The perfect gas mixture

Certain gaseous mixtures approximate to simple behaviour in the following respects (a) the gas mixture as a whole obeys the equation of state pV==nRT, where n is the total amount (mols) of all substances (6) two such mixtures are at equilibrium with each other through a semi-permeable membrane when the partial pressure is the same on each side, for each component which is able to pass through the membrane (c) there is no heat of mixing. The molecular conditions which must exist in order that the mixture shall have 

The above properties may be taken as defining the perfect gas mixture. Alternatively, we can proceed as in the previous section and put forward a definition in terms of the chemical potential. Proceeding in this manner a gaseous mixture will be said to be perfect if the chemical potential of each of its components is given by the following relation, in which is a function of temperature orUy, 

The defining equation (3 18) can be put in a more compact form by means of the partial pressure Pi. Thus 

This choice of definition of the partial pressure makes the sum of all the partial pressures equal to the total pressure, even if the mixture is not perfect. Thus P=P =P (3-21) 

It may be noted from (3 23) that F is the same for all components and is equal also to F/n, by (3 23) and (3 24). The pure gas i at the pressure p would also have a volume per mole, Vi, equal to RT/p. Thus 

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The differences between the gas-phase and solution algorithms appear from this point on. To derive equation 3.3, the perfect gas mixture was assumed, and A related to an equilibrium constant given in terms of the partial pressures of the reactants and the activated complex [1], This Kp is then easily connected with A H° and A .S ". As stated, the perfect gas model is a good assumption for handling the results of the large majority of gas-phase kinetic experiments. 

The important models with which we are concerned are the perfect gas, the perfect gas mixture and the ideal solution (gaseous, liquid or solid). These may be defined in either of two ways which are entirely equivalent (1) in terms of limiting experimental laws such as the gas equation and Raoult s law (2) in terms of expressions for the chemical potentials of the various components. These expressions are as follows ... 

The perfect gas mixture has been defined in accordance with... 

Departures from the above equations due to deviations from the law of the perfect gas mixture do not usually exceed a small percentage except at pressures above atmospheric or if there is association in the vapour phase, such as occurs in the case of formic and acetic acids. A procedure for applying the Duhem-Margules equation allowing for deviations from the gas law has been described by Scatchard and Raymond. ... 

The use of a gas mixture presents a two-part problem. If the state of the mixture is such that it may be considered a mixture of perfect gases, classical thermodynamic methods can be applied to determine the state of each gas constituent. If, however, the state of the mixture is such that the mixture and constituents deviate from the perfect gas laws, other methods must be used that recognize this deviation. In any case, it is important that accurate thermodynamic data for the gases are used. 

Start with a basis of 1 lb-mole of the natural gas at T = 80°F = 540°R and P = 40 psig = 54.7 psia. The volume percent and mole percent compositions are identical for a perfect-gas mixture. 

The mole is particularly useful when working with gas mixtures. It is based on Avogdro s law that equal volumes of gases at given pressure and temperature (pT) conditions contain equal number of molecules. Since this is so, then the weight of these equal volumes will be proportional to their molecular weights. The volume of one mole at any desired condition can be found by the use of the perfect gas law. 

A consequence of mechanical equilibrium in a perfect gas mixture is that the pressures developed by each species sum to give the mixture pressure. This is known as Dalton s law, with the species pressure called the partial pressure, / , ... 

Subtracting Equation (3.47) from Equation (3.42) gives, for a perfect gas mixture with uniform properties in the control volume,... 

The conditions that apply for the saturated liquid-vapor states can be illustrated with a typical p-v, or (1 /p), diagram for the liquid-vapor phase of a pure substance, as shown in Figure 6.5. The saturated liquid states and vapor states are given by the locus of the f and g curves respectively, with the critical point at the peak. A line of constant temperature T is sketched, and shows that the saturation temperature is a function of pressure only, Tsm (p) or psat(T). In the vapor regime, at near normal atmospheric pressures the perfect gas laws can be used as an acceptable approximation, pv = (R/M)T, where R/M is the specific gas constant for the gas of molecular weight M. Furthermore, for a mixture of perfect gases in equilibrium with the liquid fuel, the following holds for the partial pressure of the fuel vapor in the mixture ... 

