Clapeyron equation integration

Once a point on the coexistence line has been found, one can trace out more of it using the approach of Kofke [177. 178] to numerically integrate die 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). 

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 ... 

Two estimates will be made using vapor pressure data from the CRC Handbook [63] and the integrated form of Clausius-Clapeyron equation ... 

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. 

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,... 

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] ... 

All partitioning properties change with temperature. The partition coefficients, vapor pressure, KAW and KqA, are more sensitive to temperature variation because of the large enthalpy change associated with transfer to the vapor phase. The simplest general expression theoretically based temperature dependence correlation is derived from the integrated Clausius-Clapeyron equation, or van t Hoff form expressing the effect of temperature on an equilibrium constant Kp,... 

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] ... 

What would be the form of the integrated Clausius-Clapeyron equation if the heat capacity of the vapor were given by the equation... 

In this case, this equation is commonly referred to as Clausius-Clapeyron equation (e.g., Atkins, 1998). We can integrate Eq. 4-7 if we assume that Avap/7, is constant over a given temperature range. We note that AvapHt is zero at the critical point, Tc, it rises rapidly at temperatures approaching the boiling point, and then it rises more slowly at lower temperatures (Reid et al., 1977). Hence, over a narrow temperature range (e.g., the ambient temperature range from 0°C to 30°C) we can express the temperature dependence of p by (see Eq. 3-51) ... 

Equation (7.35c) is known as the Clausius-Clapeyron equation, which may also be expressed in integrated form... 

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... 

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). 

Substituting this formula into the Clausius-Clapeyron equation and integrating by parts... 

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. 

Integrating the Clapeyron equation assuming that A/ -and A Vf are constant,... 

Equation (11) is an integrated form of Clausius-Clapeyron equation. If the integration is carried out indefinitely (without limits) then we can write the vapour-pressure equation (9) as,... 

It is evident that Equation (2.68) is analogous to the well-known Clausius-Clapeyron equation for a one component gas-liquid system. Integration of Equation (2.68) between the limits of equilibrium pressures and temperatures of p,/72 and Tlt T2 gives ... 

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. 

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... 

Equation (2.5) is the Clausius-Clapeyron equation. General integration, assuming to be constant, gives... 

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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