Integration of the Clausius-Clapeyron equation

Most methods for the determination of phase equilibria by simulation rely on particle insertions to equilibrate or determine the chemical potentials of the components. Methods that rely on insertions experience severe difficulties for dense or highly structured phases. If a point on the coexistence curve is known (e.g., from Gibbs ensemble simulations), the remarkable method of Kofke [32, 33] enables the calculation of a complete phase diagram from a series of constant-pressure, NPT, simulations that do not involve any transfers of particles. For one-component systems, the method is based on integration of the Clausius-Clapeyron equation over temperature,... 

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). 

The integration of Equation (11.22) to determine the equilibrium constant as a function of the temperature or to determine its value at one temperature with the knowledge of its value at another temperature is very similar to the integration of the Clausius-Clapeyron equation as discussed in Section 10.2. The quantity AHB must be known as a function of the temperature. This in turn may be determined from the change in the heat capacity for the change of state represented by the balanced chemical equation with the condition that all substances involved are in their standard states. 

One way to develop a correlation for the vapor pressure is to apply the integration of the Clausius-Clapeyron equation using simplifying assumptions ... 

Involves an integration of the Clausius-Clapeyron equation. Also... 

Now the azeotropic pressure p is 1 atm. Consider the integration of the Clausius-Clapeyron equation for pure component A between temperature Tg and the boiling-point Assuming the latent heat to be... 

This suggests that a plot of P against 1/T should yield a line having a local slope of (-A, /R). A straight line is obtained only when is nearly constant, i.e., over a narrow range of temperatures. An integrated version of the Clausius-Clapeyron equation finds use in correlation of vapor pressure data ... 

It is possible, however, to simplify the calculation of the energy transfer by assuming that the vapor phase is always a saturated vapor. O Connor (Ol) has shown that the rate of approach of a superheated vapor to saturated conditions is extremely rapid when the superheated vapor is in direct contact with its liquid phase. If the vapor phase is assumed to be saturated, the temperature of the phase can be calculated from an integrated form of the Clausius-Clapeyron equation instead of from the vapor-phase energy-transfer equation. 

The vapor pressure of the liquid at the surface Pg can be evaluated from an integrated from of the Clausius-Clapeyron equation if the surface temperature Ts is known. 

The form of the Clausius-Clapeyron equation in Equation (5.5) is called the integrated form. If pressures are known for more than two temperatures, an alternative form may be employed ... 

We employ the integrated form of the Clausius-Clapeyron equation when we know two temperatures and pressures, and the graphical form for three or more. 

One of the critical issues in vapor pressure methods is the choice of the procedure to calculate the vaporization enthalpy. For instance, consider the vapor pressures of ethanol at several temperatures in the range 309-343 K, obtained with a differential ebulliometer [40]. The simplest way of deriving an enthalpy of vaporization from the curve shown in figure 2.4 is by fitting those data with the integrated form of the Clausius-Clapeyron equation [1] ... 

Calculate the number n of moles of HCl in the solution dispensed. Give S and 5 for the initial and final volumes, and give a limit of error (95 percent confidence) for n. The heat of vaporization of a liquid may be obtained from the approximate integrated form of the Clausius-Clapeyron equation. 

Solids Below the triple point, the pressure at which the solid and vapor phases of a pure component are in equilibrium at any given temperature is the vapor pressure of the solid. It is a monotonic function of temperature with a maximum at the triple point. Solid vapor pressures can be correlated with the same equations used for liquids. Estimation of solid vapor pressure can be made from the integrated form of the Clausius-Clapeyron equation... 

In order to extend the method of calculating ideal solubilities of gases to temperatures above the critical point, when the liquid cannot exist and direct determination of the vapor pressure is not possible, it is necessary to estimate a hypothetical vapor pressure by suitable extrapolation. This is best done with the aid of the integrated form of the Clausius-Clapeyron equation. If the vapor pressures at any two temperatures are known, the hypothetical value at a temperature above the critical point may be evaluate on the assumption that the heat of vaporization remains constant. 

Use the definite integral form of the Clausius-Clapeyron equation [Solution to Exercise 4.8(b)]. /pi AvapH ( I 1 ... 

The integral and differential heats of adsorption are determined by measuring the adsorption isotherms for a given system at different temperatures. From the data, the equilibrium pressures necessary to obtain the same coverages at the different temperatures are determined. From the slope of the plots of In = const versus /T, the differential isosteric heats of adsorption for a given coverage are determined by the use of the Clausius-Clapeyron equation ... 

One such method uses a linear relation between temperature and the molar heat of vaporization which is estimated using the Kistiakowsky Constant. The integrated form of the Clausius-Clapeyron equation may be expressed... 

Over a limited temperatme range around an equilibrium condition (po,Tb), where AH a constant, an integrated form of the Clausius-Clapeyron equation apphes... 

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). 

From the knowledge of the quantities (known by independent measurements) in Clausius-Clapeyron equation, the slope of the fusion curve can be evaluated and integrated to get pf(T). For example, in the 5-20 mK range, the Clausius-Clapeyron equation gave temperature values with an 1% accuracy [52,53] ... 

The Clausius-Clapeyron equation" is an integrated version of the Clapeyron equation that applies to equilibrium between an ideal gas vapor phase and a condensed phase, with the conditions that the volume of the... 

The Clausius-Clapeyron equation gives the variation of the vapour pressure p of a liquid with absolute temperature T. To derive the relationship involves integration of the expression... 

Suppose now that the heat of vaporization of a substance is independent of temperature (or nearly so) in the temperature range over which vapor pressures are available. Equation 6.1-2 may then be integrated to yield the Clausius-Clapeyron equation... 

The indefinite integral, Eq. (4.16), is known as the Clausius-Clapeyron equation. Unfortunately, a plot of In p versus l/T over a significant range of l/T does not give a straight line. Consequently, Eq. (4.16) often is modified one result is the Antoine equation discussed in Sec. 3.3. A definite integral of Eq. (4.15) is... 

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