Mixtures of perfect gases

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. 

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 ... 

SOL. 19. 1. Prigogine, Pression, diffusion et superfluidite dans les melanges de gaz parfaits (Pressure, diffusion and superfluidity in mixtures of perfect gases), Colloquium Ultrasonore Trilingen, Cl. Sciences, Acad. Roy. Flamande, 1952, pp. 285-295. 

The majority of applications considered in this writing concern mixtures of perfect gases where the equation of state is... 

Systems lor which /t, has this form possess remarkably simple properties. Moreover, mixtures of perfect gases (i.e., gases under cundilions which can be approximated with sufficient accuracy by the ideal gas law) and very dilute solutions have these properties. 

PARTIAL PRESSURE. The pressure exerted by each component in a mixture of gases. In a mixture of perfect gases... 

Similarly, the viscosity and thermal conductivity can be evaluated approximately with the help of kinetic theory arguments.15 Finally, we need an equation of state relating p, p, and T. Assuming we are dealing with a mixture of perfect gases, we have... 

Thermodynamics of a Mixture of Perfect Gases.—Suppose we have a mixture of nx moles of a gas 1, n2 moles of gas 2, and so on, all in a container of volume V at temperature T. First we define the fractional concentration c, of the th substance as the ratio of the number of moles of this substance to the total number of moles of all substances present ... 

A very important expression originates from dealing with partial pressures in mixtures of perfect gases, but is used "everywhere" ... 

For a mixture of perfect gases, the chemical potential is /u = /li° i RT In C. We can relate the chemical potentials to the chemical equilibrium constant and the affinity by... 

Note that the steam/water must be at saturation temperature 298.15 K and saturation pressure 0.03 bar. A mixture of perfect gases would have shared the pressure pro rata to their mole concentrations. Hence CO2 pressure = 0.97 bar. 

Hence, is a strictly additive quantity only in mixtures of perfect gases, that is to say, in cases when the configuration of a molecule does not statistically depend on the configuration of the remaining molecules of the system. In dense multi-component systems, as a result of correlations between molecules of a given component or between those of different components of the system, the quantities and are non-zero,... 

The second is the one discovered by Berthollet and which is known under the name of law of the mixture of gases To keep in equilibrium at a given volume and pressure a mixture of perfect gases, it is necessary to submit it to a pressure equal to the sum of the pres-sures which would he maintained separately, at the same volume and temperature, by each of the mixed gases. 

These several laws completely characterize, from the thermodynamic point of view, the properties of a mixture of perfect gases they lead in fact to the following proposition, which allows calculating all these properties when those of mixed gases are known ... 

The internal potential of a mixture of perfect gases is always equal to the sum of the internal potentials which it would be proper to attribute to each of the mixed gases if it occupied alone, at the same temperor ture, the entire volume of the mixture. 

PVT, Vapor Density, and Virial Coefficient. In these three techniques the pressure (of a known volume and weight of gas) is measured over a temperature range. The data may be treated in two ways by assuming that the real gas is a mixture of perfect gases (each gas being one of the species), or by describing the behavior of the real gas by a virial equation. 

As an example, let us consider a system consisting of a mixture of perfect gases the total volume of the system is an extensive variable given by cf. chap. X) ... 

We have already obtained expressions for and for a mixture of perfect gases (2.10 and 2.19). Furthermore... 

A simple example of a system of this kind is a mixture of perfect gases capable of reacting chemically, in which there is a Maxwellian distribution of molecular speeds, but where the concentrations are not those corresponding to chemical equilibrium among the components. 

Systems for which has the form (7.1) possess remarkably simple properties. Moreover these systems are of considerable practical importance since, as we shall show later (cf. chap. X, and chap. XX), mixtures of perfect gases and very dilute solutions have just these properties. It is therefore useful to have a collective term to describe systems satisfying (7.1) they are called ideal systems. 

If the gas mixture could be considered as a mixture of perfect gases, then we should have... 

