the concentration approaches zero. For pure ideal gases the fugacity is the same as the pressure, and for ideal gas mixtures the fugacity of one species is equal to that species partial pressure. 

Two pure ideal gases 1 and 2, both at the same temperature T and pressure P, are mixed isothermally with final compositiony, andy2- What is the entropy of mixing per mole of mixture formed ... 

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