Equation of Clausius-Clapeyron

A plot of the logarithm of the reduced pressure against the reciprocal reduced temperature is shown in Fig. 8.1. The equation of Clausius - Clapeyron reads as... 

From this, the equation of Clausius-Clapeyron is derived ... 

The enthalpy of melting A/ sl can be calculated with the equation of Clausius-Clapeyron and with the change of specific volume from Vg to Vl ... 

With Diihrmg s rule, it is possible to estimate the vapor pressure curve of a solution on the basis of the known vapor pressure curve of the pure solvent and of the known boiling temperature of the solutioa The pure substance s vapor pressure curve can for instance be determined by the equation of Clausius-Clapeyron (is illustrated in Fig. 2.1-10) ... 

As in low molecular systems, the nematic phase exhibits orientation order between rod-like units, but no positional order. Due to orientation of mesogenic units also spacers are oriented by stretching. Molecules are oriented with respect to director. Thus, the isotropic - nematic transition is a disorder-order transition from a phase without long range order to a phase with long range orientation order but still liquid-like. It is an equilibrium transition since there is no undercooling. This transition is characterized by = S -S <0 and < 0. We may relate it to equation of Clausius-Clapeyron. It follows ... 

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

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

Problem 6 Give the thermodynamic derivation of Clapeyron equation and Clausius-Clapeyron equation. Discuss their applications also. 

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,... 

Calculated from least squares fit of Clausius-Clapeyron equation ... 

Fortunately, there are a number of good vapour-pressure measurements available for mercury which, combined together, give us the heat of evaporation with an accuracy of about 1 pro mille. If we limit ourselves to pressures not exceeding a few tenths of an atmosphere we may use the equation of Clausius and Clapeyron,... 

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 equation of Clausius and Clapeyron in integrated form for the dependence of the vapor pressure p on temperature T reads as... 

Chen, L. Grant, D.J.W. Extension of Clausius-Clapeyron equation to predict hydrate stability at different temperatures. Pharm. Dev. Technol. 1998, 3, 487 94. 

His essential contributions to Chemical Kinetics, besides the part previously cited in the first part of this chapter, culminated in the discovery of the relation between the rate constant and the equilibrium constant (Van t Hoff, 1884). He interpreted the Chemical Equilibrium as the balance between opvposite reactions so he related equilibrium constant to the ratio of the rate constants of the direct and reverse reaction. From an application of Clausius-Clapeyron equation Van t Hoff found the dependence of the equilibrium constant K from the absolute temperature T ... 

Specifying p (or T) eq 9.38 permits the calculation of the equilibrium temperature (or p) as function of ij. The curve T 1 ) at constant p or p l ) at constant T is called Equilibrium-Flash-Vaporization (EFV) curve. Knowing T, p and with eqs 9.36 and 9.37 the distribution functions for the phases L and Vmay be calculated.An analytical solution of the integral of eq 9.38 is only possible for 1 = 0, that is when the feed and liquid phases are equal, when the following assumptions hold (1), the distribution function of the given liquid phase has to be Gaussian and (2), the vapour pressure function p x, T) has to be calculated with a combination of Clausius-Clapeyron s equation and Trouton s rule. 

In Chapter 6, we introduced some important concepts that we can apply to systems at equilibrium. The Clapeyron equation, the Clausius-Clapeyron equation, and the Gibbs phase rule are tools that are used to understand the establishment and changes of systems at equilibrium. However, so far we have considered only systems that have a single chemical component. This is very limiting, because most chemical systems of interest have more than one chemical component. They are multiple-component systems. 

Application of Clausius-Clapeyron Equation to Phase Changes... 

The integration of the equation (13) and of Clausius-Clapeyron equation yields... 

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). 

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... 

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). 

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 ... 

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. 

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 ... 

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. 

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" ... 

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

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 ... 

This equation is known as the Clausius-Clapeyron equation. Rudolph Clausius (1822-1888) was a prestigious nineteenth-century German scientist B. P. E. Clapeyron (1799-1864), a French engineer, first proposed a modified version of the equation in 1834. 

In using the Clausius-Clapeyron equation, the units of AH and R must be consistent. If AH is expressed in joules, then R must be expressed in joules per mole per kehrin. Recall (Table 5.1, page 107) that... 

As pointed out earlier, the equilibrium constant of a system changes with temperature. The form of the equation relating K to T is a familiar one, similar to the Clausius-Clapeyron equation (Chapter 9) and the Arrhenius equation (Chapter 11). This one is called the van t Hoff equation, honoring Jacobus van t Hoff (1852-1911), who was the first to use the equilibrium constant, K. Coincidentally, van t Hoff was a good friend of Arrhenius. The equation is... 

Clausius-Clapeyron equation An equation expressing the temperature dependence of vapor pressure ln(P2/Pi) = AHvapCl/Tj - 1/T2)/R, 230,303-305 Claussen, Walter, 66 Cobalt, 410-411 Cobalt (II) chloride, 66 Coefficient A number preceding a formula in a chemical equation, 61 Coefficient rule Rule which states that when the coefficients of a chemical equation are multiplied by a number n, the equilibrium constant is raised to the nth power, 327... 

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