If we now compare equations (10.35), (10.36), (20.1) and (20.3) we see that the chemical potentials in a mixture of perfect gases, and in an ideal solution, depend in the same way on the mole fractions, but in very different ways upon the applied pressure. Both systems are ideal. 

In the first place the latter equation is satisfied for any gaseous phase which behaves as a mixture of perfect gases, whereas the van t Hoff equation is obeyed if the solution is not only ideal but also at the same time very dilute. The existence of deviations from van t Hoff s equation does not in any way prove that the solution is non-ideal. 

We may now replace the chemical potential of the solution by the value given in (20.1), and assume the gas phase to behave as a mixture of perfect gases so that equation (10.34) can be used for the chemical potential of the vapour. Hence... 

In a mixture of perfect gases, since the molecules take up no space and do not interact, each gas behaves as if it were alone in the container. The total pressure is therefore just the sum of the pressures that each of the gases would exert if it were alone in the same volume. These pressures are called the partial pressures of the gases. If nA molecules of perfect gas A and nB molecules of perfect gas B are mixed we may write the total pressure P as P = PA + JPB. The contributions of A and B to the total pressure will depend simply on the number of moles of each of these substances present. Thus the partial pressures PA and PB are given by... 

Suppose the initial partial pressures to be 1 bar for each reactant. (The partial pressure of one component of a mixture of gases is defined as the pressure it would exert if it were alone in the available space. For an ideal mixture of perfect gases, the total pressure is the sum of the various partial pressures.) At 25°C, AG° for this reaction is -100.4 kJ/mol, and so the driving force is considerable, and reaction takes place over the catalyst with great vigour. The temperature of the reaction vessel is kept constant. As the reaction occurs, the partial pressures of ethylene and hydrogen drop, and their reduced active mass , or... 

The partial pressure of a gas in a mixture of gases is the pressure the gas would exert if it occupied alone the same container as the mixture at the same temperature, It is a limiting law because it holds exactly only under conditions where the gases have no effect upon each other. This can only be true in the limit of zero pressure where the molecules of the gas are very far apart, Hence, Dalton s law holds exactly only for a mixture of perfect gases for real gases, the law is only an approximation. 

Partington, A Short History of Chemistry, 3rd ed., p. 169, Dover Publications, New York (1989). In a lecture at the Royal Institution in 1810, Dalton attributed the origin of this atomic theory to attempts to explain his law of partial pressures (1801 1802), which states that the pressure exerted by a mixture of perfect gases is the sum of the pressures exerted by the individual gasses occup3fing the same volume. The partial pressure of a gas is the pressure a gas would exert if it occu pied the container alone and if it behaved perfectly. Dalton s law of partial pressure is a more general form of Henry s law, which states that the amount of gas absorbed by a liquid is proportional to the pressure. 

In our equilibrated system A + B <-> C + D, there exists a certain equilibrium concentration cz of transition states. In this entire chajiter, concentration in mixtures of perfect gases will be expressed as a number density, i.e., number per unit volume. The rate of reaction is equal to the number of systems per unit volume crossing the barrier per unit time ... 

The second part of the problem now consists in the evaluation of the enthalpy of formation of AB from A and B. Since the system is a mixture of perfect gases, designation of a standard state for the enthalpy is superfluous so that superscripts zero can be omitted. Since the reaction between A and B considered here is a collision between hard spheres, there is no energy of interaction between the colliding parmers and the energy of formation A o at 0°K of the complex AB from A and B must be equal to zero ... 

If in a mixture of perfect gases with mole numbers N, N2,. .. each component behaves as if it were independent of the others, the equation of state of the component i is given as... 

The validity of (3.76) actually goes far beyond mixtures of perfect gases. Systems in which the chemical potentials of the components can be expressed by (3.76) are called ideal systems. A special case of ideal systems are dilute solutions. A statistical derivation of (3.76) for dilute solutions may be found in LANDAU-LIFSHITZ, Theoretical Physics, Vol. V (1968). In dilute solutions, the mole fraction of a solute is approximately given as x. = N. /N where is the number of moles of the solvent. This enables us to rewrite the chemical potential of a solute approximately in terms of its concentration c. as... 

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