Amagat equation

An adsorbed film obeys a modified Amagat equation of state, t(t = qkT (see Eq. ni-107). Show that this corresponds to a Freundlich adsorption isotherm (Eq. XI-12) and comment on the situation. 

This behavior suggests the use of an equation employed by Amagat for gases at high pressure the two-dimensional form is... 

The results of Amagat s and Raveau s work may be summed up in the statement that, whereas the theorem of corresponding states holds good very approximately, the equation of van der Waals gives results quite inconsistent with the experimental values, especially near the critical point. 

For many gaseous solutions, even if the gases are not ideal, the partial molar volumes of the components are equal to the molar volumes of the pure components at the same total pressure. The gases are said to obey Amagat s mle, and the volume change on mixing is zero. Under these conditions, the gaseous solution behaves ideally in the sense that it obeys the equation... 

For the regions of high pressure Amagat proposed the equation... 

These two equations are applicable to mixtures of ideal gases as well as to pure gases, provided n is taken to be the total number of moles of gas. However, we must consider how the properties of the gas mixture depend upon the composition of the gas mixture and upon the properties of the pure gases. In particular, we must define the Dalton s pressures, the partial pressures, and the Amagat volumes. Dalton s law states that each individual gas in a mixture of ideal gases at a given temperature and volume acts as if it were alone in the same volume and at the same temperature. Thus, from Equation (7.1) we have... 

Two further crude approximations have been used for the virial equation of state. The first is that the virial coefficients combine linearly. This combination of constants results in an equation of state that is additive in the properties of the pure components. In such a mixture Dalton s and Amagat s laws still hold, and the mixture may be called an ideal mixture of real gases. The assumption is probably the crudest that can be used and is... 

The adiabatic compression of saturated vapours was considered by Bruhat, who also calculated the angle between the liquid and vapour phase isochores in the entropy-temperature diagram. Amagat investigated the discontinuity in specific heats where an isothermal cuts the saturation curve. Hausen, from a complicated formula for the specific heat of steam involving two Einstein terms ( 2.IX N), calculated the heat content and entropy of steam. Leduc found the value of n for dry steam in Rankine s equation for adiabatic... 

Accurate prediction of the volume of a mixture of liquids will usually require experimental data to relate the masses in kmol to the volume in m. For example. Perry et al. (1984) give tabulations for several liquid mixtures found in process plants. In the absence of such data, the simplest relationship is the linear relationship of Amagat s law, already quoted as equation (13.94). The specific volume of liquid component i, t>i (m /kmol) will depend on the temperature, T, of the liquid in the CSTR, so that the liquid volume is given by. 

Hint. Multiply the first of equations (3) through with v differentiate to get d(pv)jdv = 0, etc. The conclusion is in harmony with M. Amagat s experiments (Ann. Chim. Phys., [5] 22, 353, 1881) on carbon dioxide, ethylene, nitrogen and methane. For hydrogen, a, in (3), is negligibly small, hence show that pv has no minimum. 

As shown by Prausnitz (1969), this relation follows from Amagat s Rule, which describes the mixing situation where the volume of the solution is the same as the total volume of the individual pure components. In this case the AV of mixing is zero, and the partial molar volume Vi is equal to the molar volume of pure i, as discussed in Chapter 9. This can be demonstrated by first rewriting equation (11.9) for a pure component i as... 

Equation (4-32) in Table 4.2, the ideal gas equation, is widely applied to pure gases and gas mixtives. This equation neglects molecular size and potential energy of molecular interactions. When each species in a mixture, as well as the mixture, obeys the ideal gas law, both Dalton s law of additive partial pressures and Amagat s law of additive pure species volumes apply. The mixture equation in terms of molal density p/M is... 

Volumetric ideal mixing (Equation 10.4) is also called Amagat s law, which we saw was connected to the Lewis fugacity rule in Chapter 8. 

We have three unknowns, and so we need three equations. The equations are Dalton s equation to estimate the partial pressure of N2, Amagat s law to estimate the partial volume of N2, and, finally, the ideal gas law to estimate the density of the mixture. 

Note that the value of JRT in the appropriate units is the value of the right-hand side of the given equation as p->0. This expression is tidcen from International Critical Tables (vol. lu, p. 8), and much of the p-F-2 data in the literature am quoted in this form. It is equivalent to the use of the Amagat unit of volume which is the volume of 1 mol of gas at 0 and 1 atm and is close to 22.4 dm, but varies slightly from gas to gas. 

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