Physical Clapeyron equation

Fundamental Property Relation. The fundamental property relation, which embodies the first and second laws of thermodynamics, can be expressed as a semiempifical equation containing physical parameters and one or more constants of integration. AH of these may be adjusted to fit experimental data. The Clausius-Clapeyron equation is an example of this type of relation (1—3). 

Clausius-Clapeyron equation, and to obtain a theoretical expression for a vapor bubble growing in a superheated liquid. The equation (F5, F6) is a second-order differential equation which is so complex as to be of limited usefulness without serious modification. Fortunately, the equation becomes enormously simpler if the inertia of the liquid can be ignored during bubble growth. Forster and Zuber give a careful discussion of the physical requirements for neglecting inertia of the liquid. These are that either the bubble must be very small or the temperature of the bubble... 

For wider temperature ranges, Hv (T) can be expressed as a polynomial or some other function of T. Integration of the Clausius-Clapeyron equation then leads to expressions given in the Handbook of Vapor Pressure (Yaws 1994) or in the Physical and Thermodynamic Properties of Pure Chemicals (Daubert et al. 1994). 

This equation gives the enthalpy of the system relative to the standard state, and the independent variable would now be (H — nH" ) rather than H itself. The quantities (H — H" ) and (ft — H ") are the changes of enthalpy when the state of aggregation of 1 mole of the component is changed from the triple-primed state to the primed state and to the double-primed state, respectively, at the temperature and pressure of the triple point. These quantities can be determined experimentally or from the Clapeyron equation, as discussed in Section 8.2. The three simultaneous, independent equations can now be solved, provided values that permit a physically realizable solution have been given to (H — nH "), V, and n. If such a solution is not obtained, the system cannot exist in three phases for the chosen set of independent variables. Actually, the standard state could be defined as one of the phases at any arbitrarily chosen temperature and pressure. The values of the enthalpy and entropy for the phase at the temperature and pressure... 

Although we shall not be concerned experimentally with measuring heats of adsorption, it is appropriate to comment that Mi for the physical adsorption of a gas is always negative, since the process of adsorption results in a decrease in entropy. The isosteric heat of adsorption (the heat of adsorption at constant coverage 6) can be obtained by application of the Clausius-Clapeyron equation if isotherms are determined at several different temperatures the thermodynamics of adsorption have been fully discussed by Hill. ... 

The isosteric heats of adsorption have been calculated from isotherms by the use of Clausius-Clapeyron equation. The detailed results 5) show that in all the cases measured physical adsorption is taking place. In this paper the heats given in Table I correspond to half-surface coverage. 

The Clausius-Clapeyron equation relates the equilibrium vapor pressure, the cell temperature, and the physical characteristics specific to the organic compound ... 

The Clausius-Clapeyron equation, one of the most famous in physical chemistry, is most applicable for this discussion. The equation states that the partial differential with respect to absolute temperature of the logarithm of a pure liquid s vapor pressure is inversely related to the liquid s absolute temperature. We again consider the liquid TCE and state the Clausius-Clapeyron equation mathematically (29) ... 

The Clausius-Clapeyron equation in physical chemistry states... 

The general relationship between the amount of gas (volume, V) adsorbed by a solid at a constant temperature (T) and as a function of the gas pressure (P) is defined as its adsorption isotherm. It is also possible to study adsorption in terms of V and T at constant pressure, termed isobars, and in terms of T and F at constant volume, termed isosteres. The experimentally most accessible quantity is the isotherm, although the isosteres are sometimes used to determine heats of adsorption using the Clausius-Clapeyron equation. In addition to the observations on adsorption phenomena noted above, it was also noted that the shape of the adsorption isotherm changed with temperature. The problem for the physical chemist early in the twentieth century was to correlate experimental facts with molecular models for the processes involved and relate them aU mathematically. 

An important physical-chemical property that characterizes the interaction of solid surfaces with gases is the bond energy of the adsorbed species. The determination of bond energy is usually made indirectly by measuring the heat of adsorption (or heat of desorption) of the gas. The heat of adsorption can be determined readily in equilibrium by measuring several adsorption isotherms. The Clausius-Clapeyron equation... 

