Clapeyron

Clapeyron-Clausius equation A thermodynamic equation applying to any two-phase equilibrium for a pure substance. The equation states ... 

Clausius-Clapeyron equation See Clapeyron-Clausius equation. 

At the saturation pressure, the viscosity variation with temperature follows a law analogous to that of Clapeyron for the vapor pressure f ) ... 

The analogue of the Clapeyron equation for multicomponent systems can be derived by a complex procedure of systematically eliminating the various chemical potentials, but an alternative derivation uses the Maxwell relation (A2.1.41)... 

Equation (A2.1.53) is frequently called the Clausius-Clapeyronequation, although this name is sometimes applied to equation (A2.1.52). Apparently Clapeyron first proposed equation (A2.1.52) in 1834, but it was derived properly from thennodynamics decades later by Clausius, who also obtained tlie approximate equation (A2.1.53).)... 

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... 

Numerous mathematical formulas relating the temperature and pressure of the gas phase in equilibrium with the condensed phase have been proposed. The Antoine equation (Eq. 1) gives good correlation with experimental values. Equation 2 is simpler and is often suitable over restricted temperature ranges. In these equations, and the derived differential coefficients for use in the Hag-genmacher and Clausius-Clapeyron equations, the p term is the vapor pressure of the compound in pounds per square inch (psi), the t term is the temperature in degrees Celsius, and the T term is the absolute temperature in kelvins (r°C -I- 273.15). 

The Clapeyron equation expresses the dynamic equilibrium existing between the vapor and the condensed phase of a pure substance ... 

Another well-known thermodynamic result, the Clapeyron equation, applies to first-order transitions (subscript 1) ... 

This expression describes the variation of the pressure-temperature coordinates of a first-order transition in terms of the changes in S and V which occur there. The Clapeyron equation cannot be applied to a second-order transition (subscript 2), because ASj and AVj are zero and their ratio is undefined for the second-order case. However, we may apply L Hopital s rule to both the numerator and denominator of the right-hand side of Eq. (4.47) to establish the limiting value of dp/dTj. In this procedure we may differentiate either with respect to p. 

Vapor Pressures and Adsorption Isotherms. The key variables affecting the rate of destmction of soHd wastes are temperature, time, and gas—sohd contacting. The effect of temperature on hydrocarbon vaporization rates is readily understood in terms of its effect on Hquid and adsorbed hydrocarbon vapor pressures. For Hquids, the Clausius-Clapeyron equation yields... 

This equation follows from equation 66, because vaporization occurs at the constant pressure Moreover, the heat of vaporization is related to the slope of the vapor—Hquid saturation curve through the Clapeyron equation ... 

The vapor pressure for the soHd at 25°C has been calculated from the value for the Hquid at 70°C and the heats of vaporization and fusion using the Clausius-Clapeyron relationship. 

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). 

Cluusius-Clupeyron Eijliation. Derived from equation 1, the Clapeyron equation is a fundamental relationship between the latent heat accompanying a phase change and pressure—volume—temperature (PVT data for the system (1) ... 

The Clapeyron equation is most often used to represent the relationship between the temperature dependence of a pure hquid s vapor pressure curve and its latent heat of vaporization. In this case, dT is the slope of the vapor pressure—temperature curve, ADis the difference between the... 

If the latent heat of vaporization is then assumed to be constant over the temperature range of interest, equation 6 can be integrated to give the Clausius-Clapeyron expression ... 

Two empirical parameters are evident in equation 7, the heat of vaporization and the integration constant, I. Experimental data indicate that the linear relationship suggested by Clausius-Clapeyron may not be followed over a large temperature range (4) therefore additional adjustable parameters have been added to equation 7 to improve its correlating abiUty. The most prominent of these is the Antoine equation ... 

Reference Substances. Use of a reference substance has its origins in the work of Clausius-Clapeyron and equation 73, a form of equation 7 ... 

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. 

Correlation Methods Vapor pressure is correlated as a function of temperature by numerous methods mainly derived from the Clapeyron equation discussed in the section on enthalpy of vaporization. The classic simple equation used for correlation of low to moderate vapor pressures is the Antoine S equation (2-27). 

Enthalpy of Vaporization The enthalpy (heat) of vaporization AHv is defined as the difference of the enthalpies of a unit mole or mass of a saturated vapor and saturated liqmd of a pure component i.e., at a temperature (below the critical temperature) anci corresponding vapor pressure. AHy is related to vapor pressure by the thermodynamically exact Clausius-Clapeyron equation ... 

Known as the Clapeyron equation, this is an exacl thermodynamic relation, providing a vital connection between the properties of the liquid and vapor phases. Its use presupposes knowledge of a suitable vapor pressure vs. temperature relation. Empirical in nature, such relations are approximated by the equation... 

As an example of how the approximate thermodynamic-property equations are handled in the inner loop, consider the calculation of K values. The approximate models for nearly ideal hquid solutions are the following empirical Clausius-Clapeyron form of the K value in terms of a base or reference component, b, and the definition of the relative volatility, Ot. 

Isosteric Heat of Adsorption The most useful heat of adsorption for fixed-bed calculations is the Isosteric heat of adsorption, which is given by the Clausius-Clapeyron type relation... 

Figure 26-65 illustrates that Eq. (26-90) provides a linear approximation to the nonlinear relationship between two-phase specific volume and reciprocal pressure (v vs. P or vs. T ). For single components, me initial slope of the curve is found using me Clapeyron equation to give ... 

Nomograph defined. This method assumes the application of the Clausius-Clapeyron equation, Henry s law, and... 

Better examples of shortcut design methods developed from property data are fractionator tray efficiency, from viscosity " and the Clausius-Clapeyron equation which is useful for approximating vapor pressure at a given temperature if the vapor pressure at a different temperature is known. The reference states that all vapor pressure equations can be traced back to this one. 

Unfortunately values of A// at sueh low temperatures are not readily available and they have to be eomputed by means of the Clausius-Clapeyron equation or from the equation given by Hildebrand and Scott" ... 

Fig. 2. Clapeyron diagram showing saturated refrigerant and isosteres...

Fig. 3 shows an idealised solar collector (generator) containing adsorbent which is connected to a condenser that rejects heat to the environment and an insulated box containing a liquid receiver and a flooded evaporator. Fig. 4 shows the p-T-x (pressure - temperature - concentration or Clapeyron diagram) for the adsorbent-adsorbate pair with typical temperatures. 

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