Pressure of a condensed phase

Dependence of the vapor pressure of a condensed phase on total pressure... 

We shall now derive an equation for the vapour pressure of a condensed phase in a form which stresses the significance of the entropy of vaporization. 

Results of experimental studies of solids under pressure can be described by the equations of state (EOS). So many equations of state involving the temperature, volume and pressure of a condensed phase have been proposed and used that a full review of them is out of question. Besides books dealing with the subject in general, many papers are more concerned with some specific element such as carbon, or aluminium, or the rare-gas solids, or the behavior of solids under extreme pressures, etc. Useful compilations of EOSs are available [267-269]. The most popular among physicists are the EOSs of Murnaghan [270], Birch [271], Vinet et al. [272,273] and Holzapfel et al. [274-277]. The latter two EOSs have similar forms ... 

Letp be the ordinary equilibrium pressure in a system composed of a condensed phase and its vapour, and let, r, Y be the specific volumes of these phases, respectively, under a pressure p. We now assume that these values are altered to p r Y, when the condensed phase alone is exposed to a pressure P + p. 

Fugacity, like other thermodynamics properties, is a defined quantity that does not need to have physical significance, but it is nice that it does relate to physical quantities. Under some conditions, it becomes (within experimental error) the equilibrium gas pressure (vapor pressure) above a condensed phase. It is this property that makes fugacity especially useful. We will now define fugacity, see how to calculate it, and see how it is related to vapor pressure. We will then define a related quantity known as the activity and describe the properties of fugacity and activity, especially in solution. 

The vapor pressure of a given phase of a substance is the pressure exerted by its vapor when the vapor is in dynamic equilibrium with the condensed phase. 

Finally, the bubble collapse stops when the pressure inside a bubble (pg) in the right hand side of (1.13) dramatically increases as the density inside a bubble nearly reaches that of a condensed phase (A bubble is almost completely occupied by the van der Waals hard-cores of gas and vapor molecules at that moment). At the same time, the temperature and pressure inside a bubble dramatically increase. 

When the standard states for the solid and liquid species correspond to the pure species at 1 atm pressure or at a low equilibrium vapor pressure of the condensed phase, the activities of the pure species at equilibrium are taken as unity at all moderate pressures. Consequently, the gas phase composition at equilibrium will not be... 

For condensed phases (liquids and solids) the molar volume is much smaller than for gases and also varies much less with pressure. Consequently the effect of pressure on the chemical potential of a condensed phase is much smaller than for a gas and often negligible. This implies that while for gases more attention is given to the volumetric properties than to the variation of the standard chemical potential with temperature, the opposite is the case for condensed phases. 

Chapter 5, vapor pressure isotope effects are discussed. There, a very simple model for the condensed phase frequencies is used, the Einstein model, in which all the frequencies of a condensed phase are assumed to be the same. From this model, one can derive the same result for the relationship between vapor pressure isotope effect and zero-point energy of the oscillator as that derived by Lindemann. 

The monolayer behavior of A-stearoyltyrosine (Fig. 16) was more complex. Under conditions (0.0liV HCl, 22 C) where the racemic material formed a condensed film having a limiting molecular area of 39 2 A, the force-area curve of L-(+)-A-stearoyltyrosine exhibited a liquid-expanded film at large areas (ca. 100-45 per molecule) followed by a transition beginning at 16.5 dynes/cm surface pressure to a condensed phase having a smaller limiting molecular area of 34 2 A . However, both these latter samples exhibited only the liquid-expanded phase on distilled water alone. 

DESI Desorption electrospray ionization (DESI) is a recently developed technique that permits formation of gas-phase ions at atmospheric pressure without requiring prior sample extraction or preparation. A solvent is electrosprayed at the surface of a condensed-phase target substance. Volatilized ions containing the electrosprayed droplets and the surface composition of the target are formed from the surface and subjected to mass analysis (Takats et al., 2005 Wiseman et al., 2005 Kauppila et al., 2006). 

The standard state of a condensed phase is taken at 1.0 bar pressure. Show that the activity of incompressible condensed matter at pressure P is from Eq. (47), a = exp Vm(P — 1 )/RT. Apply this formula to find the activity of liquid octane, with density 0.7g/cm at 25°C and 100 atm pressure. How much does the chemical potential of liquid octane change between 1 and 100 atm ... 

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. 

In this section, we will derive the vapor pressure of a spherical droplet of radius r. This vapor pressure is greater than that of a flat surface because the liquid is on the concave side of the surface and is thus at a higher pressure than the surroundings. As we saw in Eq. (61) of Chapter 6, a higher pressure increases the escaping tendency of a condensed phase. The effect of surface curvature is only appreciable for very small droplets. 

It is above Kp f which are listed for the equilibrium constants of formation when a condensed phase is present. Kp, f is determined from the standard free energy change of the gaseous system and the vapor pressure of the condensed phase. 

The activity of a condensed phase is defined as its fugacity relative to the fugacity at its standard state, i.e., pure constituent at 25 °C (298 °K) and 1 atmosphere pressure. 

A decrease in size of a condensed phase particle results in an increase in the chemical potential of the substance due to excess surface energy and the elevation of the Laplace pressure inside the particle (see Section 1.1). With a spherical particles of radius r, one can estimate the increment Aj of the chemical potential of component i of the continuous phase by the expression... 

We now proceed to calculate the thermodynamic functions of a condensed phase. First, by taking the temperature and pressure as independent variables, we may evaluate the molar enthalpy. 

The function fJ (T) which we have introduced is analogous to that introduced in the theory of perfect gases (c/. 10.11), except that here the standard pressure is zero instead of unity. The way in which the chemical potential of a condensed phase varies with pressure is, however, quite different from the logarithmic variation of p, with p in the case of a perfect gas. Here we see that, since (1 - kp) 1 the variation of jjL with p is linear and is given by the term pv (T, 0). As we have seen this term is quite negligible under ordinary conditions so that the chemical potential is independent of pressure and... 

We now proceed to calculate the thermodynamic functions in terms of temperature and volume as independent variables. Since the pressures needed to change the volume of a condensed phase appreciably e.g. by 10 per cent) are extremely high we cannot now use the approximate equation of state, but must employ the full equation (12.18). 

If it is assumed that the specific volume, u , of the phase is very nearly constant over a large pressure increase, then when the pressure difference in the exponential is small enough, the exponential will be nearly one. Thus, at moderate pressures the fugacity of a condensed phase is nearly equal to the vapor pressure. This approximation was used in Eq. (11) to write Raoult s Law. When the pressure is low enough that the fugacity coefficient is nearly 1.00, Eq. (13) reduces to Raoult s Law extended with the activity coefficient included or Eq. (11). The fugacity coefficient can be calculated from equations of state, and the activity coefficient can be found from various correlations as discussed earlier. 

Recovery of the VOC Components in the Condenser. To recover a VOC component in the condenser, the partial pressure of the VOC in the strip gas must exc the vapor pressure of the condensed phase ... 

Note the large value of 11 for the very dilute 0.0055 mol/L glucose solution in Problem 4.7. Since the density of water is 1/13.6 times that of mercury, an osmotic pressure of 102 torr (102 mm Hg) corresponds to a height of 10.2 cm x 13.6 = 139 cm = 4.5 ft of liquid in the right hand tube in Fig. 4.3. The large value of H results from the fact that the chemical potential of a component of a condensed phase is rather insensitive to pressure. Hence it needs a large value of II to change the chemical potential of A in the solution so that it equals the chemical potential of pure A at pressure P. 

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