Physical adsorption Isotherms of Ar and N2 on a-BN have been determined and analyzed. The adsorption energies distribution functions were calculated using a double Gaussian as distribution function. The Isosteric heats of adsorption were calculated employing the Clausius-Clapeyron equation. Both systems display a sharp maximum upon completion of a statistical monolayer (see Fig. 4-21). For additional Information, see [50]. 

The twin problems of cleanliness and structure can now be overcome by the use of single crystals, where both the chemical and physical states of the surface can be monitored using a range of surface spectroscopic techniques. However, single-crystal studies introduce other limitations. In particular the measurements must be carried out under UHV and it is only possible to measure the heats of adsorption indirectly. The most common methods involve either isotherm data and the use of the Clausius-Clapeyron equation or direct analysis of the temperature programmed desorption (TPD) peaks. 

Here A, B, and C are empirical parameters that are available for many fluids. Values for Antoine constants can be found in Appendix A. 1. The Antoine equation, an empirical equation, is strikingly similar to Equation (6.11). The Antoine equation brings back a similar theme to that discussed with equations of state (Chapter 4). When an empirical equation s form reflects the basic physics that it is trying to describe, it tends to work better. Why do you think the Antoine equation works better than the Clausius-Clapeyron equation in correlating saturation pressures There are several more complex correlations reported in the literature for as a function of T. These forms will not be covered in this text. 

After de Forcrand s Clapeyron, and Handa s methods, a third method for the determination of hydrate number, proposed by Miller and Strong (1946), was determined to be applicable when simple hydrates were formed from a solution with an inhibitor, such as a salt. They proposed that a thermodynamic equilibrium constant K be written for the physical reaction of Equation 4.14 to produce 1 mol of guest M, and n mol of water from 1 mol of hydrate. Writing the equilibrium constant K as multiple of the activity of each product over the activity of the reactant, each raised to its stoichiometric coefficient, one obtains ... 

The application of the second law of thermodynamics to chemistry, first occurring notably in the study of dissociation phenomena in solids, influenced Horstman in 1873 to point out that such changes are exactly similar to physical changes of state, and that the thermodynamic equation derived by Clapeyron and Clausius for changes of state are also applicable here, i.e., dp/dT QjT d - v), where p is the dissociation pressure, Q is the heat of dissociation, T is the absolute temperature, and C and v are the volumes of the system after and before dissociation. In the case in which a gas or vapor formed is supposed to behave as an ideal gas, and the volume of the solids is neglected in comparison with that of the gas, the equation becomes d log p/dT Q/RT, where R is the universal gas constant. ... 

In this chapter we have proposed a unified thermodynamic approach to the analysis of melting/freezrng phenomena in confined systems. The approach is based on the Clausius-Clapeyron relations for coexistence of solid and liquid phases jointly with equations relating the vapor pressures above bulk (with plane interfaces) and confined states of a substance. For illustration we have applied our analysis to three types of confinement plane interfaces, small particles and pores. The analysis for other types of confinement like free, wetting and adsorption films, emulsions, and so on is straightforward. Notably, the logic of derivation allows us to better understand the influence and physical content of the parameters involved in the key equations. 

The vapor pressure of pseudocomponents is also an important property when an equation-of-state approach is not used. All other approaches to process thermo-dynamics require some form of vapor-pressure correlation. The vapor pressure for pure hydrocarbons has been extensively tabulated in many component databases such as DIPPR (Design Institute for Physical Property Research, American Institute of Chemical Engineers) and significant libraries are available in modern process modeling software. Several correlations are available in the literature for the vapor pressure of pseudocomponents. It is important to recall that the vapor pressure and heat vaporization are related through the Clausius-Clapeyron [17] Equation (Equation (1-46)). This relationship imposes a constraint if we wish the model to be thermodynamically consistent In general, most of the popular correlations for vapor pressure such as the Lee-Kesler [9,10] agree well with heat of vaporization correlations and maintain thermodynamic consistency. We present the Lee-Kesler vapor pressure correlation in Equation (1.47). 

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