Equation 2.63 is valid for any homogeneous or heterogeneous reaction. The only difference is in the definition of activities. For a species in a perfect gas-phase mixture a = pi/p°, where pi is the partial pressure of species i andp° is the standard pressure (1 bar). For a real gas-phase mixture a =f/p°, where is the fugacity of i. The fugacity concept was developed for the same reason as the activity to extend to real gases the formalism used to describe perfect gas mixtures. In the low total pressure limit (p -> 0), fi = pi. 

The pressure dependence, as before, is derived not only from the perfect gas law for p, but from the density-pressure relationship in Z as well. Also, the effect of the stoichiometry of a reacting gas mixture would be in Z. But the mole fraction terms would be in the logarithm, and therefore have only a mild effect on the induction time. For hydrocarbon-air mixtures, the overall order is approximately 2, so Eq. (7.46) becomes... 

The early application of volumetric data for hydrocarbons made use of the perfect gas laws. They were not sufficiently descriptive of the actual behavior to permit their widespread use at pressures in excess of several hundred pounds per square inch. The need for accurate metering aroused interest in the volumetric behavior of petroleum and its products at elevated pressures. Table II reviews references relating to the volumetric behavior of a number of components of petroleum and their mixtures. For many purposes the ratio of the actual volume to the volume of a perfect gas at the same pressure and temperature has been considered to be a single-valued function of the reduced pressure and temperature or of the pseudo-reduced (38) pressure and temperature. The proposals of Dodge (15), Lewis (12), and Brown (8) with their coworkers serve as examples of the nature of these correlations. The Beattie-Bridgeman (2) and Benedict (4) equations of state describe the volumetric behavior of many pure substances and their mixtures with an accuracy adequate (31) for most purposes. However, at pressures above 3000 pounds per square inch the accuracy of representation with existing constants leaves something to be desired. 

The perfect-gas equation of state for multicomponent mixtures depends on the species composition. Representing the composition as either mass fraction Yk or mole fraction Xk leads to... 

Written in cylindrical coordinates and specialized for a perfect-gas mixture the conservation laws are ... 

Therefore, the partial pressure of a component i in n mol of a perfect gas mixture with a total pressure p is ... 

Then the equation of state of the mixture of gases proves experimentally to be just what we should calculate by the perfect gas law, using the total number of moles, (ni + tia + ) that is, it is... 

Mole per cent or mole fraction, weight per cent or weight fraction, and volume per cent or volume fraction may be employed to designate the composition of a solution. Avogadro s Law is not applicable to liquids, and equal volumes of different liquids do not contain the same number of molecules. Consequently, mole per cent and volume per cent are not equivalent in liquid solutions as they were in perfect-gas mixtures. To convert mole per cent to weight per cent the procedure is identical with that previously described for gases. To calculate the volume per cent of a liquid solution from the mole per cent or weight per cent the densities of the pure components must be known. 

We see then that a perfect gas mixture is an ideal system as defined in chapter VII 1. The properties established generally in that chapter can be applied directly to the present case. Perfect gas mixtures differ from other ideal systems in that the function /x (P, p) depends upon pressure in the simple logarithmic manner given in (10.36),... 

We can now make use of the above expressions for chemical potentials to calculate the affinity of a reaction in a perfect gas mixture, and to deduce the conditions for chemical equilibrium. 

We assume that the gas mixture obeys the perfect gas laws. By reference to table 8.1 we find that the standard affinity of this reaction at 298-16 °K and 1 atm. is equal to... 

I Perfect gas mixtures are ideal over the whole concentration range so that = A . 

The partial molar quantities in a perfect gas mixture are given by... 

For a perfect gas mixture all the activity coefficients are unity the values of 1 - y or In may be used as measures of deviations from the perfect gas laws. 

If the vapour does not behave as a perfect gas mixture, then we must employ the fugacity p instead of the partial pressure p c/. for example, J. N. Bronsted and J. G. Koefoed, Det. Kgl. Danske Vid. Selsk. (Mat-fys) 22, part 17 (194G). 

This equation between the partial pressures is valid no matter what are the deviations from ideal behaviour, and depends only on the assumptions that the gas phases consist of a perfect gas mixture, and that the partial molar volumes of the components in the solution are negligible,t (c/. 1). 

With the additional assumptions that the latent heats are independent of temperature, and that the vapour phase is a perfect gas mixture we obtain ... 

Furthermore, if the vapour phase behaves as a perfect gas mixture yf and y are both unity and the equations can be solved for x and xf to give... 

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